ErkDemon

Different Types of Zero

2010-09-30T23:35:00.001+01:00

It took mathematicians a while to realise that infinities came in different sizes.

The problem was an inadequacy of language. All "infinities" are infinite, but some are a little more infinite than others. For instance, "infinity-squared" gives an infinite result, but it's a stronger infinity than the infinity that we started out from ... but by deciding to assign all these different infinite results the same name — "infinity" — we created an implicit assumption that this "infinity" was a thing, a single entity rather than a family. We ended up reciting things like "infinity is just infinity, by definition". Well, if so, it was a pretty bad definition, because infinity isn't so much a value as a realm, or a concept that allows multiple members, like "integer".

Our conventional language breaks down in these sorts of situations. To try to get a handle on the infinities, we can construct an "infinity-based" number system where our reference base unit [∞] is a "reference infinity" of "one divided by zero" (we can say, "1/0 =[∞]"), and we can compare other infinities to that, so that 2/0 gives 2×[∞], and 2×[∞] / [∞] = 2. It's possible to do proper math and get sane finite results by multiplying and dividing infinities together, as long as you remember to keep track of how big each individual infinity is (and/or where it originally came from).

We do similar things with complex numbers. These have two components, a conventional "real" component, and an "imaginary" component that's a multiple of the "impossible" square root of minus one, which we abbreviate as i. Even though the imaginary components don't exist in our default number system, we can still do useful math with these hybrid numbers ... that's actually how we generate exotic mathematical creatures like the the Mandelbrot Set. The approach works. We've seen the pretty pictures.

So multiple values of infinity are okay.

But there's one last thing that we have to fix. Zero. See, it turns out that if infinities come in different sizes, then zero has to come in different sizes, too.

At first sight this seems even more crazy. We can plot a simple line going through zero, and put the tip of our pencil on the crossing point, and say there it is, right there. How can that single point have different values? Well, as with the infinities, the auxiliary values exist off the page — when different graphs all hit zero at the same position, the properties associated with the coincident points on those different graphs aren't automatically completely identical even though they show up as being at the same position. Coincident points on different intersecting lines can carry different slopes and rates of change, and can have associated vectors and other associated baggage that gets lost when we try to break a line down into instantaneous isolated unconnected values.

Zero times any conventional number gives a zero, just like infinity times any conventional number gives an infinity. But not all zeroes have the same emphasis or strength, and this can become important when you have them fighting against each other. If we're only multiplying our zeroes by normal boring numbers then the auxiliary parameters don't matter, but as with the infinities, when we start multiplying or dividing different zeroes, we have to track the strengths of the zeroes, or else we tend to end up with mathematical garbage.

One of the problems that theoretical physicists currently have is they they're coming up against a range of problems — black hole event horizons, Hawking radiation, gravity-wave and warpfield propagation — where clusters of values assigned to apparent physical properties have a habit of going to zero or to infinity and beyond, even though the underlying local physical properties are non-zero and non-infinite. To deal with those problems we either have to find ways of sidestepping the pathological math, or come up with a more complete mathematical vocabulary that doesn't freak out when we occasionally need to divide a known strength of infinity by an associated known incarnation of zero.
Otherwise, we're liable to come to bad conclusions about how certain things are "provably physically impossible" because they appear to break the math, when in fact, the real problem is that the math that we're trying to apply to the problem is too naive. If we go down that route, we can end up accidentally elevating the result of human error to the status of an accepted mathematical proof.

Which is bad.

The Decline of Theoretical Physics

2010-08-01T23:19:00.002+01:00

Progress in fundamental theoretical physics now seems to have been on hold for quite a while.

I thought that the situation was summed up quite nicely by one of the characters in "The Big Bang Theory" (an improbably funny TV sitcom about sciencey people).

Penny (cheerfully as a conversation-starter):
"So, what's new in the world of physics?"

Leonard (momentarily suprised and slightly amused that anyone would ask such a question):
"Nothing!"

Penny (taken aback):
"Really, nothing?"

Leonard:
"Well ... with the exception of string theory, not much has happened since the 1930's ... and ya can't prove string theory, at best you can say, 'Hey look, my logic has an internal consistency-y!' "

Penny:
"Ah. Well, I'm sure things will pick up."

Leonard unhappily picks his nails, broods briefly, decides that there's nothing positive he can say, and then changes the subject.

And I think that just about sums things up.

A 3D Mandelbrot

2010-06-25T00:27:00.000+01:00

Skytopia have a great set of pages on the search for a 3D version of the Mandelbrot Set. Or at least, for an interesting 3D version of the normal Mandelbrot.

It's easy enough to produce fractal solids that have a Mandelbrot on one plane, and if you plot the correct 3D shadows of the 4D Julia Set, you can find shapes that have Mandelbrots on multiple intersecting planes. But getting a Mandelbrot on two perpendicular intersecting planes, while having the transition between them being more interesting than simply spinning or rotating the thing on its axis, is more difficult.

The "normal" Mandelbrot has one "real" component and one "imaginary" component, set on the x and y axes. If you add another imaginary component on axis z, you simply get the sort of boring "spun" shape that you might produce on a lathe. If you distinguish the two "imaginary" axes by whapping a minus sign in front of one of them, you get a hybrid Mandebrot/Tricorn solid, but one of the cross-sections is then a tricorn rather than a 'brot.

From here, you can try hypercomplex numbers, number systems that support multiple distinct imaginary components and define how they should fit together. In a simple hypercomplex system, we have four components, r, i, j and k — “r” is real, i and j are identicallly-acting roots of minus one, but i-times-j gives a third creature, k, and k-squared gives plus one. So we can plot r, i, j to get a 3D Mandelbrot. Trouble is, as Skytopia point out, it's a bit boring … if we look down on the 'brot's side-bulbs, they show up as simple nubbins. There are other way to try to force Mandelbrot cross-sections, but they're a bit arbitrary, and the results tend to look like someone's cut them out of a block of wood using a Mandelbrot template.

Paul Nylander (bugman) then started looking at higher-powered counterparts of the Mandebrot, and realised that the boring hypercomplex solid for z^2 actually got pretty damned interesting when you jacked the power value up to eight (z^8). This gives a gorgeously intricate beast now referred to as a Mandelbulb, with bulbs that spawn bulbs all over the place. It also has Julia-set siblings. But it's not a standard Mandelbrot.

So what else? Well, the “standard” hypercomplex number system isn't the only option. There are alternative systems that give multiple imaginary components with slightly different interrelations. There are quaternions (tried them, didn't like them), and there are other potential configurations and a larger overarching system of eight-parameter octonions. The Mandelbrot-based solid at the top of this blog was made with one of those. The internal shape is also slightly reminiscent of a Buddhabrot.

The semitransparent voxel plot above isn't really able to show the shape properly, you can see there there's some fine floating ribs that connect some of the Mandelbrot features on the two planes that aren't being adequately captured by the plot, so I'll have to run off a larger version at some point, and perhaps experiment with some colour-coding. Some of the more exotic detail, like the floating network of ribbing, might also be an artefact of a technique I used to emphasise surface structure in the plot, so I'll need to spend some time playing with the thing and working out how much of the image is “proper” 3D Mandelbrot detail, and how much is an additional fractal contribution from the enhancement code.

But meanwhile … pretty shape!

"Tesla Turbine" Pumps

2010-05-30T22:06:00.000+01:00

When used as a pump, the Tesla turbine is one of the simplest devices that exists. Its main component is simply a spinning disc – the disc is immersed in a fluid (like air, or water), the moving surface couples frictionally with the surface of the fluid, and makes the surface layer of fluid rotate with the disc. The fluid gets thrown outwards away from the rotation axis by centrifugal forces, and new fluid moves in to take its place. You then typically build a box around the container, with an inlet tube and outlet tube. The inlet feeds fresh fluid to the central axis of the disc, and the higher-pressure "centrifuged" fluid that collects around the disk edge is collected and allowed to escape via the outlet pipe.

You spin the disc (in either direction), and fluid jets through the device.

Now sure, we can do this sort of thing with a conventional bladed propeller, but those beasties have problems. The blades chop up the air or water, and create turbulence, which in turn encourages the assembly to vibrate, and small imperfections in the rotor construction can cause imbalances (and vibrations) that are different at different speeds. So bladed designs tend to be messy and noisy and juddery, and the blades' leading edges are prone to collecting buildups of dust or muck, or being damaged by collisions with any junk that happens to be caught in the fluid stream, which in turn messes up the aerodynamics of the blade and unbalances the assembly.

If you've ever built a PC to be especially quiet, you'll know that as the months pass, it gets noisier and noisier until you have to take the thing apart to clean the accumulated muck off the leading edges of the fanblades. In the case of ship's propellers these vibrations cause more extreme physical damage: sonoluminescence momentarily creates microscopic pockets of superheated steam that can etch pits into the bronze. All this work wastes energy and causes unwanted noise and vibration, and makes for additional engineering complications.

With the Tesla turbine fan, this violent interaction with the stream doesn't happen. For conventional propellers, surface friction wastes energy, with a Tesla disc, surface friction is the useful coupling mechanism that makes the thing work.

Nowadays, if you have a tropical fish tank or an outdoor pond with an ornamental fountain, the little cylindrical pump that circulates the water or drives the fountain is probably a small centrifugal Tesla turbine. Because it's bladeless, it means that any tiny creatures that get into the pump don't risk being chopped or hit by a big nasty blade, they might have a couple of bumps on the way through, but that's it. And weeds can't snag on the propeller blades and jam the pump, because there aren't any propeller blades to snag. So it's a comparatively creature-friendly and low-maintenance type of pump, if you want something to pump water for years without requiring any attention, or mashing up the microfauna.

Recently, they've also starting to consider using Tesla pumps for pumping blood. Blood includes all sorts of delicate gunge that doesn't like being disturbed too much, or it's liable to trigger a clotting reaction or an immune response. You don't want to smash up too many of the blood cells or start banging platelets together -- traditional blood pumps use clear tubing that's "massaged" by rotors to push the blood through, which makes for a nice simple high-visibility sealed unit, but you're still "squashing" some of the blood every time the pinched region travels along the tube.

Perhaps the most surprising thing about Tesla pumps, apart from their simplicity, is how long it took us so many years to realise that these things were useful. A diagram of a conventional bladed fan gives you some indication of what a device does, but a simple smooth spinning disc in a box doesn't look as if it would do anything useful. Nikola Tesla got his turbine patent as late as 1913 claiming it as a novel device, Tesla pumps apparently started being generally manufactured in the 1970's, and a quick Google for references to radial bloodpump designs seems to only throw up results newer than 1990, most in the last five or ten years.

Sometimes we miss out on useful technologies because they require too much R&D or technical skill to get them to point where they actually work, but sometimes we also miss out on trivially-easy technologies that "work first time" because they're just too damned simple.

Rice and the Chessboard

2010-05-14T23:25:00.002+01:00

In the story, an Emperor asks his mathematician to solve a difficult problem.

In payment, the mathematician asks for a chessboard with one grain of rice on the first square, two on the next, four on the one after that, eight on the next, and so on. The emperor agrees. Then the smart-alec mathematician points out that by the time we get to the sixty-fourth square, the number of grains of rice is astronomical. It's about 10^19, or 10,000,000,000,000,000,000. In binary, that's 1111,1111,1111,1111,1111,1111,1111,1111,1111,1111,1111,1111,1111,1111,1111,1111 , which is probably the largest number that you can express as a standard integer on a modern 64-bit processor running a specialist 64-bit version of Windows. One more grain of rice and you probably get an overflow error.

If each grain of rice weighs about 25 mg, then, when we double the last-square figure to get the total number of grains on the chessboard, I think we end up with something like 460 billion metric tonnes (minus one grain).

According to the story, the Emperor's response was to point out that this created a new problem that required the mathematician's involvement. As Emperor, he couldn't go back on his word, even if the mathematician allowed him to. An Imperial Decree couldn't be rescinded. On the other hand, that much rice didn't physically exist. The solution was to point out that if the mathematician didn't exist, the debt would cease to exist, too. So the Emperor signed the mathematicians' death warrant on the grounds that pulling this sort of trick on the Emperor counted as treason, and had him executed.

Here's how to work out the result in your head, without using a calculator (or even pen and paper) :

Square 1 has one grain of rice. The next ten have 2, 4, 8, 16, 32, 64, 128, 256, 512 and 1024.

Every time that you advance another ten squares, you multiply the number on the square by 1024 (2^10), which is only slightly more than a thousand (1000, 10^3). As a first approximation, every ten-square move pretty much shifts the "decimal" version of the number three places to the left.

This means that when we move sixty squares, we're adding those three zeroes six times, giving us eighteen zeroes. That leaves just three more squares, so we go 2, 4, 8 … and write down a "guesstimate" figure of 8 ×10^18 for the number of grains on the last square.

This is an underestimate, but by how much? We treated 1024 as if it was 1000, so we have a missing factor of 1.024 that needs to be multiplied in six times to get to the proper answer. What's 1.024 raised to the sixth power? Eww. :(

Well, when we square "one-point-something", we get one, plus two times the "something", plus "the something-squared" ( (1+x)^2 = 1 + 2x + x^2 ).
If the something is very small, then something-squared is going to be extremely small, and hopefully so small that we can forget about it, and get away with just doubling the original small something.

So "1.024 times 1.024" gives 1.048, plus a little bit. Call it 1.049 .
Now we need "1.049 times 1.049 times 1.049", to get us up to that power of six.
A similar principle applies: cube something very close to one, and the tiny difference kinda triples (plus a little bit).
So we take 1.049, look at the part after the decimal point, 0.049, nudge it up to a nicer 0.05 , then triple that to give ~0.15 as the ratio that has to be multiplied into our original result to find the amount of undershoot correction.

"Eight" times the 0.1 is 0.8
"Eight" times the 0.05 should be half that, so 0.4 .
Adding them together, 0.8 + 0.4 is 1.2 (× 10^18).
That's our error .
Add that to the original guess of 8 (× 10^18), and we get our improved estimate, of ~9.2 ×10^18 .

… and if we check that against our calculator, which says that 2^63 = ~9.22 × 10^18, we were correct to two significant figures. Not bad for calculating something to the sixty-third power. Yayy Us!

Ten Things you can't do on an Apple iPad

2010-05-03T04:35:00.001+01:00

Ten Things you can't do on an Apple iPad:

Watch broadcast TV
The iPad has nowhere to plug in a DVB TV tuner dongle, and even if if it had, the iPad doesn't decode the MPEG2 video format used for standard-format DVB digital tv broadcasts. It's MPEG4-only. So you can't use it as a personal video recorder, and if you have an existing PVR, you won't be able to copy or stream the recorded MPEG2 files to the iPad. Unless your other machine's fast enough to convert to MPEG4 in real time, you'll have to transcode your files to MPEG4 first. Oh, and not all MPEG4 transcoder software produces files that play properly on the iPhone OS, so even if you do transcode, you still might not be able to watch the files.
Listen to the radio
The iPhone chipset supposedly includes an onboard hardware FM radio, which the OS doesn't make available. In theory you can plug an FM receiver module into the iPhone/iPad docking connector, but in practice, it's cheaper to buy a separate radio (or a cheap MP3 player with a radio onboard). Apple don't make a separate snap-in radio, and third-party manufacturers ave been a bit reluctant to market one in case it becomes redundant overnight, if and when Apple decide to finally enable the internal device. Apple don't want you listening to FM until they can find a way to make money from it, and with FM, it's the radio station that gets the advertising revenue, not Apple.
If you have a good internet connection, you can listen to a stack of radio stations online … as long as they don't use Flash as a delivery medium.
Major radio stations are often also available via DVB ... but that's not an option with the iPad because of point (1).
Many iPhone owners get their "fix" of radio by buying a speaker dock that includes an FM radio receiver, but fitting an iPad to one of these is a bit more difficult.
Watch DVDs
Okay, so you don't expect the iPad to have a DVD drive, but netbooks at least have the option of plugging in a cheap USB-powered optical drive to play your DVD movies. Not the iPad. And even if it had a general-purpose USB port, standard DVD video is encoded in MPEG-2, so even if you find a way to get the DVD .vob files de-encrypted and onto the iPad, it won't play them. If a relative passes you a homebrew DVD with your family's home movies, you're back into Transcoding Hell. Transcoding on a mac probably produces "Apple-friendly" MP4 files, first time, every time ... on other platforms, don't count on it.
View or edit OpenOffice files
Some organisations are trying to migrate away from using MSOffice files to more open formats, to avoid vendor lock-in. The main alternative suite is OpenOffice, which runs under Windows and Linux, can read and write all the main MS formats as well as its own "open" format, and also happens to be free. Apple don't seem to have a reader for "Ooo" files. They don't seem to much approve of open formats, and would rather you used Microsoft's apps and formats than open-source – they see open-source as a bigger threat than Microsoft.
Share photos.
Jobs says that sharing photos is "a breeze" on the iPad. By "sharing", he presumably means, "tilting the screen so that other people can see it". If you want to actually give someone a copy of a holiday picture, you'll probably have to do it on a different computer, rather than the iPad. There's currently no "file export" media option. Budget picture frames usually have have picture sorting, import/export, and USB/SD card support functions, but the iPad doesn't, it's strictly a secondary device. Any serious file organisation is supposed to be done on a parent computer, so don't expect to be able to sort your piccy collection on the iPad while sitting comportably on your sofa.
There is a USB/Cardreader accessory listed for the iPad ... the Camera Connection Kit ... but Apple currently only describe it as allowing you to import files to the iPad. To get the photos out of the iPad, you're supposed to synch to the iPad's "parent" PC or Mac, and then save them from that parent device. In which case, it'd be faster to upload the files directly to the parent machine without going via the iPad. Not exactly "breezy".
Use standard peripherals.
As well as not having internal USB, the OS 3.x iPhone apparently doesn't support much in the way of bluetooth peripherals other than stereo headphones, and apparently doesn't even support Apple's own bluetooth keyboard. Apple's "official" external keyboard for the iPad is a dedicated iPad keyboard-and-stand, which only works in portrait mode. Heath and safety regulations say that you aren't supposed to use keyboards in an office environment unless they're adjustable, and this looks like it probably isn't. But Apple seem to have realised that this restriction sucked too much, and the iPad's OS 4.0 now seems to be more relaxed, and supports Apple's general-purpose bluetooth keyboard (which costs the same as the dedicated iPad keyboard).
Unless the iPad's "OS 4.0" is a radical departure from 3.x, you probably also won't be able to zap contacts or notes or files into the iPad from general bluetooth peripherals, like you can with decade-old bluetooth-equipped Palm devices. I used to carry about a pocket-sized Targus folding keyboard and an OCR pen-scanner device with my old Palm organiser. Nothing like that seems to be available for the iPad.
Record stereo audio.
Apple want you buying music, not recording it, so while the Apple dock connector has pins for stereo in, the official iPad Apple specifications don't commit to the pins doing anything. Maybe they're connected, maybe they're not. If they are, great. But its a brave third-party manufacturer who releases a product or connector for a function that an Apple device isn't guaranteed to have – even if your gadget works now, one OS revision later it might not (see also (2) external FM radio). As a playback-only media centre, the iPad again has the problem that onboard organisation is limited – you're supposed to do all your media organising on a separate parent computer, and iTunes usually won't recognise album art originating on a PC. Often it won't recognise PC-ripped tracks and let you download replacement artwork, either. Of course, if you're sick of watching CoverFlow "flipping" blank squares, you can always buy your albums over again as Apple downloads, or rip the CD's again using a mac ...
Use unapproved software.
Apple reserve the right to decide what software you run on your machine, and there are certain sorts of applications they really don't want you to have. You normally aren't even allowed to load your own media files onto an iPhoneOS device unless the iTunes "sentry" approves – the iPx range won't emulate a basic thumb drive.
You can often upload these "unapproved" apps and use your iPx gadget as a file caddy, by hacking past the Apple firmware's protection to expose the internal filesystem over USB – "jailbreaking" – but jailbreaking doesn't always work on all models, and it's too early to know what eventual proportion of iPads are likely to be jailbreakable.
Camera functions
iPhone OS 4 is supposed to finally add proper support for camera functions, but the iPad doesn't actually have a camera. In theory it'd be easy to add support for a camera that snaps onto the dock connector, but AFAIK, no third-party manufacturer has yet produced one.
It's probably easy in theory to support a swivellable webcam that can point forwards as a camera or backwards for video calls, but that'd need the device to be held upside down with the dock connector at the top. There's no technical problem with this … except that Apple's own OS 3.x applications refuse to work in upside-down mode. On OS4, the onboard applications are supposed to work in any orientation, but it's still a bit discouraging for manufacturers to know that if they launch a camera, it won't work well on v3.x devices. There's also the possibility that if Apple do decide to embrace the idea of an add-on camera, they won't make the function ready until they have a camera of their own to sell. You could buy rotatable snap-in cameras for some Palm organisers nearly ten years ago, so the iPad's still lagging behind in this respect.
And there's some useful camera-aware apps: the Evernote notetaking apps let you snap images (memos, restaurant menus, street signs), save them with geotagging data, and apply OCR to add the text in the image to a searchable comments field. If you have a iPhone with Evernote, and someone shows you their contact details on their smartphone screen or a business card, you can snap a photo and get a text file. But without a camera, none of this cool stuff will currently work on the iPad. Evernote also has a nice voicenotes feature, but again, on the iPad ... no onboard mic.
So, no Skype video calling.
SIM-swapping.
The iPad isn't locked-in to a particular phone provider (hooray!), but the bad news is that if you've just bought a high-capacity service plan for your iPhone, and you want to transfer it to your iPad (which you expect to be using for all your serious mobile web-browsing from now on), you can't. The SIMs are physically different sizes. The iPhone uses a standard-sized SIM, the larger iPad uses a smaller mini-SIM. In theory, a mini-SIM with a holder can fit into a full-size SIM slot, but that chances are that if you're an existing iPhone owner, you won't have one of those. Apple enthusiasts have gotten used to Apple engineering-in incompatibilities with other manufacturers' products, but some have gotten a bit annoyed at what looks like a deliberate incompatibility with other Apple products.

The iPad isn't really what Steve Jobs said it was. It's not a device that's designed to sit in some middle ground between netbooks and laptops, because those two types of device can do pretty much everything on the list.

The iPad's purpose is straightforward: it's designed to kill sales of the amazon Kindle, break amazon's stranglehold on ebook sales, and let Apple add ebook and magazine retailing to their existing music-and-movies portfolio. It's a conduit.
It has to be five hundred dollars in order to crush the Kindle DX, at $500 its facilities have to be limited in order to avoid undercutting Apple's own laptop range (which starts at a thousand dollars) and it has to be based on the iPod Touch (with an updated "iPhone OS" and a bigger screen) to give it an established sales channel, because that's the "other" OS that Apple have, because that preserves separation between the iPad and the more expensive OSX-based products, and because that makes it more difficult for people to dig out and redistribute downloaded paid-for content.

Those three things pretty much define it.

'Circular' Polyhedra, and the Apollonian Net

2010-04-27T01:53:00.001+01:00

This is the nice design that I used on page 2 of the book.

Annoyingly, rather a lot of other people discovered it before me: it's indexed on Wikipedia as the Apollonian Net, after Apollonius of Perga (~262 BC – ~190 BC), and it's also referred to elsewhere as the Leibniz Packing diagram, after Gottfried Leibniz (1646-1716), Newton's rival for the invention of calculus. I've even seen it credited to the design of the floor of a Greek temple. But frankly, it's such a nice shape that I'm sure that people have been discovering and rediscovering it for millennia. Draw three touching circles, fill in the inviting gap in the middle with more circles, and when you're feeling pleased with yourself and wondering what to do next, step back and look at the whole thing, draw in a bigger circle to enclose everything (facing away from you), and repeat. That's how I got there, anyway.

There's some rather interesting geometry here to do with tangents, but I got impatient trying to get a complete derivational method, and generated the figures using a vector graphics program (CorelDraw10), driven by an automating script, using a mix of partial derivations, testing, and brute force. If you're calculating a chain of circles that might be twenty or thirty stages long, successive rounding errors tend to screw up these diagrams when you calculate them "properly"(look at the overlap of the smaller circles in the Wikipedia vector graphics version), and my priority was to make sure that the circles really did fit, so I used a hybrid approach where I used trig to get each circle into the ballpark of its proper destination w.r.t. its parents, and then a successive approximation method with error correction to tweek and nudge and jiggle everything snugly into place.

The Apollonian Net makes more sense when you stretch it over the surface of a sphere, so that the four largest "primary" circles are all the same size, and are explicitly equivalent. They then form the intersection of the sphere with the four faces of a tetrahedron, giving the fractal-faceted solid that I used as a vignette on page 378.

There are two main ways to construct this solid:
1: Start with a sphere and grind four flat circular faces into it that correspond to the four faces of an intersecting tetrahedron, then keep grinding maximum-sized circular facets into the remaining curved parts, ad infinitum.

2: Start with a tetrahedron, and lop off the four points to give a shape with four regular hexagonal faces, and four new triangular faces where the tips used to be. Then continue lopping off the remaining points, ad infinitum. Each wave of cutting creates a new face at each cut, and doubles the number of sides on all the existing faces. If we cut at a depth that'll keep these polygons regular, then with an arbitrarily-high number of cuts, the faces converge toward perfect circles, and the point-mesh of the resulting peaks converges downwards to settle onto the surface of the sphere used in method 1.

Either way works.

This sort of duality is common when we construct standard polyhedra – the network of relationships in a regular polyhedron tends to be another regular polyhedron, so we can usually get to a regular shape by starting from either of its two relatives. Four of the five Platonic solids pair up nicely like this, and the last – the tetrahedron – is a special case whose "dual solid" partner is another tetrahedron. But we normally only consider these sorts of dualities when considering combinations of regular polygons with finite numbers of rectilinear sides with each other, and don't include the infinite-sided fractal shapes that show up when one of the parent solids is an infinitely-faceted sphere (which, in some ways, almost counts a a sixth Platonic solid).

We don't have to start with a tetrahedron, we can make these fractal solids from any regular polygon (cube, etc). But the tetrahedral and icosahedral versions probably look the nicest. I find the cube-based version a bit disappointing, but I grew up with rounded-cornered dice with circular faces, so perhaps I'm just a bit blasé about the solid that corresponds to the "six-circle" version of the Apollonian net.

From here, we have three immediate ways to generate new families of solids:
(1) We can choose different starting solids,
(2) we can vary the number of cuts or cutting stages (from zero to infinity), to produce finite-sided solids that look more like cut gemstones, and
(3) we can vary how the cutting is done. If we make our cuts too shallow, then the facets are distorted away from circularity, and the overall shape isn't a sphere, but has flat-topped bulges where the original polyhedral points used to be. If we cut too deep, we get bulges in the shape of the original solid's "dual" sibling, with each bulge tipped by an edge.

Another cool thing about these nets is their topological transformability. With the "closed" version, every circle has three parents of the same size of larger, including the four primary circles (who count as each other's parents). You can transform between the different versions of the net by warping and resizing, while still keeping everything as circles.

This lets us get to tilings that don't automatically suggest standard polyhedra, such as the "two-large-enclosed-circles" version that I used for the "fractal Yin-Yang" symbol on page 145, and the asymmetrical versions on page 224. And once I'd written the scripts and code to generate these figures, I had a few more blank bits in the book to fill, so I knocked up the "triangular boundary" version on page 370 which, actually, has some other interesting proportions. The "triangle" version includes parts that represent the limiting case of the edge of the Apollonian Gasket when we zoom in so far that the outer circle tends toward a straight line. Filling these voids then gives the special-case Ford Circles tiling.

Some serious people have worked on this subject. You can also Google Descartes' Theorem (after René Descartes (1596-1650), and Soddy Circles. Lester Ford and Frederick Soddy only produced their papers in 1936 and 1938, so the Apollonian Net involves math research that extends across more than two thousand years, and isn't finished yet.

It would have been nice to meet the person who designed that floor, though.

Ultra-high resolution photography

2010-04-18T23:48:00.000+01:00

The "jitter" method (earlier post) can also be used for ultra-high-resolution photography.

People want higher-resolution cameras, but the output resolution of a camera is usually limited by the number of pixels in its sensor. Some digital cameras have a "digital zoom" function, but this is a bit of a cheat: it simply invents extra pixels between the real pixels by smudging the adjacent colour values together. Conventional digital zoom doesn't actually give you any additional information or detail, it just resizes a section of the original image to fill the required size.

A second problem with cameras is camera shake. If you're holding the camera in your hand, then a tiny movement of the camera can result in the image being panned across the sensor while the CCD imaging chip is doing its thing, giving a blurred photograph. The smaller the pixel elements, and the greater the optical zoom, the worse this gets. We can try clamping the camera and taking a shorter-exposure image (so that the camera doesn't have as much time to move), but shorter exposures lead to more random "noise" per pixel, due to the reduced sampling time.

But with enough processing power, we can use jitter techniques to solve both problems:
In our earlier "audio" example, we deliberately added high-frequency noise to an audio signal to shift the sampling threshold up and down with respect to the signal, and we took multiple samples and overlaid them to achieve sub-sample resolution.
With digital photography we can use "positional" noise: we vary the alignment of the camera sensor to the background image, take multiple samples, and overlay those (aligned to subpixel accuracy), to generate images that have higher resolution than the camera sensor. In some ways, this is a little like the Nipkow disc approach used in early television systems, that often used a swept array of less than a hundred sensor elements provide a passable image ... in this case, we're not sweeping a line strip of sensors at right angles, but an entire grid of pixel elements, and using their random(-ish) offsets to extract real intermediate detail.

Instead of camera shake being a problem, it becomes Our Friend! The individual images will be noisier, but when you recombine a secondsworth of images, the end result should have noise levels comparable to a single one-second exposure – and since you might not normally try to take a one-second exposure (because of camera stability issues), static scenes might sometimes end up with reduced noise as well as enhanced resolution.

So, if we have a programmable camera, in theory it's possible to design an "ultra-resolution" mode that fires off a series of short-exposure images while we hold the camera, and then makes us wait while its processor laboriously works out the best way to fit all the shots together ... or saves the individual shots to their own directory, to be assembled later by a piece of desktop software.
If we were able to design the camera from scratch, we'd probably also want to include a gadget to deliberately nudge the CCD sensor diagonally while the component shots were being taken. If the software's smart enough, the nudging doesn't have to be particularly accurate, it just has to give the sensor a decent spread of deliberate misalignments. A cheap little piezo device might be good enough.

The problem with this approach is getting hold of the software: In theory, you can try aligning images by hand, but in practice ... it doesn't really seem sensible.
People are already writing algorithms for this sort of stuff – it's what allows the Hubble space telescope to take those absurdly high-resolution images of distant galaxies, and presumably the military guys also use the technique to get extreme resolution enhancements from spy satellite hardware. For analysing and aligning photos with "free-form" offsets, the necessary techniques already seem to be included in the Autostitch panoramic software, which even includes the ability to distort images to make them fit together better – it wouldn't seem to take a lot to turn Autostitch into an ultra-resolution compositor.

Amateur astronomers are now enthusiastically using the technique, and sharing resources (try using "drizzle" as a Google search keyword).
Suppose that you want to take an ultra-high resolution photograph of the full Moon – you train your camera-equipped telescope at the Moon, lock it down, and set it to keep taking ten pictures per second for an hour while the Moon gradually arcs across the sky and it's corresponding image crawls across your image-sensor ... and then feed the resulting thirty-thousand-odd images into a sub-pixel alignment program, to chew over for a few weeks and pull out the underlying detail. As long as the matching algorithm knows that it's supposed to be lining up the part of the images that contain the big round yellow thing rather than the clouds or the treetops, there wouldn't seem to be any real limit to the achievable resolution. Okay, so you have different atmospheric distortions when the Moon is in different parts of the sky, and when the air temperature drifts, but with a sufficiently-smart autostitch-type warping, even that shouldn't be a problem. If you didn't have a "rewarping" feature, you'd probably just have to decide which part of the moon you wanted the software to use as a master-key when lining up the images.

Techniques like this go beyond conventional photography and enter the territory of hyperphotography – we're capturing additional information that goes beyond our camera's conventional ability to take images, and doing things that, at first sight, would seem to be physically impossible with the available hardware. A bit of knowledge of quantum mechanics principles is useful here: we're not actually breaking any laws of physics, but we're shunting information between different domains to obtain results that sometimes seem impossible.

There's a whole family of hyperphotographic techniques: I'll try to run through a few others in a future post.

Titanic Syndrome

2010-04-10T02:53:00.001+01:00

On the 10th of April 1912, the RMS Titanic set out on her first passenger-carrying voyage. The Titanic (and her Olympic-class sister-ships were state-of-the-art. They had a double-hulled design that meant that if one hull ruptured, the ship was still seaworthy. The ship was considered to be practically unsinkable.

Four days later it was at the bottom of the ocean with the bodies of 1517 crew and passengers. The "unsinkable" ship was arguably the most "sinky" ship in human history.
It's normally difficult to assign a "sinkiness" ranking to ships, given that each failed ship only normally manages to sink once, but by sinking before it even made it to the end of its maiden voyage, and killing so many people, the Titanic flipped straight from being supposedly one of the safest seagoing structures ever built, to one of the most dangerous.

Titanic Syndrome isn't based on any specific mechanism. "Syndromes" are recognisable convergences of trends, that can sometimes associate a particular outcome with a recognisable set of starting parameters. When we notice one of these patterns, we sometimes have a good idea how things are likely to end without having to know the mechanism that gets us there.

In the case of Titanic Syndrome, the association is pretty self-explanatory: when people tell us that nothing can possibly go wrong, that everything's perfectly safe, that a plan is foolproof ... things usually turn out badly.

Why did the Titanic disaster happen, and happen so emphatically? The obvious answer is that the ship sank because it struck an iceberg, but there are additional factors that track back to that initial belief that the ship was almost indestructible. If the ship's crew had been less confident, perhaps they'd have done a better job of keeping watch for ice, or cut their speed. If the shipyard had been less confident about the ship's hull, maybe they'd have built it with better-quality materials, rather than just assuming that if one hull failed there was a spare. And if the company hadn't been so sure that lifeboats weren't really necessary, perhaps that'd have included enough for everyone, and not so many people would have had to drown when the ship went down, while they were waiting to be rescued.

In science, hyperbole is usually an indicator that something's wrong. Theories that are described as "pretty good" usually are, but theories that were told are excellent, or that can't possibly be wrong usually turn out to be already failing, unnoticed. Titanic Syndrome.

Theories that really are that good, don't need to be oversold – it's usually possible to express confidence in an established model more convincingly with quiet understatement. On the other hand, if a core theory is right, but the people involved are still trying to exaggerate the case for it (even though their actions are likely to backfire), then if they're making that mistake, they've probably been making others, too. So "cheerleading" is usually a red flag that some things in the picture are likely to be dodgy, even if the fundamentals of a theory are right.

And sometimes the "cheerleading" stops people noticing that the fundamentals are wrong. And those are the times ... when everybody's invested so strongly in something that they really don't want to believe in the possibility of problems, or start thinking seriously about fallback positions or lifeboats ... that you get another "Titanic-class" event.

General Relativity is Screwed Up

2010-04-02T23:27:00.003+01:00

With Einstein's general theory of relativity, one of the theory's harshest critics was probably Einstein himself. This was partly a matter of personal discipline, and partly – like the joke about sausages – because it's sometimes easier to like a thing if you don't know the gruesome details of how it was actually made. Einstein found it easy to be sceptical about the design decisions that had gone into his general theory, because he was the guy who'd made them. It had been the best general theory that had been possible at the time, said Einstein, but with the benefit of hindsight ... perhaps its construction wasn't entirely trustworthy.
The "iffy" aspects of C20th GR are difficult to see from within the theory, because – where the lower-level design decisions have forced a fudge or bodge – from the inside, these things seem to be completely valid, derived (and quite necessary) features. It's not until we look at the structure from the outside, with a designer's eye, that we see the arbitrary design decisions and short-term fudges that went into making the theory work the way it does.

Sure, the surface math looks pretty (with no obvious free variables or adjustable parameters), but that's because, as part of the theory's development, all the ugliness necessarily got moved down to the definitional and procedural structures that sit below the math. Change those underlying structures, and the surface mathematics break and reform into a different network that looks similarly unavoidable. So even though the current system looks like the simplest possible theory when viewed from the inside, we can't invest too much significance in this, because if the shape and structure was different, that'd look like the simplest possible theory, too.

To see how the theory might have been, we need to look at the subject's protomathematics, the bones and muscles and guts of the theory that dictate its overall shape, and which don't necessarily have a polite set of matching mathematical symbols.

Here are two interlinked examples of decisions that we made in general relativity that weren't necessarily correct:

Problem #1: Gravitational dragging, velocity-dependent gravitomagnetic effects

As Fizeau demonstrated back in ~1849 with water molecules, moving bodies drag light. General relativity describes explicit gravitomagnetic dragging effects for accelerating and rotating masses, and logic pretty much then forces it to describe similar effects for relative velocity, too. When you're buffeted by the surrounding gravitational field of a passing star, the impact gives you some of the star's momentum – momentum exchange means that the interaction of the two gravitational fields acts as a sort of proxy collision, and the coupling effect speeds you up a little, and slows down the star, by a correspondingly tiny amount.
For a rotating star, GR915 also agrees you're pulled preferentially to the receding side – there's an explicit velocity component to gravitomagnetism (v-gm). Even quantum mechanics seems to agree. And we can use this effect to calculate the existence of the slingshot effect, which is not just theory, but established engineering.
But v-gm effects appear to conflict with Newton's First Law of Motion: If all the background stars dragged light according to their velocity, then as you moved at speed with respect to the background starfield, the receding stars would pull on you a little bit stronger than the others, slowing you down. There'd be a preferred state of rest, that'd correspond to the state in which the averaged background starfield was stationary (ish). This doesn't agree with experience.
So the v-gm effect gets edited out of current GR, and when we do slingshot calculations, we tend to use Newtonian mechanics and model them in the time domain, instead. We compartmentalise.
Summary:
Argument: The omission of v-gm effects from general relativity seems to be arbitrary and logically at odds with the rest of the theory, but it seems to be “required” to force agreement with reality … otherwise “moving” bodies would show anomalous deceleration.
I'd consider this a fairly blatant fudge, but GR people would tend to refer to it as essential derived behaviour (based on the condition that the theory has to agree with reality).

Problem #2: Gravitational Aberration

If signals move at a finite speed, the apparent positions of their sources get distorted by relative motion. We "see" a source to be pretty much in the direction it was when it emitted the signal, with a position and distance that's out of date, thanks to the signal timelag.
If gravitational and optical signals both move at about the same speed, "c", (ignoring nonlinear complications), then we expect to "feel" the gravitational signal of a body to be coming from the same position that the object is seen to occupy. Which is kinda helpful.
But it seems that under current GR, the apparent "gravitational" position of a body gets assigned to its instantaneous position, as if the speed of gravity was infinite. We say that the speed of gravity isn't actually infinite, but that moving bodies somehow "project" their field forwards and then sideways so that it looks infinite as far as the observer's measurements are concerned. In other words, it seems that under current GR, there's no such thing as gravitational aberration.
This is a bit like the sound of fingernails scratching down a blackboard. It means that there's no longer the concept of a body having a single observed position, and we get separate definitions of "apparent position" for EM and gravity. This badly weakens the theory, because it means that mismatches between the two that that we might normally look out for to show us that we've made a mistake somewhere, are the theory's default behaviour. We lose a method of testing or falsifying the model.
So why do we do it?
We...ell, the usual argument involves planetary orbits and the apparent position of the Sun as seen by an observer on a rotating planet. But that argument's complicated and perhaps still a bit unconvincing, so … the simpler argument is that if gravitational aberration existed, it'd again seem to screw up Newton's First Law. When an astronaut travels through the universe at high speed, the background stars appear to bunch together in front of them (e.g. Scott and van Driel, Am.J.Phys 38 971-977 (1970) ), and if the gravitational effect of all those stars was shifted to the front as well, then we'd expect the astronaut to be pulled towards the region of highest apparent mass-density … forwards … and this'd further increase their forward speed, making the aberration effect even worse, which'd then create an even stronger forward pull.
So again, we manually edit the effect out, say that it's known not to exist, and then do whatever we have to do with math and language to stop the theory contradicting us.
Summary:
Argument: Losing gravitational aberration seems to be arbitrary and logically at odds with the rest of the theory, but seems to be "required" to force agreement with reality … otherwise "moving" bodies would show anomalous acceleration.

Put these two arguments together, and you should immediately begin to see the problem:

If we'd resisted the "urge to fudge", it looks as if our two problems would have eventually canceled each other out anyway, without our having to get involved. They seem to have the same characteristic and magnitude, but different signs. One produces anomalous acceleration, the other anomalous deceleration. Put them together and the moving astronaut doesn't accelerate or decelerate, because the stronger rearward pull of the fewer redshifted stars behind them is balanced by the increased number of stars ahead, which are blueshifted and individually weakened. Instead of our imposing N1L-compliance on general relativity as a necessary initial condition, the theory works out N1L all by itself, as an emergent property of curved spacetime.

So in these two cases, we seem to have corrupted the "deep structure" of the current general theory of relativity not once but twice, by trying to solve problems sequentially rather than letting the geometry generate the solutions for us, organically. Both "deleted" effects turn out to be necessary for a "purist" general theory … but once we'd fudged the theory once to eliminate one of them, we had to go back and fudge the theory a second time to eliminate the second effect that would otherwise have balanced it out.

And in doing that, we didn't just "double-fudge" a few details of the theory, we broke important parts of the structure that should have allowed it to expand and blossom into a larger, more tightly integrated, more strictly falsifiable system that could have embraced quantum mechanics and dealt with properly with cosmological issues. General relativity should have been a tough block of dense, totally interlocking theory, with independent multiply-redundant derivations of every feature, rather than the thing we have now.

The fudging of these two issues also changed some of the theory's physical predictions:

Losing gravitational aberration gave us a different set of observerspace definitions that altered the behaviour of horizons. Losing v-gm meant that we got different equations of motion, once again a different behaviour for black holes, and no way of applying the theory properly to cosmology without generating further cascading layers of manual corrections reminiscent of the old epicycle approach to astronomy. It also created a statistical incompatibility with quantum mechanics.

So general relativity in its current form seems to be pretty much screwed. GR1915 was fine as an initial prototype, but it should really have been replaced half a century ago – in 2010, it's an ugly, crippled, mutated, limited form of what the theory could, and should have been by now. But because people fixate on the math rather than on the structure, they can't see the possibility of change, or the beauty of what general relativity always had the potential to become. And that's why the subject's been almost stalled for pretty much the last fifty years, it's because Einstein died, and too many of the surviving physics people who did this stuff couldn't see past the mathematical and linguistic maze that'd developed around the subject, they didn't "get" the design principles and the dependencies between the choice of initial design decisions and the characteristics of the resulting model, and they didn't appreciate the design aesthetics.

And I find that sad on so many levels.

3D Audio, and Binaural Recording

2010-03-28T03:48:00.001+01:00

One of the dafter things they teach in physics classes is that because humans only have two ears, we can only hear location by comparing the loudnesses of a sound in both ears, and that because of this we can only hear "lefty-rightiness", unless we start tilting our heads.

It's wrong, of course: Physics people often suck at biology, and (non-physicist) humans are actually pretty good at pinpointing the direction of sound-sources, without having to tilt our heads like sparrows, or do any other special location-finding moves.

And we don't just perceive sound with our ears. It's difficult to locate the direction of a backfiring car when it happens in the street (because the sound often reflects off buildings before it reaches us) ... but if it happens in the open, we can directionalise by identifying the patch of skin that we felt the sound on (usually chest, back, shoulder or upper arm), and a perpendicular line from that "impact" patch then points to the sound-source.
For loud low-frequency sounds, we can also feel sounds through the pressure-sensors in our joints.

But back to the ears ... while its obviously true that we only have two of them, it's not true that we can't use them to hear height or depth or distance information. Human ears aren't just a couple of disembodied audio sensors floating in mid-air, they're embedded in your head, and your head's acoustics mangle and colour incoming sounds differently depending on direction, especally when the sound has to pass through your head to get to the other ear. The back of your skull is continuous bone, whereas the front is hollow, with eyeballs and eyesockets and naso-sinal cavities, with Eustachian tubes linking your throat and eardrums from the inside. You have a flexible jointed spine at the back and a soft hollow cartilaginous windpipe leading to a mouth cavity at the front, and as sounds pass through all these different materials to reach both ears, they get a subtle but distinctive set of differential frequency responses and phase shifts that "fingerprint" them based on their direction and proximity.

To make the colouration even more specific, we also have two useful flappy things attached to the sides of our heads, with cartilaginous swirls that help to introduce more colourations to sounds depending on where they're coming from. Converting all these effects back into direction and distance information probably requires a lot of computation, but it's something that we learn to do instinctively when we're infants, and we do it so automatically that – like judging an object's distance by waggling our eye-focusing muscles – we're often not aware that we're doing it.

The insurance industry knows that people who lose an external ear or two often find it more difficult to directionalise sound. Even with two undamaged eardrums, simple tasks like crossing the road can become more dangerous. If you've lost an ear, you might find it more difficult working on a building site or as a traffic cop, even if your "conventional" hearing is technically fine.

Binaural, or 3D sound recording:

We're good enough at this to be able to hear multiple sound sources and pinpoint all their directions and distances simultaneously, so with the right custom hardware, a studio engineer can mimic these effects to make the listener "hear" the different sound-sources as coming from specific directions, as long as they're wearing headphones.

There are three main ways of doing this:

1: "Dummy head" recording

This literally involves building a "fake head" from a mixture of different acoustic materials to reproduce the sound-transmission properties of a real human head and neck, and embedding a couple of microphone inserts where the eardrums would be. Dummy head recording works, but building the heads is a specialist job, and they're priced accordingly. Neumann sell a dummy head with mic inserts called the KU100, but if you want one, it'll cost you around six thousand pounds.

Some studios have been known to re-record multitrack audio into 3D by surrounding a dummy head with positionable speakers, bunging it into an anechoic chamber and then routing different mono tracks to different speakers to create the effect of a 3D soundfield. But this is a bit fiddly.

2: 3D Digital Signal Processing

After DSP chips came down in price the odd company started using them to build specialist DSP-based soundfield editors. So for instance, the Roland RSS-10 was a box that let you feed in "mono" audio tracks and it'd let you choose where they ought to appear in the soundfield. You could even add an outboard control panel with alpha dials that let you sweep and swing positions around in real time.

Some cheap PC soundcards and onboard audio chips have systems that nominally let you position sounds in 3D, but the few I've tried have been a bit crap, their algorithms probably don't have the detail or processign power to do this properly.

At "only" a couple of thousand quid, the Roland RSS10 was a cheaper more controllable option for studio 3D mixing than using a dummy head in a sound booth, and Pink Floyd supposedly bought a stack of them. There's also a company called QSound that do this sort of thing: Qsound's algorithms are supposed to be more based on theoretical models, Roland's based more on reverse-engineering actual audio.

3: "Human head" recording

There's now a third option: a microphone manufacturer called Sound Professionals had the idea that, instead of using a dummy human head, why not use a real human head?.

This doesn't require surgery, you just pop the special microphones into your ears (making sure that you have them the right way round), and the mics record the 3D positioning colouration created by your own head's acoustics.

The special microphones cost a lot less than a Neumann KU100, and they're a lot easier to use for field recording than hauling about a dummy head – it's just like wearing a pair of "earbud"-style earphones. The pair that I bought required a mic socket with DC power, but I'm guessing that most field recorders probably provide that (they certainly worked fine with a Sony MZ-N10 minidisc recorder).

Spend a day wandering around town wearing a pair of these, and when you listen to the playback afterwards with your eyes closed, it's spooky. You hear //everything//. Birds tweet above your head, supermaket trolley wheels squeak at floor level, car exhausts grumble past the backs of your ankles as you cross a road, supermarket doors --swisssh-- apart on either side of you as you enter.

"Human head" recording isn't quite free from problems. The main one is that you can't put on a pair of headphones to monitor what you're recording, real-time, because that's where the microphones are: you either have to record “blind” or have a second person doing the monitoring, and you can't talk to that person or turn your head to look at them (or clear your throat) without messing up the recording. If you move your head, the sound sources in the recording swing around in sympathy. Imagine trying to record an entire symphony orchestra performance while staring determinedly at a fixed point for an hour or two. Tricky.

The other thing to remember is that although the results might sound spectacular to you (because it was your head that was used for the recording), it's difficult to judge, objectively, whether other people are likely to hear the recorded effect quite so strongly. For commercial work you'd also want to find some way of checking whether your “human dummy” has a reasonably "standard" head. And someone with nice clear sinuses is likely to make a better recording that someone with a cold, or with wax-clogged ears.

Another complication is that most people don't seem to have heard of "in-ear" microphones for 3D human head recording, so they can be difficult to source: I had to order mine from Canada.

Media

For recording and replaying the results: since the effect is based on high-frequency stereo colourations and phase differences, and since these are exactly the sort of thing that MP3 compression tends to strip out (or that gets mangled on analogue cassette tape), it's probably best to try recording binaural material as high-quality uncompressed wav files. If you find by experiment that your recorder can still capture the effect using a high-quality compressed setting, then fine. The effect's captured nicely on 44.1kHz CD audio, and at a pinch, it even records onto high-quality vinyl: the Eurythmics album track "Love you like a Ball and Chain" had a 3D instrumental break in which sound sources rotate around the listener's head, off-axis: if you look at the vinyl LP, the cutting engineer has wide-spaced the tracks for that section of recording to make absolutely sure that it'd be cut with maximum quality.

Sample recordings

I'd upload some examples, but my own test recordings are on minidisc, and I no longer have a player to do the transfer. Bah. :(

However, there's some 3d material on the web. The "Virtual Barber Shop" demo is a decent introduction to the effect, and there are some more gimmicky things online, like Qsound's London Tour demo (with fake 3D positioning and a very fake British accent!). When I was looking into this a few years back, the nice people at Tower Records directed me to their spoken word section where they stocked a slightly odd "adult" CD that included a spectacular 3D recording of, uh, what I suppose you might refer to as an adult "multi-player game". Ahem. This one actually makes you jump, as voices appear without warning from some very disconcerting and alarming places. I'm guessing that the actors all got together on a big bed with a dummy head and then improvised the recording. There's also a couple of 3D audio sites by binaural.com and Duen Hsi Yen that might be worth checking out.

So, the subject of 3D audio isn't a con. Even if the 3D settings on your PC soundcard don't seem to do much, "pro" 3D audio is very real - with the right gear, the thing works just fine. It's also fun.

Virtual Lego

2010-03-19T18:30:00.002Z

Someone's finally come up with the "killer application" for VR and computer-augmented reality.

It's buying Lego.

You walk into a participating Lego shop, pick up a box of Lego, and walk over to the big screen. A video camera shows you your image. You hold out the box in front of you, horizontally, as if you're holding a tray.

The software sees the box, recognises which product it belongs to, and calculates the exact position of the box corners in three dimensions.

It then retrieves a 3D computer model of the assembled Lego model from its database, and projects a virtual reality image of the completed masterpiece onto the screen as if the completed Lego masterpiece is sitting on top of the box clutched in your little sticky hands.

You rotate the box, and on the screen, the 3D model rotates. Tilt the box and it tilts. Move the box around and you get to see the final Lego construction from different angles, complete with perspective effects.

Oh, and the computer-generated Lego image is also animated. If it's a garage, the little Lego cars scoot about, if it's a building, the little Lego people are wandering about doing their own thing, "Sims"-style, and if its a tipper truck, the truck drives about the top of the box, tipping stuff.

It's very, very cool.

The Caltech Snowflake Site

2010-03-14T23:17:00.001Z

While I was finishing off yesterday's snowflake post, I came across Caltech's excellent snowflake site at www.snowcrystals.com (Kenneth G. Libbrecht).

Lots of photos, lots of useful information. Caltech even have their own snowflake creation machine, that, instead of electrostatically levitating the snowflakes as they grow, or using a vertical blower, applies an electric field to grow narrow ice-spikes, and then lets the snowflakes form at the spikes' tips (which means that the central mount is probaby rigidly aligned to the resulting flake with atomic precision, and doesn't seem to affect the growing process).

If you're in the UK, and you've mocked train companies for blaming their electrical locomotive failures on "the wrong kind of snow", well, it turns out that snow crystallisation has a slightly crazy dependency on both temperature and airborne water content, forming a range of very different shapes, from the classic branched hexagon "christmas card" forms, to hexagonal plates or long hexagonal tubes (snowflake chart).

The CalTech site explains the wide variety of snowflake forms by this temperature-dependence: the idea being that snowflakes form symmetrically because the conditions across the flake are the same at any given time, and that the extreme variety of shapes is a function of the varying environmental conditions that the whole snowflake experiences as it falls through different regions of sky. It might go through a "spiky dendrite" phase, then change temperature and start trying to grow plates, and then go back to "dendrite" mode, and the exact amount of time spent in these different phases then dictates the shape that emerges.

If the identical patterning of the arms is purely a result of the identical (varying) growing conditions across the whole flake, then we don't require any additional mechanism for regulating symmetry. In that case, we'll expect individual snowflakes to accumulate diverging asymmetries as they grow, due to gradients of temperature or water availability or light or airflow across the flake. This'd seem to make the formation of extremely regular crystals a bit unlikely.
But the CalTech site argues that actually, most natural snowflakes are pretty irregular, and that people generally overestimate the degree of symmetry because the artsy folks who photograph them (presumably including CalTech!) give a misleading impression by carefully selecting out the "best" (most regular) flakes to photograph and publish.

That explanation seems to be a bit at odds with the current suggestion of how triangular snowflakes form, though: if triangular snowflakes grow because of airflow over the flake creating an asymmetrical growing environment, breaking the hex pattern, then if there wasn't an additional internal regulating symmetry-mechanism, there'd be no obvious reason why the resulting aerodynamically-disfigured flake should have 120-degee rotational symmetry. Airflow and a moisture gradient flowing across the flake in one direction might allows a bilateral left-right symmetry for the two sides of the flake that are experiencing the same growing conditions ... it doesn't explain why the conditions at the leading point of the falling tri-flake (falling point-first) should be identical to that at the two trailing side-points, or why points on the sides of those two trailing spurs points should be equivalent, when the airflow is hitting them at different angles. If triangular flakes are due to sideways airflow, then it means that the flake seems to be fighting to retain some sort of symmetry despite significant asymmetrical disruptive forces that ought to be destroying it. That'd increase the odds of there being a significant internal symmetry mechanism in play.

Of course, it may be that our explanation of triangular snowflakes is simply wrong, that airflow isn't disrupting the hex pattern, and that instead chemical contamination (or some other factor) is causing the alternative triangular crystal structure. But that'd still mean that something in our current understanding of snowflakes is wrong or incomplete. Even if yesterday's wacky suggestion about the quantum mirage effect is midguided, we'd still not know why snowflake formation is so sensitive to environmental conditions, or what the (non-aerodynamic) explanation of triangular snowflakes might be.

So again, more research needed.

The Caltech site's debunking of "mysterious" causes of snowflake symmetry is in the "Myths and Nonsense section" at http://www.its.caltech.edu/~atomic/snowcrystals/myths/myths.htm . The page says that there aren't any special forces at work here regulating symmetry, that most snowflakes are asymmetrical and "rather ugly", and that the published examples (including the ones on the site) are atypical, because "not many people are interested in looking at the irregular ones". In other words, if you look through the published work, you get a misleading impression due to publication bias. Well, yes ... quite possibly. But since the idea of what counts as "significant" symmetry might be a bit subjective,and since the datasets aren't available for us to look at, it's difficult to take this as a definitive answer until there's been actual experimental testing done.

Water is wierd stuff, and it keeps catching us out. I remember when people used to debunk ice spikes as an obvious example of psudoscience, and now those are understood, studied, and have their own page on the CalTech site. A lot of "crazy" ideas about water do turn out to be just as dumb as they first appear, but a few turn out to be correct. The trouble is, it's not always immediately obvious which are which.

Snowflake Engineering, Quantum Mirages and Matter-Replicators

2010-03-13T23:56:00.002Z

One of the most impressive things about snowflakes is that we still don't really understand how they work.

We understand how conventional crystals grow – normal crystals assemble into large, faceted, regular-looking forms because the flat facets attract new atoms more weakly than the rougher, "uncompleted" parts of the structure, which provide more friendly neighbours for a new atom to bond with. So if you have an "incomplete" conventional crystal, it'll preferentially attract atoms to the sites needed to fill in the gaps, to produce a nice large-faceted shape that tries to maximise the size of its facets, as far as it can bearing in mind the original random initial distribution of seed crystals.

But snowflakes do something different. Their range of forms makes their growth appears pretty chaotic, but they also manage to be deeply symmetrical. It'd seem that the point of greatest attraction on a region of snowflake doesn't just depend on the atoms that are nearby, but also on the arrangement of atoms on a completely different part of the crystal, which might be some way away, and facing in a different direction, on a different spur. The sixfold symmetry of a snowflake suggests that when you add an atom to the point of one of the six spurs, the other five points become more attractive ... add an atom to the side of a spur, and we're dealing with twelve separate sites (twenty-four if the atom is off the plane). Add an atom to a side-branch, and a copy of the electrical-field image of that single atom is transmitted and reflected and multiplied and refocused at potentially tens of corresponding sites on the crystal surface. And that's for every atom in the crystal.

This would be beyond fibre-optics, and beyond conventional holography. It'd be multi-focus holography, and the holographically-controlled assembly of matter at atomic scales to match a source pattern – making multiple copies without destroying the original. It'd be using holographic projection to assemble multiple macroscopic structures that are atom-perfect copies of an original. And that idea should make the hairs on the back of your neck start to stand up.

The closest thing I've seen in print to this is the quantum mirage effect described in Nature, 3 Feb 2000. Researchers assembled an elliptical quantum corral of atoms on a substrate, and placed another atom at one of the ellipse's two focal points. They then examined the second focal point, and found that the atom's external field properties seemed to be projected and refocused at the second point, to give a partial "ghost" of the source atom [*][*][*]. You could interact with the ghost even though it wasn't there. Presumably your actions on the "ghost particle" copy would be transmitted back to the source, which'd be recreating the ghost behaviour by a process of electrical ventriloquism, using the elliptical reflecting wall to "throw" its voice to the ghost location.

Something similar may be happening in a perfectly-symmetrical monocrystalline snowflake as it grows. Maybe the crystal's regular structure just happens to not just split the image of the atom into multiples, but refocus them with phase coherence at all the key symmetry points. Maybe we could try adding a few metal atoms to one part of a snowflake crystal and seeing if matching atoms are preferentially attracted to the other corresponding sites.

A possible clue is the phenomenon of triangular-symmetry snowflakes.
It's been suggested that these form in nature when an asymmetrical snowflake falls corner-first, with the airflow disrupting regular hexagonal crystal formation (see also Wired). But since the remaining triangular symmetry is still so strong, this hints that perhaps the strongest linkage between crystal sites is in triples, with a secondary slightly weaker triplet attraction producing the hex.

Okay, so I suppose there might be problems in attempting to use giant snowflake crystals as matter-photocopiers ... for snowflake formation, every copied pattern forms an extension of the crystal, if you use the crystal to try to copy other things, then the "irregular" matter being copied is liable to disrupt of the focusing. You might only be able to copy layers an atom or two thick (at least, to start with).

But a giant atom-perfect monocrystalline snowflake would be an awfully fun thing to play with if you had a chip-fabrication lab with goodies like force-sensing tunnelling microscopes.

And to me, that was the one thing that could have justified building the International Space Station. The ability to build a giant, heavy-duty zero-gravity snowflake, hopefully one big and chunky enough to withstand eventually being brought back to Earth immersed in liquid helium for further study (what does Bose-Einstein condensate do when it's in in contact with a hex crystal?). That had to be worth a few billion in research money, and would have given the public something pretty to look at when it came time to tell them what the money had bought. We haven't done it yet, but maybe ...

Kylie Minogue and the Gorilla Experiment

2010-03-05T21:39:00.005Z

To a large extent, we see and hear what we expect to see and hear. As newborns we're hit with a tidal wave of experiential data, a screaming torrent of raw sensory information that we have to learn how to deal with, and our brains' main coping strategy is to scrunch itself up until it's found ways of shutting out most of the din.

As infants, we initially lose neurons at an alarming rate until the remaining pathways can mimic (and to some extent synchronise with and predict) external datapatterns. We construct progressively more complex predictive mental models for how the outside world works, and increasingly live within our own models. We experience what we expect to experience, unless there's such a glaring mismatch that it can't be ignored.

It's a matter of data-reduction and enhanced reaction-times. We coast along, our experience being steered by sensory data but not dictated by it. If you're sitting on a chair, you don't suddenly jolt every few seconds and exclaim, "Chair!" – once the chair's been accepted you assume that it's still there until you're told otherwise. This internal secondary reality also compensates for the significant processing delays that happen in our brains – so that we think that we experience the world in real-time – by starting to react unconsciously to our internal models' predictions, before we're consciously aware of what we've seen. We live our lives from moment to moment in a state of continual anticipation.

Sometimes random data tickles our expectation-engine – when a black bin-bag blowing in the wind in the corner of an alley momentarily triggers an expectation of seeing a black cat, we don't just interpret the movement as possibly belonging to a cat, we actually see and remember the cat (until we look a second time and realise that it's just a refuse bag, and the rogue memory gets shredded).

These models act as perception filters and error-correction filters for what our brains allow us to register as reality. Information that's not compatible with the model (or not relevant) simply doesn't register on our consciousnesses, it gets stripped out as anomalous data and jettisoned before we have a chance to become fully aware of it.

The usual example for this is the basketball experiment, conducted by Daniel Simons and Christopher Chabris in the 1990s, but unfortunately, if I explain what the experiment is, it'll spoil it for you. If you don't already know about it, don't read anything else about it until you've watched this video and tried to count just the number of basketball passes made my the people in the white shirts. Then read the analysis.

The Gorilla Effect is now considered a classic, but what most psychologists might not realise is that in 1991, someone had already done a large-scale version of the experiment, using the UK's music broadcasting networks.

In '91, Kylie Minogue was still widely seen as a squeaky-clean pop songstress, freshly out of Neighbours, warbling heavily-processed Stock Aitken and Waterman lyrics over generic (and slightly cheesy) SAW chunka-chunka backing tracks.And that's when someone at the Minogue team decided to slip the f-word into one of the singles, three times, to see who noticed. Nobody did.

The single was called "Shocked" and charted at number 6.

" Shocked by the power, ooh-ohh, shocked by the power of love.
You got me fucked to my very foundations, shocked by the power, shocked by the power ..."

Whattt???

Uncharacteristically for SAW lyrics, “fucked to my very foundations” was actually a pretty great line for a pop song. Alliterative an' everything. I'd have been proud of it. And maybe that's why someone decided to leave it in.

Whether it was an ad-lib, like Atomic Kitten's alternative “You can lick my hole again” soundcheck version of their single, I don't know. But that's the version of "Shocked" that actually got broadcast, over and over again, on TV and on the radio. In a country that was obsessed with the F-word being used on music programmes, in which the Sex Pistols had made their careers by effing on Bill Grundy's show, and Jools Holland was suspended for accidentally let it slip on a live trailer for "The Tube" in 1987, and every Madonna single was eagerly being pored over by the UK press for possible naughty words or double-entendres that people could declare themselves outraged by, la Minogue got away with repeatedly standing up on Top of the Pops [a bit after ~7pm], and apparently singing her little heart out about how she was "fucked to my very foundations", three or four times per appearance, without anyone hearing it.

If you get hold of the more recent "Ultimate Kylie" compilation, the audio's different. They've either changed the recording or used a different version in which The Kylie is definitely singing "rrucked", with a pronounced "rr" rather than "fucked", with an "ff". But go back to contemporary broadcast recordings of the single ( thanks, YouTube! ), and yep – it's different.

The "Kylie" version of the gorilla experiment might be one of the biggest mass-media psychological experiments ever to take place, but unless you can get hold of contemporary recordings of radio and TV broadcasts, you might be forgiven for thinking that it never happened.

The Magic of Richard Feynman

2010-02-26T15:38:00.003Z

Richard Feynman (1918-1988) was one of the more colourful and charismatic characters in US physics.
He's remembered as one of the greatest physics minds of the Twentieth Century, which sometimes leaves non-physicists wondering exactly what it was (apart from Feynman diagrams) that he actually did to get that reputation. How did he end up being regarded as some sort of god amongst physicists, he never actually discovered anything that most people will have heard of?

One of Feynman's hobbies was stage magic. He was a keen practical joker, and was fascinated by the way that people are led to believe certain things, or why they end up acting in certain predictable ways. He was fascinated by fallibility, and predictable fallibility, which is one of the reasons why he was such a a great choice when they were picking people for the Rogers Commission, to investigate the reasons for the 1986 "Challenger" space shuttle disaster. Feynman understood the concept of system failure, both at the organisational and personal level, and he liked to play with people, including other physicists.

Stage magic often works through a process of misdirection. The practitioner demands with every element of their voice, facial gestures and body language that the audience may like to look over //here//, to the extent that we find it almost impossible not to look at their selected spot – perhaps an inch or so away from their extended, waggling fingertips – while with their other hand over //there//, they perform the mechanics of the actual trick.

A magician might announce before performing their stunt: "Look at this table. It's a perfectly ordinary table. It really, really is." And they bang on the table with their fist, and walk around it, and hit it with a stick, and mark an X on it with white chalk ... and you're concentrating so hard on the table to try to find why it's NOT an ordinary table, that you fail to notice the large black velvety cloth hanging above it, or the trap door behind it. The table is, in fact, completely ordinary. It's a double-bluff.

That's misdirection. You don't necessarily tell the audience something that's untrue or misleading, you give them a series of false clues, and let them work out the wrong story for themselves.

Another factor that makes stage magic effective is the way that people apply Occam's Razor. Technical stage tricks often require ludicrous amounts of preparation, absurd amounts of technical expertise or physical dexterity, and improbable investments in custom hardware. The assistant just happens to be double-jointed, or has an identical twin sister, or a false leg. At some subconscious level, the watcher's mind runs through a set of absurdly complicated and tortuous conspiracy theories that might explain what they're seeing and gives up, deciding that it's simpler to assume that the magician really can fly or make tigers disappear. The audience reasons that this isn't true (it's "only a trick"), but at a gut level they've already suspended disbelief enough to enjoy the show.

And so, to Feynman's magic trick.
One of the recurring stories about Richard Feynman goes something like this:
A physicist is working on a difficult problem. The physicist contacts Feynman. Feynman's secretary replies that Feynman is very busy, but could maybe schedule a meeting at some nebulous future date.
Several months pass. The physicist is contacted unexpectedly by the secretary to say that the secretary has just spotted that Mr Feynman now has a gap in his schedule, at quite short notice, and would the physicist still like to make use of it? The physicist eagerly agrees.

The physicist walks into Feynman's office.
"So,", says Feynman, "My secretary's just told me that you're working on some sort of interesting problem, but you'll have to forgive me, I've been really busy for the last couple of days, and haven't had the chance to look into it. Could you explain it to me? Oh, and could you start from scratch and make it simple, because, you know, this really isn't my field, and I'm not really up to speed with this subject. Start from the beginning."

The visitor is flattered and walks up to the board and starts explaining the nature of the problem. He pauses.

"So", says Feynman, "Let me see if I've got this right ..."

Feynman stares at the board and frantically marks symbols up while talking through what he's doing, until he has an equation.

"So your starting point would be something like that, yes? Okay, now tell me what you did next."

The visiting physicist is dumbfounded. What Feynman has just written on the board is the solution. And it's not just the solution, it's the solution to a more general version of the problem than the one that the visitor has been struggling with for months, or years. And Feynman's just done it in about three minutes flat.

The physicist leaves, ego totally destroyed, knowing that RF is in a totally different league to lesser mortals like himself.

Now, the "reveal".

If you were a suspicious stage-magician type, what you might suspect happened would be something like this: Physicist contacts RF's secretary, mentioning something about the problem. Secretary tells Feynman. RF researches the problem and all the relevant papers on the subject, and finds out how far the physicist has gotten. The secretary sends a stalling letter. Feynman adds the problem to his stack of other outstanding problems, playing them off each other, trying to cross-fertilise the different issues and bounce ideas between them, considering it a break from the problems he's actually trying to work on for himself. Finally, he works out the solution, and at this point, his secretary sends out the letter saying that RF now has an unexpected gap in his schedule.

Physicist arrives, RF plays dumb and asks them to outline the problem, RF "solves" it in three minutes flat, apparently using only the tools that the visitor has just provided.

Of course, this scenario still required RF to have been a damned good theoretical physicist. It also required RF to have had a wicked sense of humour, and to have done an awful lot of tough background work each time he pulled his stunt, just to create a few brief minutes of surprise for his "audience".

But that's exactly what stage magicians do.

"Relativity in Curved Spacetime", PDF eBook

2010-02-24T19:38:00.002Z

I've just provisionally put Relativity in Curved Spacetime online as an eBook, to see what happens. It's the full fixed-layout PDF file for the book, with an added "bookmark pane" PDF index and some annotations. If you're curious about the page layouts or you'd like a single-sheet PDF listing of the book's contents, click on the links.

I've initially priced the thing at USD $4-99, which comes out as about three quid in British Pounds. That's about a third of what Apple are going to be charging for ebooks.

If you want a nicely-bound hardcopy, and don't fancy printing off nearly 400 sides of paper, you can still buy the paperback and hardback. Otherwise, the PDF version's on Payloadz.com .

My Website Sucks

2010-01-29T12:00:00.004Z

I know why it sucks ... it's not because I don't know how to write a proper website ... I do ... it's because it's a personal site, and I kinda tinker with it and add things from time to time, and experiment ... and because I've been using HTML for too long.

I was designing the site for someone else, I'd be less indulgent and more brutal with it. I'd insist that the owners had a clear brief of exactly what they wanted the site to do, and how to judge success. It'd be focused and lean and mean. I'd decide a visual theme, and a hierarchy, and apply it strictly. But when it's your own site, the tendency is to drift and add things and sections and use the pages as a sandbox for playing with different techniques until you end up with an indulgent hodge-podge of themes and style ideas that don't really gel.

If it was someone else's site, I'd tell them to delete the whole thing and start again. Don't just fiddle with the layout, start with a blank sheet of paper and a pen, doodle a brand new layout based on CSS, set up some default templates and rebuild the site from the ground up.

When you drift and add bits and pieces haphazardly, you end up slipping into old habits. I started writing webpages before we even had html tables. My first site (Erk's Relativity Pages) was a 300-page monster written entirely in Windows Notepad, and back then, a site designer had to learn all sorts of odd layout tricks (like using invisible GIFs as spacers) to produce efficient layouts. When tables were implemented by Netscape (and then by MS), we redesigned our pages to suit, with nice orderly auto-resizing panels – they were a pain to begin with, but the quirks and incompatibilities smoothed out with time, and we ended up using them everywhere. Tables became the answer to everything, from navigation panels to equation-setting. Then there was a craze for breaking a page up into sections and writing those sections as separate webpages embedded in frames. I managed to avoid that one (since I could see the long-term search-engine problems), but for a few years, using frames everywhere was supposed to be the mark of a "pro" designer. And then a couple of years later, the importance of search-engine optimisation became obvious, the fashion swung into reverse, and any frame-based sites began to look terribly dated.

Back in the 1980's and 1990's, the way to produce a flashy (but legible) site was to use a dark background with light text. The old CRT monitors tended to be strongly curved, with a display area that didn't extend quite to the edges, so a dark background made your page appear larger. With low-res CRT displays, "inverted" light-on-dark text was often easier to read, because the the outward blurring of light from the letters produced a sort of natural antialiasing effect. With dark text on pale backgrounds, the surrounding light tended to bleed over the characters, making them more difficult to read. Adding background patterning made the pages look more exciting, made the screen defects less distracting, and helped the user forget that they were staring at a fairly nasty little computer screen.

In 2010, things have flipped. Legibility isn't a problem on modern LCD displays, and because the screens are now flat, stark white rectangles actually look good. The monitor glass is thinner, so "snow blindness" due to light-scattering from large bright areas isn't so much of a problem, and you no longer need to add a faint background texture to pale or white backgrounds to disguise the "bitty" red, green and blue phosphor dots of a low-res CRT screen.

Nowadays, we practically squander space. On large screens, we use column layouts that waste most of the screen display, so that the central vertical column corresponds to what the user sees if they try to view the site from an iPhone. The web in the 1980s was content-starved, and you'd try to impress visitors with how much you had on your site and how much you could cram onto a small screen. In 2010 the visitor is spoilt for choice, so now designers try to keep things minimal and direct their visitors as quickly as possible to the information they actually want, otherwise they'll just click back to Google and try somewhere else.

After tables and frames, we now have Cascading Style Sheets. CSS is genuinely cool, and I really ought to rip up the existing pages and redo all their elderly table-based layout completely using CSS. Trouble is, it'll require a certain amount of work, and the immediate result will be that certain existing things (like same-height panels) won't work so well. There are bodges and workarounds, CSS isn't quite perfect yet.

The site's "look" also badly needs an overhaul. It was originally going to just be a few pages supporting the book, with a navy blue block across the top and down the left side referring to the book cover art (front and spine). On the subsequent pages, that morphed into a "program window" theme, with a title bar and an icon in the top left corner. I never quite worked out what to do with the spine. It's now an inconsistent mess, with pages on almost unrelated subjects like fractals, and should really be torn down and rebuilt.

Relativity theory is in a similar mess. A number of themes have come and gone, and left their mark on the subject. There are artefacts and traditions in the way that theory is presented that don't really make sense in the new context, and older methods that aren't compatible with newer principles. We teach special relativity as having destroyed aether theory, but we still teach SR using the length-contraction idea, which was an old aether theory concept borrowed from Lorentzian electrodynamics.

In theoretical physics, we probably have a feeling deep down that we know that we really ought to be tearing up the current system and starting again. But it'd require a lot of work without an immediate payoff, and some of the things we currently do would stop working for a while as the new system found its feet. The current system is bodgy and patched and held together with string and duct tape, but we know how to use it, and over time the bugs and fudges have started to feel like old friends. We invested a lot of time in special relativity (like website designers spent a lot of time learning the quirks of HTML tables), and now that we know that system, we tend to use it everywhere. With special relativity, we've gone further and actually redefined some key parts of relativity theory in such a way as to make SR inevitable and unavoidable, and this lock-in frees us from having to make awkward upgrade decisions.

So while it may seem that I'm sometimes a bit harsh on the theoretical physics community for being welded to obsolete and archaic systems that don't really make sense in the C21st, I do actually sympathise and empathise with their problem. They ought to rip up their SR-based structure and redesign, just as I ought to rip up my table-based webpage layout and redesign. But there's a difference between knowing that you ought to do something, and actually rolling up your sleeves and starting work, especially when there's no external deadline forcing your hand, and you always seem to have other more pressing things demanding your time.

So to help the theoretical physics community, here's a time-point. The book came out in late 2007, and sketches out the principles and the rough shape of the suggested next-generation replacement for our current general theory of relativity. This is early 2010, and the book's now been out for two years. That book is the roadmap to what comes next. So perhaps we can have a concerted start on plotting out at least a rough preliminary schedule for a replacement to general relativity, some time in 2010?

Meanwhile, I'll try to think of a way of cleaning up the website.

Einstein's Cosmological Constant

2010-01-22T22:14:00.001Z

Back in 1916, Einstein was still working to the assumption that the universe should be neat and tidy, and since he was now using a more mathematical approach, this meant "infinite and unchanging".
If you were solving the equations of general relativity, and getting solutions in which the universe appeared to be unstable, then you could throw those away. Chaos was bad. Order was good. Stability was good. Static solutions were better than dynamic ones.

Since it seems that gravitational mass is always positive, gravitational effects are cumulative, and over a large enough region, the combined background curvature should be enough to curve space right back on itself. The combined attraction also ought to be trying to make the universe contract, so we've appreciated for a while that unless there was some other effect in play, the universe should either be expanding and slowing, or collapsing in on itself (see: Erasmus Darwin, 1791).

Einstein wanted his universe to be pretty much flat at very large scales, so he got rid of the effects caused by cumulative curvature by adding an additional squiggle to the equations: an invented long-range repulsive effect whose purpose was to counteract the cumulative long-range effects of gravitation, allowing a tidy, constant, unchanging, static universe. If the rest of the equation generated long-range curvature effects and evolution over time, the upper-case Greek letter Lambda (Λ) represented the necessary compensating effect that might exactly cancel these effects.

Einstein referred to this as the Cosmological Constant.

Unfortunately, Einstein had made his model too tidy. A few years later, Edwin Hubble successfully measured a distance-dependent trend in the spectral shifts of light from a range of galaxies (Hubble shift), and we realised that the complicating large-scale effects that Einstein thought he'd eliminated with his Cosmological Constant seemed to be physically real. After taking some time to think the matter over, Einstein agreed that a Riemann-type solution (without Lambda) gave a cleaner and more natural implementation of General Relativity. He later described his early decision to invent the Constant to force large-scale flatness onto GR as "The biggest blunder of my career".

End of story.

However, the subject seemed to kick off again in the 1990's when a lot of headlines started appearing in in the popular science press (and in scientific papers) to do with the idea of dark energy, and the idea that the universe seemed to be expanding faster than GR1915 predicted – these articles usually declared that "Einstein's Cosmological Constant" was back, and had excited-sounding researchers competing to see who could give the best quote about Einstein having been "right all along".

This wasn't really true: Einstein's Cosmological Constant had been a mathematically-derived thing that only had one allowable value, and whose justification was to set the strengths of a range of effects in the model (large-scale curvature, distance-dependent redshifts, change in size over time) to zero. It had been there for purely logical reasons, in the context of a static universe, because a static universe seemed to need it. It existed to explain an assumed physical equilibrium that turned out not to exist, in a universe that wasn't ours. It was derived from bad assumptions, but at least it was derived.

The modern counterpart was almost the opposite. The antigravitational "dark energy" cosmological constant applied to an expanding universe that seemed to be expanding too fast for GR1915, and the effect initially had no fundamental logical, mathematical, geometrical or theoretical basis. It was, essentially, a parameter describing the extent to which the result of our GR predictions "missed" the actual data.
More recently, some researchers have tried to put the dark energy idea onto a more "theoretical" footing by arguing that perhaps the constant might not have a fixed arbitrary value, but might be a measure of the universe's expansion. That'd make the "modern" CC less fudgey, but it'd also mean that, as well as the thing not being Einstein's, it wouldn't be a constant, either.

So why did we initially get all those news stories announcing things like: "Eighty years later, it turns out that Einstein may have been right ... So he was smarter than he gave himself credit for." [*] ?

Putting it brutally, it was about PR. Attaching Einstein's name gave a false sense of historical provenance and a false sense of respectability. It let researchers use Einstein's name as a shield to deflect awkward questions about the apparent arbitrariness of their new expansion effect, and it turned a fairly boring and slightly negative story about GR failing to agree with the evidence into a snappy human-interest story about the throes of the scientific process coming out right in the end, and Einstein being right, and GR being right.

The "Einstein's Cosmological Constant returns: Einstein was right after all!" stories generated a lot of news headlines, and let researchers give interviews to magazines and appear on the telly and improve their departments' media profiles. Suddenly there were a lot of editors and journalists wanting quotes on the cosmological constant, because they wanted to print the same reader-grabbing "Einsteiney" headline, but didn't want to put their name on the claim, as reporters, because it was dodgy. So they rang round the universities and found a bunch of cosmologists happy to give the right quote if it meant getting their name in a magazine or getting onto the telly.

The story was junk. It was researchers collectively gaming the news media, and manufacturing and repeating a story that they knew would work, in order to get more media exposure. And unfortunately, that's the sort of behaviour that makes the general public more inclined to distrust scientists.

Clever, Bright, and Smart

2010-01-15T10:22:00.001Z

There's no single scale that adequately describes someone's abilities. People can excel at some types of task and be hopeless at others, and we have a range of different words for different types of aptitude.

Three of the most popular ones are clever, bright and smart.

Cleverness is about tool manipulation. It's about having a library of information and methods at your disposal that you can call upon to attack a problem. It's about the toolset. "Clever" researchers tend to be great at solving well-known types of problems, or well-defined problems that are attackable with existing approaches. It's a matter of going through the toolset until you find something that works. Clever people tend to be good at technical subjects that involve absorbing a lot of jargon and detail. They're not always so good at solving or understanding problems that aren't well defined, or seeing the bigger picture, or starting with a blank page.

Brightness is about being able to appreciate larger patterns and relationships that don't necessarily conform to an existing approach or definition. Bright people tend not to be so dependent on clearly-defined goals or methodology, and can take a more "free-form" approach to work, where the project parameters and characteristics emerge as the project progresses.
A computer programmer needs to be clever, but a software designer needs to be bright.

"Clever vs. Bright" is like comparing soldier ants with butterflies. The soldier ant, working with other soldier ants, manages to overcome a lot of problems even if each individual ant doesn't really know where they fit into the larger scheme of things. The butterfly arguably has the better world-view, but can't always do very much with it.

Smartness is about being able to understand and exploit opportunities to gain advantage and achieve goals. It's possible to be clever and bright without being smart. Having "smarts" means that you learn from experience and think ahead strategically, to plan how the workings of a system can allow you to achieve your desired outcome.

Smart people are often also bright and clever, but they're also smart enough to realise that their success doesn't depend on cultivating those other skills to the same extent, because once they've become moderately successful, they can "hire in" clever and bright people to do that part of the work, and delegate. Successful entrepreneurs tend to be smart, and bright, and clever, but their focus is on being smart.

Military R+D usually wants researchers who are extremely clever, but not necessarily too bright or smart. A "bright" employee might query what their work is to be used for, notice how their research fits together with others to produce a device that they aren't supposed to know about, or query the legality or ethics or consequences of the project they're involved in. A smart researcher might realise that the market value of their work is more than their current employer is paying, leave to take a better job when they realise that the project is in trouble, or try to wrest control of the project from the existing managers.

Now, this is where it gets complicated:

People who describe themselves as smart (outside a limited peer group) usually aren't.
Smart people tend not to publicly identify themselves as as smart, because it's usually not a smart thing to do. Clever people sometimes describe themselves as smart, because nobody's actually told them what the words mean, and they're not bright enough to work it out for themselves. They follow the lead of the other clever people in their group that they've heard describing themselves as smart. The bright people also don't normally describe themselves as smart, because they only hear the word being used self-referentially by people with poor social skills who are "clever-only", and they decide that they don't want to be lumped in with them.

So if you're studying monkeys in a zoo that are picking grubs out of a log that have been put there by the zookeeper, the clever monkey will become adept at using a stick to extract the grubs, the bright monkey will watch the zookeeper and only go grub-hunting when the log's just been refilled, and the smart monkey will congratulate the other two on their cleverness, assume a management position and a share of the grubs, and then patent the stick.

Different skills.

Relativity Four Point Zero

2010-01-07T16:30:00.002Z

Okay, here starts a new decade. I've started a simple site sketching out the basic principles of the suggested revised general theory:

http://www.fourpointzero.org

I figured that if the work of Galileo & Newton counts as "Relativity v1.0", special relativity changed some key equations and counts as v2.0, general relativity altered and added some fundamental principles and did away with SR's concept of global lightspeed constancy, and therefore counts as v3.0, then since this isn't compatible with the current textbook definitions of GR (because it eliminates the "compulsory" SR component), it counts as another "discontinuous" iteration and earns a further major version number, 4.0 .

You can't get to 4.0 without breaking a few eggs. That's what makes it 4.0 .

I was thinking of giving the new site ~twenty-six sections listed in alphabetical order, one page per letter, but I think I might just stop at five or six (the current pages A-E seem to work quite well as a logical progression). I'm trying not to fall into my usual trap of writing realms of material that most people won't want to read, and keeping things pretty minimalist, so there's a lot of the more juicy stuff left out. I think this sort of "skeleton" overview probably serves a useful purpose, so don't expect a lot of updates to the "4.0 org" site, unless other people get involved.

New Year's Eve

2009-12-31T20:44:00.001Z

Okay, that's it. First decade of the new century over, and we've got almost nothing good to show for it, physics-wise.
That's bad. We only get ten of these per century. One down, only another nine to go before 2100. If we're burning through resources at the current rate, we can't afford to waste decades like this if we want to actually achieve something significant this century before we get hit by a resources crash.

So a suggested schedule. Let's officially notice the idea of a no-floor implementation of GR by at least late 2010, and see if we can get rid of dark matter and dark energy. Let's have the quantum gravity guys working on acoustic metrics as a low-velocity approximation have the guts to come out and actually suggest that this might be the basis of a real theory, and not just a toy model. Let's stop issuing press releases claiming that the current version of general relativity is the wonderfullest theory and has never ever failed us, let's acknowledge the problems and let's sit down and write a proper general theory from scratch, stealing that "acoustic metric" work.

Instead of setting a schedule that puts the next theoretical breakthroughs maybe eighty or a hundred years from now because we aren't clever enough to understand string theory, let's get off our arses and do the things that we do know how to do. Kick off with the no-floor approach, and when we're energised by the success of that, converge the acoustic metric work with a GR rewrite .. and suddenly the next generation of theory only looks about five years away. If we're very lucky, two and a half. If we can't get enough people onboard fast enough, maybe eight to ten.

Unless we take that first step of exploring the idea that change might be possible and might be a good thing, we won't get anywhere except by dumb luck and/or massive public spending on hardware. If we're not careful, and we don't change the way we do things, next thing we know it'll be 2020 and we still won't have achieved anything.

So let's write off the 00's as a big double-zero. Let's pretend that the Bush years and Iraq and the financial crash never happened. We don't need multi-billion-dollar hardware for this, we only need to be able to think, and to be a bit more adventurous than we've been for the last few decades. Lets redo general relativity properly and get a theory that we can be proud of without having to spin results, one that actually predicts new effects in advance rather than retrospectively, and has the potential to lead us into genuinely new physics territory.

Tomorrow is 2010. Let's start again.

Differential Expansion, Dark Matter and Energy, and Voids

2009-12-30T22:10:00.010Z

Normally with a field theory, you have some idea where to start. You start by defining the shape and other properties of your "landscape" space, and then you add your field to that context, and watch what it does when you play with it.
But in a general theory of relativity (which is forced by Mach's Principle to also be a relativistic theory of gravity), the gravitational field is space. The field doesn't sit inside a background metric, it is the background metric.
So with this sort of model, we've got no obvious starting point – no obvious starting geometry, and not even an obvious starting topology, unless we start cheating and putting in some critical parameters by hand, according to what we believe to be the correct values.

We make an exasperated noise and throw in a quick idealisation. We say that we're going to suppose that matter is pretty smoothly and evenly distributed through the universe (which sounds kinda reasonable), and then we use this assumption of a homogeneous distribution to argue that there must therefore be a fairly constant background field. That then gives us a convenient smooth, regular background shape that we can use as a backdrop, before we start adding features like individual stars, and galaxies.

That background level gives us our assumed gravitational floor.

We know that this idea isn't really true, but it's convenient. Wheeler and others tried exploring different approaches that might allow us to do general relativity without these sorts of starting simplifications (e.g. the pregeometry idea), but while a "pregeometrical" approach let us play with deeper arguments that didn't rely on any particular assumed geometrical reduction, getting from first principles to new, rigorous predictions was difficult.
So while general relativity in theory has no prior geometry and is a completely free-standing system, in practice we tend to implicitly assume a default initial dimensionality and a default baseline background reference rate of timeflow, before we start populating our test regions with objects. We allow things to age more slowly than the baseline rate when they're in a more intense gravitational field, but we assume that the things can't be persuaded to age more quickly than the assumed background rate (and that signals can't travel faster than the associated background speed of light) without introducing "naughty" hypothetical negative gravitational fields (ref: Positive Energy Theorem).
This is one of the reasons why we've made almost no progress in warpdrive theory over half a century – our theorems are based on the implicit assumption of a "flat floor", and this makes any meaningful attempt to look at the problem of metric engineering almost impossible.

Now to be fair, GR textbooks are often quite open about the fact that a homogeneous background is a bit of a kludge. It's a pragmatic step – if you're going to calculate, you usually need somewhere to start, and assuming a homogeneous background (without defining exactly what degree of clumpiness counts as "homogeneous") is a handy place to start.

But when we make an arbitrary assumption in mathematical physics, we're supposed to go back at some point and sanity-check how that decision might have affected the outcome. We're meant to check the dependencies between our initial simplifying assumptions and the effects that we predicted from our model, to see if there's any linkage.
So ... what happens if we throw away our "gravitational floor" comfort-blanket and allow the universe to be a wild and crazy place with no floor? What happens if we try to "do" GR without a safety net? It's a vertigo-inducing concept, and a few "crazy" things happen:

Result 1: Different regional expansion rates, and lobing

Without the assumption of a "floor", there's no single globally-fixed expansion rate for the universe. Different regions with different "perimeter" properties can expand at different rates. If one region starts out being fractionally less densely populated than another, its rate of entropic timeflow will be fractionally greater, the expansion rate of the region (which links in to rate of change of entropy) will be fractionally faster, and the tiny initial difference gets exaggerated. It's a positive-feedback inflation effect. The faster-expanding region gets more rarefied, its massenergy-density drops, the background web of light-signals increasingly deflects around the region rather than going through it, massenergy gets expelled from the region's perimeter, and even light loses energy while trying to enter, as it fights "uphill" against the gradient and gets redshifted by the accelerated local expansion. The accelerated expansion pushes thermodynamics further in the direction of exothermic rather than endothermic reactions, and time runs faster. Faster timeflow gives faster expansion, and faster expansion gives faster timeflow.

The process is like watching the weak spot on an over-inflated bicycle inner tube – once the trend has started, the initial near-equilibrium collapses, and the less-dense region balloons out to form a lobe. Once a lobe has matured into something sufficiently larger than its connection region, it starts to look to any remaining inhabitants like its own little hyperspherical universe. Any remaining stars caught in a lobe could appear to us to be significantly older than the nominal age of the universe as seen from "here and now", because more time has elapsed in the more rarefied lobed region. The age of the universe, measured in 4-coordinates as a distance between the 3D "now-surface" and the nominal location of the big bang (the radial cosmological time coordinate, referred to as "a" in MTW's "Gravitation",§17.9), is greater at their position than it is at ours.

With a "no-floor" implementation of general relativity, the universe's shape isn't a nice sphere with surface crinkles, like an orange, it's a multiply-lobed shape rather more like a raspberry, with most of the matter nestling in the deep creases between adjacent lobes (book, §17.11). If there was no floor, we'd expect galaxies to align in three dimensions as a network of sheets that form the boundary walls that lie between the faster-expanding voids.

And if we look at our painstakingly-plotted maps of galaxy distributions, that's pretty much what seems to be happening.

Result 2: Galactic rotation curves

If the average background field intensity drops away when we leave a galaxy, to less than the calculated "floor" level, then the region of space between galaxies is, in a sense, more "fluid". These regions end up with greater signal-transmission speeds and weaker connectivity than we'd expect by assuming a simple "floor". The inertial coupling between galaxies and their outside environments becomes weaker, and the influence of a galaxy's own matter on its other parts becomes proportionally stronger. It's difficult to get outside our own galaxy to do comparative tests, but we can watch what happens around the edges of other rotating galaxies where the transition should be starting to happen, and we can see what appears to be the effect in action.

In standard Newtonian physics (and "flat-floor" GR), this doesn't happen. A rotating galaxy obeys conventional orbital mechanics, and stars at the outer rim have to circle more slowly than those further in if they're not going to be thrown right out of the galaxy. So, if you have a rotating galaxy with persistent "arm" structures, the outer end of the arm needs to be rotating more slowly, which means that the arm's rim trails behind more and more over time. This "lagging behind" effect stretches local clumps into elongated arms, and then twists those arms into a spiral formation.
When we compare our photographs of spiral-arm galaxies with what the theory predicts, we find that ... they have the wrong spiral. The outer edges aren't wound up as much as "flat-floor" theory predicts, and the outer ends of the arms, although they're definitely lagged, seem to be circling faster than ought to be possible.

So something seemed to be wrong (or missing) with "flat-floor" theory. We could try to force the theory to agree with the galaxy photographs by tinkering with the inverse square law for gravity (which is a little difficult, but there have been suggestions based on variable dimensionality and string theory, or MOND), or we could fiddle with the equations of motion, or we could try to find some way to make gravity weaker outside a galaxy, or stronger inside.

The current "popular" approach is to assume that current GR and the "background floor" approach are both correct, and to conclude that there therefore has to be something else helping a galaxy's parts to cling together – by piling on extra local gravitation, we might be able to "splint" the arms to give them enough additional internal cohesiveness to stay together.

Trouble is, this approach would require so much extra gravity that we end up having to invent a whole new substance – dark matter – to go with it.
We have no idea what this invented "dark matter"might be, or why it might be there, or what useful theoretical function it might perform, other than making our current calculations come out right. It has no theoretical basis or purpose other than to force the current GR calculations to make a better fit to the photographs. Its only real properties are that its distribution shadows that of "normal" matter, it has gravity, and ... we can't see it or measure it independently.

So it'd seem that the whole point of the "dark matter" idea is just to recreate the same results that we'd have gotten anyway by "losing the floor".

Result 3: Enhanced overall expansion

Because the voids are now expanding faster than the intervening regions, the overall expansion rate of the universe is greater, and ... as seen from within the galactic regions ... the expansion seems faster than we could explain if we extrapolated a galaxy-dweller's sense of local floor out to the vast voids between galaxies. To someone inside a galaxy, applying the "homogeneous universe" idealisation too literally, this overall expansion can't be explained unless there's some additional long-range, negatively-gravitating field pushing everything apart.

So again, the current "popular" approach is to invent another new thing to explain the disagreement between our current "flat-floor" calculations and actual observations. This one, we call "Dark Energy", and again, it seem to be another back-door way to recreating the results we'd get by losing the assumed gravitational background floor.

So here's the funny thing. We know that the assumption of a "homogenous" universe is iffy. Matter is not evenly spread throughout the universe as a smooth mist of individual atoms. It's clumped into stars and planets, which are clumped into star systems, which are clumped into galaxies. Galaxies are ordered into larger void-surrounding structures. There's clumpiness and gappiness everywhere. It all looks a bit fractal.

It might seem obvious that, having done the "smooth universe" calculations, we'd then go back and factor in the missing effect of clumpiness, and arrive at the above three (checkable) modifying effects, (1) lobing (showing up as "void" regions in the distribution of galaxies), (2) increased cohesion for rotating galaxies, and (3) a greater overall expansion rate. It also seems natural that having done that exercise and having made those tentative conditional predictions, that when all three effects were discovered for real, the GR community would be in a happy mood.

But we didn't get around to doing it. All three effects took us by surprise, and then we ended up scrabbling around for "bolt-on" solutions (dark matter and dark energy) to force the existing, potentially flawed approach to agree with the new observational evidence.

The good news is that the "dark matter"/"dark energy" issue is probably fixable by changing our approach to general relativity, without the sort of major bottom-up reengineering work needed to fix some of the other problems. At least with the "floor" issue, the "homegeneity" assumption is already recognised as a potential problem in GR, and not everyone's happy about our recent enthusiasm for inventing new features to fix short-term problems. We might already have the expertise and the willpower to solve this one, comparatively quickly.

Getting it fixed next year would be nice.

Black Holes are Rude (in French)

2009-12-29T15:37:00.005Z

English-language physics textbooks (before the mid-1970's) tend to give the impression that everyone had agreed that black holes couldn't radiate. It was supposed to be mathematically proved. Done deal.

But there's a slight geographical cultural bias. Not all countries' research communities adopted the idea of the perfectly-non-radiating black hole with the same enthusiasm. The French theoretical physics community in particular seemed not to like black holes very much at all.

And this was probably at least partly because in French, the term for "black hole" – "Trous Noir" – is slang for "anus".

Now, imagine what that must do to a serious talk on black hole theory delivered in French. To have to give a 45-minute lecture on how things that disappear into a black hole can't be retrieved, including topics like the proof that that "black holes have no hair", and its relationship to the hairy ball theorem. How the heck do you teach this subject without your students snickering?

So the French approach circa 1960 seemed to be to hunker down and wait for the new fashion to blow itself out (err...), after which normality could be restored. And it happened. The Wheeler black hole got assassinated by Stephen Hawking in the 1970's with his presentation on Hawking radiation.

But the English-speaking physics community kept using the term "black hole", even though technically, horizon-bounded objects under QM were now known NOT to be black holes in the Wheeler sense of the word. They weren't black, or holes. Maybe we kept the phrase because we didn't want to admit we'd screwed up, maybe we kept it because of the historical habit of physicists to completely ignore the literal meanings of words when it suits them, and maybe ... we simply liked upsetting the French.

Thanks to Hawking radiation, if you teach black hole theory in French you now have the unenviable job of addressing a room full of students on the subject of black hole emissions, and hoping that nobody thinks its funny to start making quiet comedic fart noises at comically appropriate moments.

Perhaps the smart thing to do is to take this opportunity to come up with a whole new name for a "QM black hole". Call it something like an "Etoile Hawking" (a "Hawking Star"). It's two extra syllables, but it solves the problem.

Nuclear Fusion and the Road to Hell

2009-12-27T13:14:00.000Z

The running joke in the nuclear fusion community is that commercial fusion is thirty or forty years away ... and always will be. The "forty years" rule isn't based on any technical issues, it's based on politics. If you say "a hundred years", then no politician is going to fund you. They want to see results in their lifetime. If you say "twenty years", then people expect you to already have prototype plans drawn up. If you say "thirty", then you get a ten-year grace period, and THEN people expect to start seeing blueprints. "Fifty years" doesn't have enough urgency ... the economy might be different in five decades. And it sounds like a made-up number. But "forty years" conveys a sense that we need to get started NOW. It dangles the carrot just far away for a politician to hope that they're doing the right thing, it won't come to fruition on their career, but they'll see the results in their lifetime.
Which, of course doesn't happen, but by then we have a new crop of politicians that we can give the "forty" schtick to.

So "forty years" is an ongoing collective collective sales pitch by the fusion community to get money for their big conventional tokamak projects from their respective governments. The guys involved sincerely believe that fusion power is the future of the human species, and that the system WILL work one day, and that it HAS to be funded for us to progress. They seem to be using the "tobacco industry" principle – that if you testify that you believe that something is correct, then as long as you can force yourself to believe it at that particular moment, it's difficult for anyone to call you to account for lying. The unattributed quote in the New Scientist editorial after the funding round in 2006, when someone was asked whether they honestly believed the estimates being given for timescales by the fusion community was: "We have to, or the politicians wouldn't give us the money".

The tokamak guys probably reckon that this doesn't technically count as scientific fraud, because they're only misleading politicians, and not other scientists. It's just gaming the system, nobody's getting hurt, right? And anything that gets additional money for science is good, yes?

And that's where the rot sets in. Because "big tokamak" research is so damned expensive, it means that once you've started, you're committed. To spend years of your life on a project and billions of dollars and THEN have it cancelled would be worse than not having started. So you find yourself in a "double or quits" situation, where you have to keep the lie going, and find yourself having to do other bad things to protect the structure you've built.
It basically has you by the balls.

There's quite a few other potential ways to do nuclear fusion, and although lots of them look flakier than the idea of using a nice big solid tokamak, they're also a hell of a lot cheaper to research. So you'd think that the sensible course of action woudl be to put a little money into those alternatives as a side bet, in case the "big toroidal tokamak" idea doesn't pan out, or in case one of those cheap ideas suddenly starts working.

But if you're a "BTT" guy, the side-projects can't be allowed conventional funding or credibility. That's why, when the Cold Fusion story hit, those involved were immediately being written off as con-artists or delusional incompetents by people who knew nothing about palladium-hydrogen geometries – the threat wasn't that the CF guys might successfully con a measly five or ten million in funding from the government, it was that governments might start considering a "mixed basket" approach to fusion research, and if you have five cheap fusion research programmes and one very expensive one, the temptation is to drop the one that costs so much more when funding gets tight. So once you're chasing Big Fusion, it becomes imperative for the success of your mission that there are no other options for a funding committee to look at. Ideally, you want all that other research stopped.

This is the road to damnation. You wake up one day and find that you're no longer the heroic researcher battling the corrupt political system to save a project from cancellation – you're now part of the corrupt political system suppressing other people's fusion research. And it's not the politicians at fault - it's you. Somewhere along the line, you morphed from Anakin Skywalker to Darth Vader, and became one of the Bad Guys.
The only way to justify your actions – and save your scientific soul – is to come up with the goods and save humanity. But this means that you can no longer consider even the possibility that the BTT approach might not be the right way to go, because that'd mean that you'd lose your one shot at salvation.
So what happens if one day you realise that there's something your colleagues overlooked that seems to make the entire project unworkable? Do you go public and risk being responsible for shutting down everything you and your colleagues have devoted your careers to, or do you keep quiet in the hope that you're wrong? If you decided to go public, how far might some of your colleagues go to stop you? Things get nasty.
This is why we have stories about people selling their souls to the devil, and finding out, too late, that they've killed the very thing they wanted to save. They're cautionary tales about human nature and temptation that are supposed to help us to do the right thing in these situations.

The fusion guys have been getting away with it so far, because we all hope that they'll actually be able to come up with the goods. But the public is getting increasingly sceptical about how far they can trust scientists, and the fusion community has to take it's share of the blame for that.

Let's suppose that the global warming argument is correct, and that in 15-20 years time the Earth's weather systems shift in a way that's not terribly convenient for our current city locations, or that we end up bankrupting ourselves in a last-ditch attempt to cut down on carbon emissions. Who're we going to blame?
The climate change people will blame the politicians for not listening to the scientists and planning ahead ... but the politicians will be able to say that they did take the best available scientific advice, and did plan ahead. And spent the money on the big fusion programmes. They didn't properly fund development of next-generation fission reactors that'd be more palatable to the general public than the current monsters, because they were told that fission reactors would be obsolete by now. They didn't do more to fund clean coal, because our power stations weren't still supposed to be burning fossil fuels past the end of the Twentieth Century. They didn't do more to fund wave and wind and solar power research, or try to make society more energy-efficient, because by now we were supposed to be enjoying practially unlimited energy "too cheap to meter". The concentration of strategic oil reserves in Middle-Eastern countries like Iraq and Iran wouldn't matter so much by now, certainly not so much that we'd be prepared to go to war over them. The forty-year estimates back in the 1960's meant that we simply didn't need to prioritise these things. The fusion guys had assured us that we didn't need to, all we had to do was write them a cheque.

Like most people, I hope that the guys can show us that we're wrong, and really can get this to work on a reasonable timescale.

Otherwise ... Welcome to Hell.

Fibonacci Fractals

2009-12-21T23:12:00.001Z

This fractal's based on the Fibonacci Rose.

The original Rose has two identical interlocking spiral arms. If we delete one of them, we're left with a simple spiral chain of triangles. Each triangle has three sides – one side connects the triangle to its larger parent, one connects it to its single smaller child triangle, and the third side is unused.

Adding child triangles to two sides gives us the fractal – a characteristic cauliflower-shaped branching structure whose adjacent bunches have corners that just touch.

At larger scales, this looks just like one of the family of fractal shapes that we get by using the Golden Ratio to calculate triangle sizes, that let us zoom infinitely far in or out and always get the same shape.

With the "Fibonacci" versions we can zoom out infinitely far, but as we zoom in, there's a range where the proportions start to shift perceptibly away from the Golden Ratio, and then, suddenly, the branching sequence hits a dead end, and stops.

The Tetrahedral Triple-Helix

2009-12-20T04:57:00.003Z

Mathematicians playing with geometrical solids tend to concentrate on the finite ones. Those provide a nice satisfying sense of closure, and they're cheaper to build with straws and pipecleaners than the infinite ones.

This is an interesting shape that doesn't fall into that category. It's a simple rigid stack of tetrahedra that generates a "column" with a triple-helix. The odd thing is, you'd expect an architect somewhere to have already used this on a structure somewhere ... but I don't recall ever seeing it.
Maybe I missed it.

The sequence rotates through [~]120 degrees and [nearly] maps onto itself every nine tetrahedra (that is, the tenth [nearly] aligns with the first). If you want to follow one of the spiral arms through a complete [~]360-degree revolution, that takes 9×3=27 tetrahedra, (#28 corresponds to #1) .

Oh, and it has a hole running right down the middle.

I'll try to upload some more images in another post.

"Snowflake" Fractal

2009-12-19T22:18:00.000Z

If you want something that looks more like a snowflake than the previous hexagonal carpet, you could always use the "Koch Snowflake" fractal, which is gotten by repeatedly adding triangles to the sides of other triangles.
But every single general text on fractals seems to include the Koch. I mean, don't get me wrong, it's a fairly pleasant shape, but after the nth "fractals" text slavishly copying out exactly the same fractal set-pieces, you start to think ... guys, could we have a little bit of variation pleeeeaaase?

So here's a different snowflake. This one's built from hexagons. Each hexagonal corner forms a nucleation site that attracts a cluster of three smaller hexagons, and their free corners in turn attract clusters of three smaller ... you get the idea. The sample image has been drawn with about six thousand hexagons.

The resulting "snowflake" outline is really very similar to the Koch, but the internal structure's a bit more spicy. A suitable design for Christmas cards for mathematicians, perhaps.

Hex "Fractal Carpet"

2009-12-18T14:22:00.003Z

It snowed here today! Wheee!

In honor of the White Hexagonal Fluffy Stuff, here's a nice fractal carpet made of hexagons that illustrates how an infinite number of copies of a shape can converge on a larger fixed-area version of the same shape.

This one's generated from about five and a half thousand hexagons, but obviously, you can keep going infinitely far.

The construction rule's simple. You start with one hexagon (with sides of length "one"), and then add half-size hexagons to any free corners. Then repeat, infinitely (with sides of length "one half", "one quarter", and so on).

What the process converges on is a larger completely-filled hexagon with sides of length "three", so the final area is exactly nine times the original.

If you wanted to get even more recursive, you could try copying the entire hexagonal shape into every hexagon that you used to draw it (and then repeat that). Which would look rather cool. But take rather longer to calculate.

Mongolia, Stochastic Quantum Mechanics, and Spacetime Curvature

2009-12-17T23:22:00.002Z

One of the arguments in the book was that quantum mechanics can describe velocity-dependent distortions in spacetime associated with the motion of a massed particle with respect to its surroundings.

It's a simple enough idea:
For a fundamental particle, QM says that the particle's effective position has a degree of uncertainty to it. It ought to be smeared out over the surrounding region of spacetime as a probability field. Its energy and momentum are smudged. Although any attempts to sample that energy and momentum are going to give quantised results that jump about a lot, if we average the results of a large number of similar possible measurements to produce a smoothed, idealised map of the probability-field for the particle's distribution of massenergy, we end up with a density-field surrounding the particle's notional position that expresses the distribution of massenergy through space that – functionally and definitionally – would appear to have the properties of a gravitational field (because that's effectively what a massenergy field is). The momentum is similarly smudged, giving us a polarised field-distortion component that expresses the particle's state of motion. The shape of the field tells us the probability-weightings for the likelihood of our being able to make certain measurements at given locations.

One interpretation might be that the underlying stochastic processes are truly random, and that the shape and dimensionality of classical physics appears as an emergent feature.
Another might be that the shape represents classical physics principles operating below the quantum threshold, but being drowned out by signal and sampling noise (until we average out the noise to reveal the underlying structure).

Trouble is, if we take these QM averaged-field descriptions seriously, they imply that the correct classical geometrical model for particle interactions isn't flat spacetime – the existence and state of motion of a particle corresponds to a deviation from flat spacetime, and the greater the relative velocity between particles, the more significant the associated gravitomagnetic curvature effects become.

With this approach to quantum mechanics, observer-dependence doesn't have to be some "spooky" effect where the same experiment has physically-diverging results depending on how the observer looks at it, due to reality having an odd, probablistic multiple-personality disorder ... we get different predictions when we change the position and speed of our observers because the presence and properties of those observers physically alters the shape of the experiment, in ways that can cause quantised measurements of the geometry to come out differently. It's a non-Euclidean, nonlinear problem. At those scales there's no such thing as a perfect observer, so to describe how the experiment should play out in isolation, and then to repeat it with different embedded observers charging across the playing-field, is to carry out different experiments.

It's not a difficult argument, but "particle physics" people have a tendency to argue in favour of special relativity by saying that we know for a fact that curvature plays no measurable role in high-energy physics, and mathematicians have a habit of trusting physicists when they say what their experiments show ... so I didn't know of any examples of QM people discussing velocity-dependent curvature when I wrote the book.

Anyhow, earlier this year I stumbled across one. Funny story: I was looking at my Google Analytics statistics, and obsessing about how nobody from Mongolia seemed to have been visiting my website, and then I happened to visit a "citation statistics" site that included a world-map, so naturally I zoomed in to find out what fundamental theoretical research had been coming out of Mongolia. As one does. The search only gave one result, so I clicked on it.

And then I choked into my coffee as this thing came up:
"Space-time structure near particles and its influence on particle behavior"
International Journal of Theoretical Physics [23] 1031-1041 (1984)
Kh. Namsrai, Institute of Physics and Technology, Academy of Sciences, Mongolian People's Republic, Ulan-Bator, Mongolia

Abstract: "An interrelation between the properties of the space-time structure near moving particles and their dynamics is discussed. It is suggested that the space-time metric near particles becomes a curved one ... "

DOI 10.1007/BF02213415

The paper also appears as a chapter towards the end of Namsrai's rather expensive book, Nonlocal Quantum Field Theory and Stochastic Quantum Mechanics (Springer, 1985)

11.8.2:

" ... Physically this relationship means that by knowing the space-time structure near the particle we can calculate its velocity (generalized) and, on the contrary, by the value of the particle velocity one can try to build the space-time structure near the moving particle. Thus, it seems, there exists a profound connection between these two concepts and they enter as a single inseparable entity into our scheme. "

The "stochastic" approach looks at QM from a "shotgun" perspective, superimposes the result of a large number of potential measurements ... and arguably generates a spacetime that's "curved" in the vicinity of a "moving" particle in such a way as to describe the particle's velocity. The curvature generates the velocity, and the velocity generates the curvature. Which was kinda what I'd been saying. But with a lot more advanced math to back it up.

Of course, the irony here is that Namsrai's paper and book describe the emergent classical properties of apparently random processes ... and I only found the piece (which is exactly what I needed to find), by using an apparently random method.

Spooky! ;)

ReactOS: Windows without Windows

2009-12-15T22:44:00.004Z

ReactOS is a free, open-source replacement for the Windows XP operating system. Okay, the ReactOS guys will probably take issue with that statement and argue that as far as the latest codebase specifications are concerned, the target platform has now moved on, and is now W2k3/Win7 ... but for most prospective users, ReactOS is effectively an XP-substitute.

Lots of people liked XP, but since Microsoft no longer sell XP retail and the official MS mainstream support for XP ended back in early 2009, it's good to have an alternative. It's also good to have an alternative supplier to Microsoft – Windows 7 is supposed to be fine, but the gap between XP sales being cut off and Win7 appearing rather undermined Microsoft's image as a vendor who could be relied on for continuity of supply. MS temporarily cut off huge numbers of users who'd invested in XP and couldn't use Vista, because it wanted to prop up Vista sales. It's difficult to commit to a single operating system if the sole supplier is in the habit of pulling stunts like that.

ReactOS aims to become the Windows that Windows never quite managed to be (to steal the old IBM OS2 slogan, "A better Windows than Windows"). Written from scratch (to avoid legal issues) the developers haven't had to compromise their code to meet pesky release deadlines, so ReactOS is very, very fast and very, very small. It should run Windows software on a netbook faster than XP, using fewer resources. It's the mythical lean-and-mean "de-bloated Windows" that Windows users were asking for for years.

The downside of this perfectionist "no-deadlines" approach is that ReactOS is also very, very late.

The ReactOS 0.3.11 release was finally finished today (Twitterlink).
ROS 0.3.x is "pre-beta" software, which translates as "It runs, but not all software, and not on all hardware, and expect the occasional crash" Lots of programs are already supposed to run on ROS just fine, others may stall in certain situations when they try to call a Windows function whose ROS counterpart hasn't quite been implemented yet, or where a subsystem is still a work in progress. These problems should become less and less common as the ROS version numbers increase and the remaining holes in ReactOS get filled in. Issues that affect the more popular programs are being fixed first.

The final goal is 100% Windows compatibility, for everything. And that's not just for software but for hardware too. So if you have a scanner or printer that already comes with a Windows driver, that driver should (eventually) install "as-is" under ROS. That's worth repeating: no ReactOS-specific drivers are required. So you won't have the problem that Linux had, of having to wait around for a company to pay someone to write a special driver to get your niche hardware to run on a "non-Windows" OS. If there's already a Windows driver, it should (eventually) install under ReactOS, with no special steps needed.

Problems? Well, the lateness issue. The ReactOS project been going for a while now, and it only really started looking sensible this year. It would have been great if ROS could have been gotten to "retail-level" solidity before Win7 came out ... while people were still desperate for any version of Windows that wasn't Vista ... but that didn't happen. Windows has accumulated a LOT of different subsystems, from video to networking, all of whose features would need to be replicated in order for all programs to run under ROS in exactly the way they do under Windows.
Certain things have been deprioritised. ROS doesn't yet have a "fancy" desktop, and it can't write to an NTFS filesystem. But, like an amoeba collecting DNA from other organisms it eats, ReactOS is picking up tricks from other open-source projects – ROS is now sharing code with the WINE project (which lets you run Windows code in a bubble inside Linux), so further improvements to WINE help ROS, and vice versa.

Currently, one of the main issues for ROS is getting the thing to install on a disparate range of real-world PC hardware with its limited set of default bundled drivers. Once ROS is up and running you can start downloading and installing more specific Windows drivers, but if the included drivers aren't compatible with your hardware, you're stuck. Many ROS enthusiasts are running the OS in a "virtual machine" bubble under another operating system, so default support for real-world hardware hasn't been so much of a priority until recently.

Version 0.3.9 installed fine on my old (~2001) Sony laptop, but wouldn't install on an ASUS netbook, because the netbook had a serial ATA drive that the included ROS drivers didn't understand (to be fair, original-release XP doesn't install on the ASUS either, for the same reason).
At version 0.3.10, they tackled the SATA problem by bundling in a "universal" ATA driver from a different open-source project ... which turned out to have a few compatibility issues of its own, with the result that 0.3.10 refused to install on either of my test machines.
Version 0.3.11 is probably going to be the first version where the OS really starts to be functional to most people, with 0.4.x starting to develop and polish some of the user-interface components and secondary features.

If you can actually install ReactOS on your hardware, what sort of advantage is there to running programs under it instead of under XP? Well, you know how XP takes at least 45 minutes to an hour to install? Because of its small footprint, ReactOS 0.3.9 installed onto my blank laptop in five minutes flat. And by "five minutes" I really do mean five "stopwatch" minutes (as in "approximately three hundred seconds").

That's fast.

Momentum and Kinetic Energy

2009-12-14T11:00:00.002Z

Kinetic energy is a slippery subject under relativity theory, so I try to give it a wide berth wherever possible. :) Here's how it behaved under C20th theory:

Under "early SR", kinetic energy seemed to appear in the equations in a quite an intuitive form … if we converted a body to light, and moved past the body at velocity v, different parts of the light-complex would be redshifted and blueshifted by Doppler effects, but the overall summed energy (according to the LET/SR relationships) would always end up being increased by the Lorentz factor, 1:SQRT[1 - vv/cc]. We could interpret the increase rather nicely by arguing that, since we could use SR to describe a “moving” body as being Lorentz time-dilated, this should translate into an apparent increase in the object's inertia (and its effective inertial mass), and by applying the E=mc^2 equation to that enhanced “relativistic mass” value, we arrived at the appropriate “enhanced” value for the total energy of the light-complex under SR. This additional energy due to motion wasn't the traditional “half em vee squared”, so our original schoolbook arguments and derivations for KE weren't quite correct in the new context except as a low-velocity first approximation.
With “modern”, “Minkowskian” SR, the subject of special relativity evolved. It cast off some earlier concepts from Lorentzian electrodynamics and Lorentz Ether Theory, and ended up as the theory of the geometrical properties of Minkowski spacetime. The idea of relativistic mass got downgraded, and some of the more mathy people started to say that the concept of relativistic mass was “bad” and shouldn't be taught. What was important (they said) were just two things: (1) the rest massenergy of a particle, and (2) its path through spacetime. Everything else was a derivative of these two things, and the conserved property wasn't either momentum or kinetic energy, but a new hybrid thing, called momenergy. But they kept the extra Lorentz-factor when calcuating things like SR momentum.
So with the appearance of “modern SR”, our concept of what kinetic energy was (and what it ought to be) changed again. Taylor and Wheeler's “Spacetime Physics” (2nd ed., 1992) has a useful chapter on momenergy (§7) that works through this “Minkowskian” approach.
Under general relativity things got even more slippery, because we arguably moved from the concept of “simple” kinetic energy towards that of physical, recoverable kinetic energy that could be expressed as a change in shape of the metric. Defining the nominal energy contained in a region gets difficult unless we also define the properties of other neighbouring regions that it might interact with. Even if we know all the masses and velocities involved, the effective resulting energy can also depend on their distributions and arrangements.

If we switch to the “redder” equations suggested in the book, things change again. Because each ray is now redder than its SR counterpart by the Lorentz factor, that “nice” SR result that the totalled energy of the emitted light increases with velocity by the Lorentz factor vanishes. Now, the sum of all the energies of the rays gives exactly the same value regardless of the relative speed between the observer and the experiment. Adding the ray-energies together gives a fixed value that represents just the rest massenergy of the original body. So (as someone asked, after the Newton and E=mc^2 post) where did the kinetic energy component go?

Well, in this system, the thing that describes the original body's ability to do work due to its motion isn't just the total summed energy of the rays, but the total ray-energy multiplied by the asymmetricality of its distribution.
Suppose that we instead took an electrical charge and distorted its field – it'd now have have two energy components: the default energy associated with the electric charge, and an additional energy due to the way that the effect of that charge was distorted (like the energy bound up in a stretched or squashed spring). If an electrical charge is seen to be moving, its field seems distorted due to aberration effects, and it therefore carries an additional chunk of energy, even though the "quantity of field" is the same. Similarly, in our gravitomagnetic model, we have a moving gravitational charge whose "quantity of field" is the same for all velocities, but whose velocity-dependent distortion carries an additional whack. When we then convert the body to trapped light, the total energy of all the individual rays corresponds to just the "rest field" or the "rest massenergy", and the original body's kinetic energy shows up as the apparent additional energy-imbalance across the light-complex, due to the fact that it's moving.

Thought-experiments: If a “stationary” body is converted to light, the resulting distribution of light-energy is completely symmetrical. No asymmetry means no equivalent kinetic energy. Add energy symmetrically to the light-complex and then convert it back into matter, and because the resulting body still has zero overall momentum, the added energy has to translate into additional rest mass rather than KE. Add energy to the complex asymmetrically, and the imbalance means that the resulting mass now has to be “moving” wrt the original state in order to preserve our introduced asymmetricality, and appears as kinetic energy. Remove energy from the original balanced complex, asymmetrically, and you again create motion and KE in the resulting object (although the reconstituted body now has a smaller rest mass thanks to the energy that you stole).
In this model, the kinetic energy for a simple “moving” point-particle doesn't show up in a simple calculation of summed energy values for the equivalent light-complex. It appears as the energy-differential across the light-complex caused by the way that that rest energy is redistributed in the light-complex due to the original body's motion.

To find the asymmetry of energies in a light-complex, we can use a vector-summing approach, which allows ray-energies to cancel out if they're aimed in opposite directions, leaving us with a residual measure of the differential energy ... which relates to the net momentum of the light-complex.

So under the revised model, there are two energy values to consider, the "quantity of field" associated with a particle, and the angular distribution of how that first energy-field is arranged. The first one's the rest massenergy, the second's the kinetic energy.

This isn't the usual way of doing things, but it arguably gives us a more minimalist logical structure than the one used by special relativity. Under SR, the motional energy of a particle shows up in the geometry twice – first in the gross quantity of energy, and then a second time in the redistribution of that Lorentz-increased energy due to velocity – and we have to do some odd-looking four-dimensional things to cancel one from the other. With this system, the quantity only appears once, as the asymmetrical distribution of the fixed quantity of rest-energy.

To me this looks like it might be a more elegant way of doing physics.

Academic Research

2009-12-11T14:39:00.005Z

A group of scientists crash on a desert island on the way back from a conference. Quickly, they each decide to start doing something to help the group survive until they're rescued, using their own individual skills. In the group are an engineer, a biologist, a mathematician and a tenured research scientist.

"I know," says the engineer, "I'll build a shelter, and start looking for a water source. Maybe start digging a well."

"Okay," says the biologist, "I'll check whether the water's likely to be drinkable, and I'll do a quick audit of the local flora and fauna, and see what's edible and what's poisonous. I'll also keep a look out for signs of any wild pigs that we might be able to trap and eat."

"Fine," says the mathematician, "I'll stocktake our supplies and work out how long they're likely to last and how we should ration them, and I'll also try to work out where we are, what side of the island we ought to use for our signal beacon, and the best location for it, for maximum visibility."

All this time, the academic researcher has been shaking his head, until he can't contain himself any longer. "You're all crazy!", he shouts.
The others look at him, blankly, and wait for him to explain.
"You guys honestly don't have a clue, do you?", says the academic. "All of these plans are totally impractical. Let me point out a few basic facts here that you all seem to have forgotten. We're lost. We're probably hundreds of miles from the nearest shipping lane. Nobody knows we're here. We have no working communications equipment, and even if we did, we have no electricity to power it."
"So how the hell are we supposed to submit the research proposals for all this work?"

Quantum Gravity

2009-12-06T18:30:00.001Z

Quantum Gravity is one of those cool project names that doesn't necessarily have a set definition.

This is because the Quantum Gravity guys know roughly where they're trying to get to, but are still exploring different potential routes that might be able to get us there. It's not necessarily about "quantising gravity" in the graviton sense (although some people are following that path) – more generally, it's any research that tries to work out how the heck we can reconcile quantum mechanics with general relativity.

Currently, the situation with gravitational horizons is that quantum theory lets us prove, absolutely definitely, that horizons radiate (Hawking radiation), while Einstein's general theory lets us prove, absolutely definitely, that they don't. We can superimpose QM "sprinkles" onto a GR background to retrofit the desired QM effect onto a nominal GR metric, but that's really a "hand-crafted" response to the problem. It'd be better to have a single overarching theory with a single consistent set of rules that let us incorporate the best parts of both models in a single internally-consistent scheme, and that's the end result that quantum gravity research tries to take us towards.

A common approach to QG research is seeing what happens when we add additional dimensions, since this increases the range of geometrical possibilities beyond those already defined for the individual models that we're trying to unify ... the hope being that we might find one special extended geometry that has a special claim to owning the sub-theories that it needs to include. The string theory guys tend to do this sort of work (although some folk feel that they may have gotten a little carried away with the extent of their attempt to catalogue all the possible solutions!).
Another is to look at ways of extending the QM approach of carrier particles as mediators for force to include gravitation, and see what happens. So we have things like Higgs Field research. But since classical gravitation is usually considered to be a curvature effect, not everyone agrees that we need a separate "curvature particle", since the idea seems to be a little self-referential.
We now also have people studying acoustic metrics as a way of generating curved-surface descriptions of Hawking radiation.

Different approaches to QM can suggest different approaches to QG. The "stochastic" approach to QM suggests that perhaps we can average the random fluctuations in a model to produce an effective classical geometry, and perhaps we might even be able to leave the dimensionality of these fluctuations unspecified, and see whether our own 3+1 dimensional universe emerges naturally over larger scales once everything's smudged together in the correct statistical proportions.

So, lots of approaches. Folk are working on a lot of potential theories, in the hope that one of them might end up being (or leading to) the winner ... or perhaps a few different methods might succeed if they turn out to be dual to each other. It's probably a bit like being back in the early days of quantum mechanics, where there were multiple approaches and nobody really knew which were good, which were bad, which were dead ends, and which were equivalent.

Meanwhile ... while we're waiting for "the theory of quantum gravity" to arrive ... we have some nice logos to look at.

Panoramic Photography

2009-11-30T23:11:00.001Z

If you're wondering how to produce panoramic images and "360×360"-degree "bubble" images like the one used for the Airbus 380 website, there are three main methods:

Use a special panorama camera. These typically mask off the lens so that only a thin vertical slit is exposed. The camera is mounted on a tripod with a clockwork mechanism that pans it across the view, and the rotation process winds the film onto the spool, past the slit. This used to get used a lot for school group photographs.
You spend a vast amount of money on a special custom-made fisheye lens. In the (1980's?) a photographer made a bit of a splash using one of these to produce fish-eye cityscapes. People hadn't seen anything quite like it.
Nowadays: you use pretty much any digital camera with a bit of auto-compositing software. Some cameras even have a crude landscape-stitching facility built-in (even my mobile phone does it!)

If you want to do this properly, and you can run Windows software, the Google search term to remember is Autostitch.

It's downloadable from the University of British Columbia's site, and the Windows demo version is free for non-commercial use. They just ask that if you upload anything to the web, you use "autostitch" as one of your tags, so that they can see what people are doing with it. The user-interface is pretty much non-existent – lots of scary number-boxes for people with wierd lenses – but really all you have to do is leave the defaults as they are, click "File / Open", select the pictures you want assembled, and then click the "Open" button and go make a cup of tea. When you come back, you'll probably have a perfect panoramic image. If you want a friendlier front-end, they license the library of routines to commercial software companies, who'll be happy to sell you a less clunky implementation. They've licensed it to George Lucas' Industrial Light and Magic, and there's even now a version of Autostitch for the iPhone.

Autostitch automatically works out the right order to arrange your photos, compensates for lens distortion, compensates for tilt and zoom, assigns angle values to pixels, works out how best to mesh them together, downgrades "inconsistent" parts of individual photos so they don't contribute as much to the final picture, adjusts the colour balance and exposure of each photo to merge in with its neighbours, and then smoothly crossfades everything together.

Here's what Autostitch really generated for the above photo, before I cropped it to remove the black gaps (where hadn't taken a photo for the program to use) ... it's not something that you'd normally want to show someone, but it shows how much mathematical cleverness is going into the fitting process – this is NOT just simple "image-tiling" software:

The authors' research paper ("Automatic Panoramic Image Stitching using Invariant Features", Matthew Brown and David Lowe) goes into this in a lot more detail, with sample pictures.

If you use a 360-degree sequence of images, the Autostitch output will rotate seamlessly from side to side in your graphics-editing software, and if you take a 100% bubble sequence, there are Flash applications that can re-generate the view you'd see by looking at any horizontal or vertical angle (if you still haven't tried the Airbus demo, go now!).

If you don't need a wraparound view, and you have a tripod and a notepad and some patience, you can use a larger zoom setting on your camera than normal and take a LOT of overlapping pictures of a static scene. This lets a cheap five-megapixel camera easily generate monster hundred-megapixel images ("output image size" is one of the more understandable boxes on the "scary parameter page"). It's good to have a couple of "test goes" at this technique, so that on that one fateful day when you're confronted with a perfect picture that's too tall or wide to fit into your camera view, you can rattle off a sequence of overlapping snaps, knowing that Autostitch should be able to assemble them when you get home.

The one real limitation of Autostitch is that it's currently only set up for merging photos taken from a single point. There are some situations where you might want to auto-undistort and tile a sequence of photos taken from different locations: for instance, if you were photographing a mural painted on a long wall, you'd probably want to take a series of images from different locations along the wall and have them all assembled side-by side to form a long strip.
Autostitch doesn't yet do that. But maybe they might add the feature if enough people ask.

The Airbus A380 cockpit

2009-11-27T22:40:00.002Z

This is a cut-down still of part of the Airbus A380 cockpit, taken from airbus.com. It's from a panoramic viewer, on a page that uses Flash to let you pan and tilt and zoom in and out of a view in any direction, so that you can really explore the cabin in detail, in high-res. If you want to look out of the window, look backwards, or look up at the ceiling while it spins, you can do that. The mouse scrollwheel zooms you in and out. It's nice. The 360-360 photography is by gillesvidal.com.

The A380 is a very nice plane, with a famously-great cockpit control surface layout. It has a comfortable, relaxing, reassuring look to it (as opposed to some of the more traditional layouts with lumpy panels and dials everywhere all screaming "Look at MEEE!"). It doesn't look scary – as a newbie, you can look at this user-interface and half-kid yourself that you might actually be able to fly it.
My concern when I first heard about the Airbus' screen-based system was: what happens if a screen develops a fault, and you lose a whole bank of virtual instrumentation? Well, the A380 panels tackle that problem brilliantly – you notice how the eight main portrait-format screens all seem to be the same size? Well, they're completely interchangeable. You're supposed to be able to pop out any of the main screens and swap them round, live. There's a couple of little grey rectangles below the bottom two corners of each screen panel, presumably those are the finger-latches. And apparently you can completely change the layout, so if one panel's connection points are messed up, you can watch its data somewhere else. I like this plane.

So let's explore ...
Twin side-joysticks and QWERTY keyboards. I don't know what those two rounded plastic bulges are ... perhaps they're calming devices, for the pilots to put their hands on in moments of stress. Or maybe they're there so that if you get thrown towards the panel, you have something to grab onto that doesn't accidentally result in you pressing an Important Switch by accident.

Three spare seats at the back (for parties), and an overhead camera (so that you can remember what you did the next morning). Fun Wagon!

Note the video camera views, on the centre screen. Useful for parking, and also for reminding yourself which airport you're at. Also for checking that you still have the right number of engines, that none of them are on fire, that all your control surfaces are present and correct, and that your wheels haven't fallen off. Without cameras (or a periscope), it's not always easy to know if your wheels are really down, because planes tend not to have glass bottoms. The central panel showing the video views is the obvious "spare" section of control surface to use in flight for additional functions if further equipment is retrofitted that needs its own display space (like customised additional avionics – rocket launchers, anyone?). There's a pull-out shelf thing in front of each seat that gives the pilots additional keyboards and pop-up screens for general flight admin and map-browsing.

Very Importantly: what looks like three cup-holders per side, left and right, away from the important controls, plus another five at the back left. It's deeply important to have enough cup-holders (one for fresh coffee, one for water, and one for soup, or perhaps noodles?). That's assuming that the holes aren't for something more boring. There's a clunky laptop-py thing at the back, for system-level stuff.

I like the documentation holder on the back of the door, made out of two types of sticky tape. But what's that panel in the door, with the nasty scratch gouged in it? Is it a “people” version of a cat-flap? I also like the design of the door-hinges, with the hinge protruding inside the cabin, and the screws accessible. That means that the cabin crew can remove the door from its hinges from the inside, if it jams (say, after a crash). Someone's put a lot of thought into this.

Twin microphones (for karaoke duets? Pilot-copilot comedy banter?). Between the "emergency power" and "oxygen" switches overhead (up above the left windscreen variable-speed wiper knob), there's also an intriguing switch marked “Entertainment”. Hmm.

Rear right, there's what looks like a locked cabinet marked CDROM. Well, if the Batmobile has one, I suppose the 380 ought to have one, too.

"Escape rope" compartments on both sides. Down to the rear left, by the fire extinguisher (whose sign I initially misread as “portable fire eating”), there's a hatch set into the floor. I've seen this hatch drawn on a schematic with a ladder poking through that exits through the front wheel port. I guess this means that if you're a pilot and you have a panic attack before takeoff, you can pop down through the floor and run away across the airfield without the passengers realising that you've gone.

The seat covers have large tags facing each other saying "Pilot" and "Copilot", which might be useful for resolving cabin arguments. Point at the tag. 'Nuff said. Also handy for avoiding those embarrassing "But I thought YOU were supposed to be flying the plane!" moments.

So, a very nice vehicle.

The only design decision here that I'd query is the upholstery. Pinstripe? Hmmm. But perhaps there's a reason for that, too ... perhaps striped material doesn't show sweat stains so easily. You don't want to be settling down into your seat for a long-haul flight, and be too conscious of the big sweaty patch left by the previous pilot. Eurgh. I wonder how often they change the covers?

With the addition of deep-pile furry tiger-pattern seat covers, vibro-back-massagers built into the seats, a proper entertainment system with giant speakers, and a couple of foot spas, I'd give this cabin 10/10.

The Relativistic Ellipse

2009-11-23T23:29:00.000Z

This is an especially cool diagram for relativity theory, but it's rather hard to find in print. There's a limited version of it in Moreau's 1994 "Wave front relativity" paper, and I put it in the book (chapter 8), but I can't think offhand of anywhere else you're liable to find it.

It's simply an ellipse with lines radiating from one focus and converging on the other.

Imagine that you have a point-source of light giving off pulses. Surrounding the point-source is a spherical mirror, which catches the outgoing spherical EM wavefront and bounces it directly back to the source. All parts of the reflected wavefront arrive back at the source at the exact same moment.
This tells us (a) that all parts of the surface are at 90 degrees to the source, and (b) that all parts of the surface are at the same distance from the source.

=Relativistic Aberration=

Now let's replay the same situation, but imagine how it would have looked to us if we were whizzing past the experiment in a spaceship (but not so close that we actually disturbed the light in any significant way).

Now, the geometry seems to be different. We're forced to agree that the reflected wavefront still converges on the emitter (because nothing within the experimental region has physically changed), but since the light takes a finite time to go out and come back again, as far as we're concerned, the experimental hardware has been moving while the light was out doing its thing.
For us, the light was being emitted from one position and refocused at another.

And the shape that does that is an ellipse.

If we look at the shape of the relativistic ellipse, we find that the outgoing rays are angled forwards ... they have to be in order for them to be able to keep up with the "moving" source. And if we measure the angles of these rays on the diagram, it gives us the textbook relativistic aberration formula used by special relativity (and also by Newtonian optics, old ballistic emission theory, and any other relativistic model).

=Velocity-rescaling, distance and time under Special Relativity=

The thing that's slightly counter-intuitive about the diagram is that if the radius of the sphere is half a light-second, and if it's supposed to take exactly one second for the light to return to its starting point (so that the bouncing light makes a clock that supposedly ticks every second), you might expect the distance "v" that the object moves in one second to simply be the distance between the two points. Slightly perversely, under SR, it isn't. The relative proportional velocity v/c (velocity quoted as a fraction of the speed of light) has to be the ratio between the focal point distance and the stretched, longest dimension of the ellipse. So if the distance between the focii is half the length of the ellipse, we can say that the velocity is half lightspeed
(in the diagram above, it's 0.8c).
But since the ellipse is stretched, the distance between the points (if v is defined as a particular fraction of the speed of light) is stretched, too. If we're to follow SR and say that lightspeed is a fixed global reference, then the distance between bounce-points is somewhat more than v metres.

Under special relativity, the width of the ellipse is assumed to be constant regardless of velocity, the ellipse is stretched by the Lorentz factor (calculated from our proportional velocity), and the "point-to-point" distance ends up elongated by the Lorentz factor, too.

Under special relativity we explain the extra distance by invoking Lorentz time dilation. We suggest that the particle travels further than expected in our coordinate system in one of its own seconds, for a given nominal velocity, because its clock is running slow (so for us, it travels for more than a second,and crosses more than v metres). Or we can argue that if an observer moving with the experiment sees a piece of paper with the diagram drawn on it passing by with the same proportional velocity of v, that for them, the distance between the marks is v metres, because their measurements indicate that the moving paper is Lorentz length-contracted. The ellipse looks like a giveaway that lightspeed isn't globally fixed, but if we assume that it is, and need to explain why the ellipse somehow doesn't really count as an ellipse, we end up with the traditional SR length-contraction and time-dilation explanations.

Contract the elongated ellipsoid by the magical gamma factor, and its outline turns neatly back into the original sphere.

=Doppler shifts=

The next thing that we can do is to look at the length of the lines. Turns out that, if we're doing the SR version of the exercise, each ray elongates or shrinks by precisely the right ratio for special relativity's relativistic Doppler effect. The forward and rearward distances are stretched and squashed by the ratio SQRT[(c-v)/(c+v)], and the 90-degree-aimed ray is stretched in length by SQRT[1 - vv/cc].
That's the Lorentz transverse redshift prediction of special relativity.

=Ellipses are Cool=

So this one little diagram tells you almost everything that you need to know about special relativity. Once you've drawn it with the appropriate proportions for a given velocity, all you have to do is read off the angles and distances with a protractor and ruler to find SR's physical predictions about the appearance of a moving body, as seen from any angle.

If you'd prefer not to rely on any "odd" theory-specific definitons of velocity, distance or time whenbuildign the ellipse, all you have to do is draw in two rays from a focus, with lengths rescaled by the theory's particular Doppler shift predictions, and the rest of the diagram constructs itself. Along with the Minkowski lightcone diagram, it's probably one of the most powerful diagrams in special relativity.

So why isn't it in the books?

We-ell, perhaps the problem with the diagram is that it makes people think. Which leads to troubling ponderings, because it turns out that the diagram doesn't have to be used with special relativity. It'll compute the SR relationships if we deliberately stretch the point-to-point distance by the Lorentz factor, or if we use the SR "relativistic Doppler" relationships to define the reference wavelength-distances, or if we decide that lightspeed has to be defined as globally constant for all participants ... but if we're only interested in the principle of relativity, and we're not prepared to commit to these extra SR-specific things, the ellipse also lets us plug in other assumptions, and lets us see the their consequences.

For instance, we know that old Newtonian optics was technically a "relativistic" theory (although nobody could get NO to work properly with wave theory). We know the forward and rearward wavelength changes associated with that theory, so we can draw in these two wave-distances from one of the focal points, and construct the rest of the ellipse around these maximum and minimum radii. What we end up with is an exact duplicate of the SR ellipse, with the same proportions and aberration angles, but with an additional Lorentz magnification. All the NO wavelengths are longer than their SR counterparts by a Lorentz ratio. So transverse redshifts aren't unique to special relativity.

And then you notice some other things. The SR ellipse can be compacted back into its original circular outline just by contracting it on one axis. This is analogous to tilting the diagram off the page to produce a contracted "shadow", which gets us into the subject of Minkowski spacetime, tilted planes of simultaneity, and other cool things. The SR family of ellipses actually represents constant-width tilted cross-sections through a constant Minkowski lightcone and can be visualised as projected conic sections.

The SR version of the constructed ellipse is the only one that has this special property.
This tells us that if we require spacetime to be "flat" in moving-body problems, the SR relationships are the only ones that work. We're still freetoargue argue about the correct philosophical interpretation and presentation of the theory, and about whether the interpreted contractions and clock-changes are physically real or not (and about what wemean by "physically real" in the context of SR), but the defining Doppler characteristics of the theory – the things that dictate the final physical predictions and equations ofmotion, regardless of interpretation – are set, locked and non-negotiable once we've decided that we won't be implementing curvature as part of the model. According to the ellipse, Relativity (limited to simple inertial motion) plus flat spacetime gives SR. It's airtight.

If we now go back to the enlarged Newtonian version of the ellipse, we find that the rules are different. The enlarged NO wavelengths can't be fitted back into the original sphere without distorting the centre of the ellipse out of the page. Instead of a tilted-and-rescaled cross-section through a fixed geometry (Minkowski spacetime) we end up with a geometry whose shape dynamically changes when there's relative motion between physical masses. Instead of a purely "projective" tilt, we have a real physical change of shape. The causal structure of the metric now depends on the presence and motion of physical bodies embedded with it. We end up with a gravitomagnetic theory, with a different form of lightspeed constancy to SR. And that's why nobody, including Einstein, could put together a sane-looking reference model for Newtonian optics that didn't go crazy when you tried to treat it as wave theory. Newtonian optics simply doesn't work in flat spacetime. The wavelengths don't fit.

I still think that it's a shame that they don't teach the relativistic ellipse in physics classes. It's a powerful tool, and a really handy device for demystifying special relativity. But perhaps it's too powerful, and perhaps if you're trying to convince a class that SR is the only possible answer, a tool that suggests the existence of alternative approaches spoils the narrative.

Social Media and the Gooeyverse

2009-11-16T01:53:00.001Z

Some "social media" guys have taken to bitching about the term "social media". They feel that it doesn't adequately describe what they do. It doesn't make them feel sufficiently ... special.

Well, there's two ways you can go. Either you replace the term "social media" with something more specific, like "Interactive social network enablement systems", or you go the other way, and invent a brand new buzzword that's shorter, snappier, and more subversive-sounding.

In which case, I vote for "Goo".

Goo is the amorphous, gelatinous set of shifting organic interconnections and linkages that binds and connects us all together. It's not a restrictive "web" or a fixed "net"-work. It's goo.
A locus of goo, like Twitter or Facebook, becomes a gooball. The connections between gooballs, the part of the internet that deals with crosslinks between social media hubs, becomes the intergoo. The global user environment for goo becomes the Gooeysphere, the view from within is the Gooeyverse. "Intelligent" software support for goo ("web 2.0"-based automatic book recommendations, and so on) becomes smart goo. Goo generators like Blogger or Wikipedia, that let users spawn new collections of goo connections are hypergoo.

Corporate blogging and twitterring becomes Blue Goo.

It also ties in well with the idea of links between software and wetware ... and the fact that a Graphical User Interface – the thing that you look at when you use a piece of software – is referred to as a GUI (pronounced "gooey") doesn't hurt, either.

All good clean fun.

Trouble is, how are you going to explain to your mother that you have a career in goo?

"Social media" doesn't sound so bad now, does it?

Villarceau Coils, Slinkies, and Ring-packing

2009-11-11T22:58:00.000Z

Computer graphics are fine, but the problem with programming a simulation of something is that often you only get out what you put in. You lose the element of surprise. So sometimes, if you you want to find out what something really does, you build one. Available technology and spare parts permitting, of course.

For the "Villarceau Coil" blogpost, I figured that it was worth making a physical model. A good hardware place nearby sells middle-sized keyring loops for 15 pence each, so I went in with a few quid and came away with a pocketful. Then it was just a question of clipping the things together.

There were a couple of things that I hadn't expected:

Thing Number One was that a "keyring Villarceau coil" is a bit like a Slinky. You put it on the palm of your band, or on a flat surface, and tilt the surface, and the thing kinda ... slinks ... downhill. It reacts to the uneven pressure on its base, rings rotate and slither past each other, the torus squirms and turns inside out, and the thing scuttles off down the slope with a slightly guilty air about it, like an octopus running along a seabed.

From a science-fiction/xenobiology point of view, the coil makes an interesting template for a possible alien lifeform. With a soft toroidal body and a hard set of spiny Villarceau rings, an animal could burrow or shred predators or food by turning itself inside out. It could start as a skinny beastie with maybe three rings, and grow more rings as it got bigger and fatter. It'd solve the problem of how to reconcile a hard exoskeleton with the ability to change size. Young could be gestated as full-size rings within the fleshy body. Giving birth would probably have to be be kinda fatal, though, unless the rings each had a notch somewhere. :(

Anyhow ... Drop the coil, and it "splashes" when it lands, then reforms back to the torus shape ... or if you've used a lot of rings, into a pair of interlinked tori. You can pop it on your finger, and pass it from hand to hand, one finger at a time, by tilting your finger to point downhill towards the destination finger, so that the v-coil slinks down the finger, turning itself inside out as it goes.

So basically, a fun executive toy. Three quid well spent.

Thing Number Two was that the rings have a natural tendency to nest (as in, "what Russian Dolls do", not "what in birds do") .... except that, in this case, the self-similar "dolls" are all made from components that are exactly the same size. Which is a slightly wierd situation.

So if you start with maybe just three rings for a skinny torus, and you add more rings to force the thing to be fatter, you find yourself using a lot more rings than you thought. They start to form nesting toroidal layers. Since every torus that you can produce using the Villarceau configuration has exactly the same major radius as a single ring, they all fit neatly inside one another.

It's kinda reminiscent of the way that electron orbits stack up around an atomic nucleus.

And since every ring in the set of nested tori has exactly the same configuration to all the others, you can reach in and pull an inner ring and with a bit of shuggling turn it into an outer ring (while one of the outer rings shuffles back inward to take its place).

This is a FUN shape. You could probably write an entire book about it.

String Theory and The Goodies

2009-11-05T04:44:00.002Z

The 1990s "string theory" boom was an example of what happens when a critical mass of researchers realise how to game the system. If enough people start churning out work on the same subject, and eagerly citing each other in multi-author papers, they levitate up the scientific citation indexes together. You get a bubble. It's in most players' interests to keep inflating the bubble – the more new people they attract to the subject, the more seniority they have in a growth area of science. It becomes like a pyramid scheme. So in the 1990s we had some pretty absurd claims being made for string theory and what it was going to do for us, and when.

I wasn't unsympathetic to string theory as such - It seemed like something that needed to be researched ... just not by everybody. The talk was that string theory was The Future, and sometimes it seemed as if anyone who wasn't already committed to some other line of "mathematical physics" research was scrabbling to get onto the bandwagon. There were people doing worthy work on the subject before the boom, and once the bubble burst, those people would probably continue doing worthy work. What was wrong was the hype.

And all though this time, I kept remembering the old 1970's episode of The Goodies, where the guys use their advertising agency to promote string as the wonder product that's good for everything. Here's just the clip of the Goodies "String" song, courtesy of YouTube:

"String, string, string, string, ev'rybody needs string!"

It also seemed to me that we'd been here before. String theory was supposed to be a ToE, a "Theory of Everything", but in the 1990s, it actually seemed to be more of a ToA, a "Theory of Anything". It sounded like a great way of being able to remodel any given physical theory, but didn't seem to offer any clues as to what sort of theory we should be trying to model. It sounded was a bit like Jean Luis Borges' fictional "Library of Babel", that contained every book ever written, and every book that might ever be written - but whose total inclusivity meant that it ultimately contained no information at all.

String theory in the 1990s seemed to suffer from the same problem that aether theory had had in the 1890s - what had made aether theory lose credibility as a subject wasn't that it gave wrong answers, or that it was limited – its problem was that it was too flexible. With enough arbitrary variables, you could construct an aether model to reproduce almost any behaviour you could possibly want – we had aerodynamic aether theories, sink-and-source aether theories, Lorentzian ether theory (LET), and so many other variants that even experts started to lose track of them. Without a guiding set of principles to eliminate possibilities, generalised aether theory as a field couldn't actually make any solid falsifiable predictions.

Aether theory had degenerated into a "Theory of Anything", and if you eventually managed to isolate a set of rules to derive a single preferred set of physical relationships from some amophous theoretical soup, then the process of logical deduction that you'd used to decide on a particular set of properties for the theory, was the theory.

And the worry with string theory was that perhaps some people in the "string" community hadn't quite understood this. Some of its more enthusiastic proponents insisted that string theory was already "discovered" – string theory was fundamental truth, and "all we had to do now" was to learn how to decode it. But without the decoding process there was only a metatheory that defined a context within which an actual physical theory might later emerge. Without an extraction process we might as well be trying to divine "ultimate truth" from tealeaves or goat entrails.

It also didn't help that the "We already have the secrets of the universe in our grasp, we just need to take a few more generations to work out how to decode them" argument, the apparent nonchalance about the lack of falsifiability, and the use of "mystical" language to try to attract public support were all things that people more usually associated with the Nostradamians and Bible Code numerologists than with the physics and math communities.

There was only so long that the string theory hype could last without the subject actually making any physical predictions, and eventually the field got hit by a nasty dose of reality, with Lee Smolin and Peter Woit both publishing critical books on what had actually been achieved. There simply wasn't a physical theory yet. Just because something was pretty didn't automatically make it physics.

I was happy to criticise when the hype was underway, but I'm uncomfortable kicking a theory when it's already down. It's too easy. Good work is probably still being done with string theory, it's just not getting headlines in New Scientist every single week as it used to. Perhaps the fashionistas will drift away and find some other fad to attach themselves to, and it'll be just the hardcore guys left, who were there from the beginning, and aren't reliant on a fortnightly press release cycle. And perhaps that's an appropriate situation.

Heck, if it gets too fashionable to knock string theory I might even have to start defending it.

Meanwhile I'm going to watch the video again. "String, string, string, string ..."

Holograms at Halloween

2009-10-31T20:15:00.002Z

I don't suppose that there's any reason why holograms have to be created on a flat sheet.
It's traditional to do it that way, and it probably makes the optics easier, but there doesn't seem to be an especial reason why all of the sheet has to be at the same angle. If you created a hologram on a curved sheet that surrounded an object, then as long as the sheet kept the same shape, it should presumably look as if the object is inside the volume (rather than appearing to be in front of or behind a flat "window"). There's also no obvious reason why you can't produce cheap printed lenticular holograms on curved sheets either, other than that it'd make the initial processing more difficult.

So, Halloween. Once we're set up for manufacturing curved holograms, the obvious application (at this time of year) is the creation of the world's most scary Halloween masks.

Put a hologram of a human skull onto a curved transparent sheet, use the sheet as a visor, fitted inside the cowl of a black cloak, and make the inside smoked or semi-mirrored, and you have a "Death" Halloween costume, where, if anyone gets too close and peers under the cowl though the sheet to try to see who's face is behind the visor, they get a rather nasty shock!

Okay, on reflection, maybe not such a great idea after all. :(
You don't want people dropping dead of heart attacks when they realise that "the death guy" appears to be wandering about with what seems to be a real, genuine, gaping skull on the top of his neck. I mean, realism is all very well, but to be striding around town leaving a trail of traffic accidents and screaming people and dead bodies in your wake would probably be taking authenticity a bit too far. Oh well.

The Villarceau Coil

2009-10-30T17:31:00.001Z

Sometimes it's fun to try to take the most ludicrously-abstract and pointless geometrical results and to try to turn them into something useful. It's a fun game, and the more abstract the thing is, the higher the chance that nobody's actually brainstormed it properly before you. The "square-cutting" exercise ended up as a possible idea for new storage media for hydrogen-powered cars, so after uploading the "Cutting up Doughnuts" post, I was scratching my head to try to think of some real-world application for the "Villarceau Circle" result, that might turn the pastry-cutting exercise into something with actual physics applications.

The best I could come up with was a variable-geometry magnetic containment device.

If we take our two interlocking Villarceau circles, and delete one of them, we're left with a simple ring that wraps once around the torus limb and its central void. This counts as a special-case toroidal winding. We can interleave a series of these single angled rings around the torus, intersecting, without any of them clashing or colliding. If current is circulated around each ring (perhaps by "breaking" the rings and wiring them in series), you have yourself a rather unusual toroidal coil.

What's unusual about it that it has variable geometry. Each circular ring-segment can be a rigid wound coil, and by tilting the angle of these coils we can create a larger torus with arbitrary proportions (major axis radius fixed, minor axis radius variable). Okay, so there's a limit to how fat or thin we'd be able to go due to the finite thickness of the rings that we're using to construct it, but essentially, we have something that looks like a toroidal accelerator and containment device, that can actually change shape while it's running.
Provided that the "open" configurations of the resulting toroidal coil aren't too open, this might let you prototype a device without having to calculate the ideal proportions beforehand - you'd be able to adjust the torus shape while the device was actually operating.

Now, suppose for the sake of argument that you wanted a containment device that allowed you to open it out, fire high-energy particles into it in low-energy mode, then close the coils, squash the plasma density to encourage some sort of reaction, and then open the coils again to allow the reaction products to spill out into the surrounding coolant. You could have a system that "breathes", and holds different shapes for different parts of its cycle.

Okay, I'm trying not to be too glib here – because nuclear physics is NOT my specialist field – but this thing would look awfully like a cross between the "cage fusor devices" and the "tokamak" configurations that people use for nuclear fusion. When it's closed you have something that looks like a tokamak, and when it's open you have something that looks (superficially) more like a fusor cage. One of the annoyances of the tokamak designs is that once you've built them, they're usually locked into particular configuration – with a Villarceau coil, the variable geometry means that you should be able to get some pretty significant changes in internal volume and field strength without having to vary the current flow to the coils. And if the internal pressure gets too great, the thing's going to have a tendency to self-adjust by opening out like a flower-bud, reducing internal pressure and temperature, and releasing excess plasma into the surrounding coolant in a semi-controlled way (rather than being all bottled up until things go more badly wrong).

Anyhow ... bottom line is, that even if this configuration is no damned use at all for conventional nuclear fusion, it'd still look damned cool as a piece of hardware.

Designers and art directors for science fiction movies take note. Remember how cool people though the Big Scary Spinny Machine was in Contact (1997)? Well, this configuration would be a really nice thing to use next time you have to design a cool fictional device for a spaceship reactor or engine pod. Shiny silver interlocking steel circles that tilt and swivel, with a whizzy blue plasma glow inside. Mmmm.

I want to see this cool thing in a movie NOW ! :) Who's going to be first?

PS: I did spent the last couple of weeks seriously consider building one of these as a toy, sticking it in a small vacuum chamber and whacking a high-tension voltage into it, as a version of those plasma balls that you find in gadget shops. I figured that with that, plus a set of circular coil units, and I might have a cool little device that could spin plasma (or bits of shiny silver paper) in an amusing way. I got as far as looking up coil formers. But sanity prevailed. Plus, I think my current landlord might take a dim view of his tenants trying to build small prototype nuclear fusion reactors on the premises.

Holographic Diamonds

2009-10-29T02:52:00.002Z

Back in January 2000, the Millennium Dome exhibition opened to what was supposed to be a display of the best of British achievement. Unfortunately the people in charge of setting it up didn't seem to have a clue how to run this sort of exhibition or what to put inside the dome, and it ended up as a bit of a national embarrassment.

One of the last-minute additions to the show was the Millennium Star diamond.

To see the diamond, you had to walk though an angled passageway that was completely pitch black apart from some slightly odd (monochromatic?) blue light, and there, in the middle, you'd see a case walled with bulletproof glass, containing the blue-lit diamond. You walk past it, perhaps pause, and then make your way out. No loitering, no photography.

Something struck me when I was in there. The thickness of the cabinet's glass meant that the diamond appeared be in different places, depending on which pane you viewed it through - that's completely normal, you usually see a similar effect with fishtanks. But the blue light confused me, because normally you only see blue-lit rooms when someone's trying to hide something. Okay, so it was a blue diamond, but still ...
The human eye is pretty bad at seeing sharp details in blue light, which is why Windows has traditionally had a blue-themed startup screen - the old splash screen used crude dithering to recreate the effect of a smooth variation in tone using the default 16-colour VGA pallette, and by doing this in blue, the eye was fooled into not noticing the effect too much. If Windows 3.1 had tried that trick in red or green or yellow, the result would have been bitty and grainy and would have looked awful. In blue, you can't see the fine detail that gives the trick away.

Now, the glass.
Bulletproof glass uses a "sandwich" of alternating toughened glass and shock-absorbing plastic sheets, so that even if you shatter every layer of glass, the shatter-patterns are different, and the pieces stay stuck together by the plastic. If someone had simply added an additional sheet of plastic film with a with a hologram of a diamond ... then how would you be able to tell? You couldn't look for alignment errors between the sheets on different panes, because the diamond woudl appear at differtent positions when viewed through the different panes anyway, due the the thickness of the glass.
Does a holographic diamond appear to refract light in the same way as a real diamond? I don't know, but if someone wanted to look for an "anomalous" spectrum effect that didn't correspond to real diamond, the use of monochromatic blue light might be a good way to stop them. And with single-colour light source, we'd also find it difficult to see any interference fringes due to misregistration of the holographic films. Optical theory says that to see those coloured fringes, the colours already have to be present in the original lightsource, andf in our "blue room", that light wouldn't be there.

Of course, for all this to work, de Beers would have to have their own in-house holography R&D department aligned with their security people, which sounds pretty unlikely. But in fact, deBeers do have very strong links to holographic reseach: They have laser systems for checking diamonds, and for laser-etching holographic security marks onto them, and slightly more peripherally, Lucent have been researching diamond as a potential holographic storage medium. DeBeers also have a holographic diamond passport scheme. So diamonds and security and holography research and lasers and de Beers all have a pretty strong overlap. There probably aren't that many companies that know more about certain sorts of holography than de Beers do.

So here's a fun, harmless little conspiracy theory to ponder that's worthy of Sherlock Holmes or Jonathan Creek: What if this diamond, which thieves tried to steal from the Dome in November 2000 in a ram-raid using a mechanical digger, nailguns and a getaway speedboat, was protected by the ultimate "stage magic-based" security system? What if the diamond, that perhaps many thousands of people would swear on oath to having seen in person ...
... was never actually there?

Cosmological Hawking Radiation, and the failure of Einstein's General Theory

2009-10-26T04:19:00.002Z

Cosmological horizons are rather arbitrary. The cosmological limit to direct observation is at different places for different observers, and if you change position, your horizon position changes to match. In that respect, a cosmological horizon is a little bit like a planetary horizon - it's different for everyone, and every physical location can be considered as being at a horizon boundary for someone.

With a cosmological horizon, we can mark out a region of space that we reckon should be directly visible, and another region beyond that shouldn't be, and try to draw a dividing line between the two that represents the horizon. The unseen region doesn't exist in an observerspace map even as space, which (in an observerspace projection) seems to fizzle out and come to a stop at the horizon limit.
As we try to look at regions further and further away, we're seeing larger and larger cosmological redshifts, and seeing further and further back in time, until we approach a theoretical limit where the redshift is total, time doesn't appear to have moved on at all since the Big Bang, and events apparently frozen into the horizon correspond to those in the vicinity of Time Zero.
In an idealised model, trying to see any further away than this means that we'd be expecting to be seeing spacetime events that originated before the Big Bang, which – in our usual models – don't exist. So the cosmological horizon is the rough analogue of a censoring surface surrounding a notional black hole singularity under general relativity. It kinda ties into the cosmic censorship hypothesis that, if any physical singularities do exist anywhere in Nature, Nature will always make physics work nicely and politely helpfully hiding the nasty singularities from view.

HOWEVER ... with a cosmological horizon, there are logical arguments that insist that we can receive signals though it.

Suppose that we have two star systems, A and B, whose spatial positions are on different sides of our drawn cosmological horizon, a couple of hundred lightyears away from each other. Let's say that B's the closer star to us – 100 ly inside our nominal horizon – and A's 100 ly outside. In an observerspace projection, we'll eventually be able to see the formation of the nearer star B (if we wait a few bazillion years) but A is off-limits.

But the nearer star B is quite capable of seeing events generated by A, and then helpfully relaying their information on to us. If A goes supernova, we should (eventually) be able to see a cloud of gas near B being illuminated by the flash. B can pass A's signals on, just as an observer at a planetary horizon can see things beyond our horizon and describe them to us, or hold up a carefully-angled mirror to let us see for ourselves.

So technically, Star A, under QM definitions, is a virtual object. It doesn't exist for us according to direct observation, but it's real for nearby observers and we can see the secondary result of those observations. B radiates indirectly through the horizon, so not only does the supposed Big Bang singularity have a masking horizon, the horizon emits Hawking radiation. If we'd bee a bit brighter back in the 1950's, we'd have been able to predict Hawking radiation by taking the "cosmological horizon" case and generalising over to the gravitational case. What stopped us from doing this was an incompatibility with the way that GR1915 was constructed.

The cosmological horizon is an acoustic horizon. It fluctuates and jumps about in response to events both in front of it and behind it. If someone near star A lobs a baseball at star B, we'll eventually see that baseball appear, apparently from nowhere, as a Hawking radiation event. And depending on how close the thrower is to the horizon, and how hard they throw the ball, we might even get a glimpse of their shoulder, as the physical acceleration of their arm warps spacetime (accelerative gravitomagnetism, Einstein 1921) making the nominal horizon position jump backwards.

For this sort of acoustic horizon to work, the acceleration and velocity of an object has to affect local optics (if the ball had been thrown in the opposite direction, we'd never have seen it).
If the local physics at a cosmological horizon generates an acoustic horizon, then that physics is going to correspond to that of an acoustic metric. NOT a static Minkowski metric. The presence, velocity and acceleration of objects must change the local signal-carrying properties of a region. Since the operating characteristics of an acoustic metric are different to those of the Minkowski metric that defines the relationships of special relativity, the local physics then has to operate according to a different set of laws to those of special relativity – the velocity-dependent geometry of an acoustic metric makes the basic equations of motion come out differently. For cosmological horizons to work as we expect, the local light-geometry for a patch of horizon has to be something other than simple SR flat spacetime, and the local physics has to obey a different set of rules to those of special relativity.

Now, the punchline: Since our own region of spacetime will in turn lie on the horizon of some distant far-future observer, this means that if we buy into the previous arguments, our own local "baseball physics", here on Earth, shouldn't be that of special relativity either.

The good news is that if we eliminate special relativity from GR, to force cosmological horizons to make sense, GR's predictions for gravitational horizons would also change. The revised general theory would predict indirect radiation effects through gravitational horizons, bringing the theory in line with quantum mechanics. Which would be a Good Thing, because we've been trying to solve THAT problem for most of the last 35 years.

The bad news is that there doesn't seem to be any polite way to do it. Disassembling and reconstructing general relativity to address its major architectural problems involves going back to basics and starting from scratch, questioning every assumption and decision that was made the first time around, and being pretty ruthless about which parts get to stay on in the final theory.

I find this sort of work kinda fun, but apparently I'm in a minority.

Cutting up Doughnuts

2009-10-16T22:54:00.010+01:00

A cool thing that I didn't know about doughnuts until someone pointed it out a few months back: no matter what proportions a doughnut has, there's always an angle that you can slice though it to produce a perfect pair of interlocking circles.

Someone mentioned this on sci.math, and pointed to "Doughnut Slicing", a webpage by John Banks and Jeff Brooks, and then, someone else pointed out that there was already a a Wikipedia article on it, under the name "Villarceau Circles" ... at which point I scooted off and tried to work the thing out from scratch, before reading anyone else's "spoilers". There's only a certain number of cool results like this, and if you read other people's work before you've had a crack at a problem yourself, that's an opportunity that you never get back.

Anyhow, it turns out that if a torus has major radius R (distance from central axis to centre of limb) and minor radius r (radius of solid limb), the magic angle A that you have to cut at to get to see the double-circle is simply

SIN A = r/R

Going back and looking at the other two webpages, it seems that, unless I missed it, the authors don't seem to have actually written that down explicitly anywhere (although they do seem to have included some more involved math).

So, one quickie download of GFA BASIC 32 (and some quickie trig) later, and the relationship's obviously right. One quick program run while my tea was cooking, generating a few hundred images of tori with radius ratios from zero to one, tilted by the appropriate angles, and I now have a sequence of pretty Villarceau images sitting on my harddrive that I'll probably string together as a YouTube animation at some point.

If you want to cut up a doughnut or bagel purchased at your local bakery to see the Villarceau circles, thread a thin stick or skewer all the way through through the central hole, and then tilt it to a maximum so that your pointy-stick is touching two different parts of the surface. That line gives you the plane that you need to cut along, and the two points where your stick touches the doughnut are the two points where the pair of circles intersect.

Gulliver's Travels, Isaac Newton, and Flying Saucers

2009-10-09T17:36:00.016+01:00

Jonathan Swift (1667-1745) anonymously published his four-part novel, "Gulliver's Travels" in 1726, at the end of a visit to London.

Most people know it for the chapters set in Liliput (where Gulliver is a giant compared to the natives), and maybe also Brobdingnag (where the natives are giants, and it's Gulliver who's considered tiny). It's a scathing social and political farce, where Gulliver's visits to other societies show different systems of government and different social orders. While in Liliput, Gulliver is considered a dangerous giant, and treats the tiny locals with callous indifference. In Brobdingnag it's Gulliver who's overlooked and considered unimportant, so the extent that he's caged and treated as a pet.

But there's also a chapter (at the end) where he visits the Houyhnhnms, a race of talking horses that Gulliver considers entirely superior to humans, who regard the local ape-decended species (the "Yahoos") as loud, primitive, warlike and violent. After living with the Houyhnhnms, Gulliver comes to see all humans as Yahoos.

And for the science fiction fans, there's a chapter about a giant flying saucer.

Really, there is. The third section of the story has Gulliver being rescued by a scientifically advanced society, based on the flying island of Laputa. The city is built on a four-and-a-half-mile-wide concave circular plate topped by buildings, along with four lakes for collecting rainwater, surrounding an astronomical observatory built into a central shaft, that also includes the levitating mechanism. It is, quite literally, a "castle in the air" inhabited by scientists.

Laputa rules over a kingdom (Balnibari), whose borders are defined by the limits of a naturally-occurring geological magnetic anomaly, and the flying city is held aloft by a giant tiltable magnet, held in place by unbreakable "adamant" cage that is of a single piece with the city's baseplate. The city rises and falls and gets sideways propulsion by adjusting the alignment of the magnet.
The flying city is a local scientific superpower, and the king's response to rogue cities below is to steer the saucer above the rebel stronghold and set it down, crushing them.

Unfortunately, I'm not aware of any illustrated editions of "Gulliver's Travels" where the illustrators tackled Laputa. Perhaps the idea was just too freaky for them. If they had, they'd have probably ended up drawing something that looked like the mothership in Spielberg's 1977 "Close Encounters" film.

The other notable thing about Swift's flying city of Laputa is that although it is ordered along entirely scientific principles, its (highly quotable) math-and-music obsessed inhabitants at the Academy of Lagado are buffoons, working on crazy and expensive projects such as the extraction of sunlight from cucumbers, constantly begging for more money for their projects as the society below them decays – it's a fairly small step to suspect that Swift was taking the mickey out of the esteemed Royal Society (then headed by Isaac Newton), and it's even been claimed that Swift emphasised this by basing all of the Lagado projects on specific Royal Society papers.

This raises an intriguing question: did Swift actually meet Newton?
It seems that when Swift had been in London in 1710, he'd been visiting a woman called Catherine Barton. Barton was Newton's half-niece, and one of the few people that Newton was close to. Barton wasn't just some peripheral nominal relative of Newton, she'd actually moved to London and moved in with Newton in about 1696 (about the time he got his job at the Mint), and kept house for him.

If Catherine Barton was living with Isaac Newton and being visited by Jonathan Swift, then Isaac Newton would have cast a rather large shadow over Swift's consciousness, even if he hadn't been /the/ Isaac Newton.

And if that wasn't enough, there was also the subject of Money.
Immediately before "Gulliver's Travels", Swift's celebrity was based on his having anonymously written and published the Drapier's Letters in ~1724-25, a series of pamphlets railing against the coining of copper currency for Ireland, which led to a widespread boycott of the new coins in Ireland and their withdrawal. One of Swift's (many) objections was an allegation that the coins were of poor quality - Newton, as Master of the Mint since the mid-1690's, had to get involved and do an assay, and reported that the allegation wasn't true.
Newton was known for his tetchiness, but Swift in particular had a reputation for being gratuitously and grossly offensive. I've got an old C19th copy of "Gulliver's Travels" that describes Swift as having "more than any other man who ever wrote in English, a liking for saying nasty things", and blames this for Swift's repeated ruination of his own career prospects. Apparently Swift wanted to be a bishop, but even as a returning hero of the Irish people, when the people in charge actually met him, it became clear that this wasn't going to happen. That edition of "Gulliver's" mentions "the deadly agitations of his private life" as being something that the C19th reader might want to enquire about in later life – but whatever this unmentionable personal train wreck was, it doesn't seem to have made it as far as his Wikipedia page.

So perhaps the two wouldn't have wanted to meet each other, especially since they both cared about the same woman. Having the the brittle, acidic, reserved Newton in the same room as the extrovert, scandalous, offensive Swift might not have been a good idea, and the fact that they both had strong ideas about currency would probably just have made things worse.

The young, exiled Francois Marie Arouet ("Voltaire") was also in London around this time, and seems to have been rather keen on Catherine, too.
Voltaire later went on to write "Micromegas" (1752), a short satire that appears to have been partly inspired by "Gulliver", in which a pair of giant aliens from Sirius and Saturn arrive on Earth and meet up with and ridicule a bunch of tiny Earth philosophers (with the exception of one guy who is a follower of John Locke). That's John Locke, the guy whose writings seem to have influenced the American Declaration of Independence, not John Locke, the character from "Lost" (a TV series about a strange island with a natural magnetic anomaly).

Another link between Voltaire's story and Swift's is that both throw in a little casual detail (known to the fictional Laputans and alien scientists) that the Mars was "known" to have two moons, and it seems natural to assume that Voltaire probably borrowed this detail from Swift. In fact, Mars has got two moons – Phobos and Deimos – but they didn't get discovered for real until 1877. That earned both writers an astronomical "credit": the only two named features on the smaller of the two satellites are a pair of adjacent craters, named "Voltaire" and "Swift".

After he'd given up on the brilliant Catherine and snuck back to France, Voltaire shacked up, long-term, with another brilliant woman obsessed with Newton, Émilie du Châtelet, who as well as being a serious respected researcher in her own right, translated, produced and reworked (with her comments) the French edition of Newton's Principia.

To the C18th coffee-house intelligentsia, a mix of physicists, philosophers and political theorists, this was a time of revolution and restructuring (not to mention a certain amount of fluidity over people's living arrangements). England had recently undergone a rapid turnover of rulers, flip-flopping from Monarchy to Republic, and back to Monarchy again, then Monarchy chosen by Parliament. Cromwell had kicked out Charles I, Charles II and James II had taken over from Cromwell, and the Glorious Revolution had then given Paliament the right to choose the monarch, which brought in William and Mary, and which they then exercised again in the Act of Settlement to shunt the succession to Anne, who'd then died, too. The Acts of Union in 1707 had then finally united England and Scotland as a single kingdom. In the politics of 1726 England there were various entrenched factions with specific ideas about how the country ought to be run, and by whom, but there was no guarantee that any one particular group would obtain ultimate control.

There was a sense that this was where we decided what the future was going to look like. Was it going to be run by royalists or republicans? Theologians or scientists? Committees or street campaigners? "Gulliver's Travels" tapped into an appetite for exploring possibilities, and showing how different systems failed. Voltaire's later story got a charge out of lampooning philosophers because at the time, philosophy was reckoned to matter. These guys were potentially the architects of the new society.

"Gulliver" can be seen as parody of how people brought up in different political and philosophical systems can believe that their own way of seeing the world and the correct order of things is right and proper, even when outsiders can see that it's ludicrous, and was, in a real sense, revolutionary. Together with a surrounding body of other philosophical and campaign literature, it helped to set up the context for debate that made the French Revolution and the American War of Independence seem possible to the people who risked their lives to make those things happen.

It's not just a kid's story about a shipwrecked guy being tied down with string by little tiny people.

The Michael Jackson Continuum

2009-10-02T00:39:00.005+01:00

Here's what happens if you take three single images of Michael Jackson - one as a kid (probably some time around "Rockin' Robin"), one during his "Thriller" period, and one when he was doing odd stuff on balconies – turn each one into a FaceGen head, and then use the program's ageing and tweening settings to generate a set of intermediate heads.

The three original heads are top left, centre, and bottom right. The rest are tweened and age-tweeked extrapolations, courtesy of FaceGen.

You should be able to get much better results than this with a more representative set of photos. A more useful source picture taken between the "Jackson Five" years and "Thriller", would have been handy, but the otherwise-usable ones that I turned up on the net all seemed to be in monochrome. :(

"Bottom left" is what FaceGen extrapolates for Michael Jackson as a fifty-something-yearold with no plastic surgery, top right runs the process in reverse, working backwards from the last picture. The rest are intermediates.

Water on the Moon

2009-09-24T20:34:00.008+01:00

In today's Times, there's a front page story saying the the Indian Chandrayaan-1 probe, carrying NASA's Moon Mineralogy Mapper has now found signs of what might be significant amounts of (presumably frozen) water on our Moon.

For anyone who wants bullet points to explain why this is potentially a game-changer, here they are:

Water + electricity = life support
Humans need water and air to survive (along with temperature control). With enough solar cells, the Moon's not short of electrical power – no pesky atmosphere to get in the way – but water and air are biggies. If the water's already there, we can tick one box, and using electricity to electrolyse water gives us hydrogen and oxygen. Oxygen lets us tick the second box. Normally we breath atmospheric-pressure air, with 20% oxygen and nearly 80% nitrogen, but we can use pure oxygen at a lower pressure, if we can deal with the additional fire risk associated with pure O2. It'd be nice to have a decent local supply of nitrogen, too, but not strictly necessary.
Water + heat + rock = building materials?
Use solar furnaces to roast moondust, or break moonrock into pulverised dust and drive off the more volatile elements, then add water ... and we might just have ourselves a form of locally-sourced readymix concrete.

You know how in films where moonbases are often all shiny white metal? To start with, they'd probably look more like adobe mud huts, or holes in the ground, with all the shiny stuff on the inside (apart from the solar panels). What you'd ideally want is big thick walls at least ten or so feet thick, on all sides, to buffer the temperature changes and block some of the radiation when the sun does annoying things with solar flares. Perhaps you'd want to maximise your protection from flare radiation without tunnelling, by by building in the bottom of a deep crater, near one of the poles ... which is also where we're hoping that some of surviving "accessible" ice might be found.

Our building materials don't have to be incredibly strong, or even airtight, we could build a crude hollow blocky mesa as our surface structure and inflate a pressurised mylar balloon inside or below for living quarters. But it'd be nice to be able to pour a bit of concrete around the balloon to minimise accidents, and it'd be handy to turn moondust into something more manageable. Other than that, we're stuck trying to stack up rocks and fill sandbags with dust. In a vacuum. Not good. Quite how you're supposed to work with concrete in a vacuum without the water immediately boiling off, I don't know, but I'm sure that some clever concrete technologists are working on it. Supercooling, perhaps?

One problem with building at the bottom of a polar crater is that having a few kilometres of rock in a straight line between you and the Sun isn't so good for solar power. So you'd probably want an array of thin foil mirrors around set up around part of the crater rim, redirecting and focusing concentrated sunlight down onto your generators. Luckily, your mirrors can be ultra-lightweight, there's no weather to damage them, and no intervening air to soak up the transmitted energy. Using reflectors minimises the amount of heavy power cabling, and also the number of solar generators, and depending on the shape of the ice formation that you're trying to exploit, an aimable solar furnace might also be handy for mining.
Hydrogen + Oxygen = rocket fuel
Hydrogen and oxygen burn rather well together to turn back into water, giving a nice roaring flame. That's the reaction that drives the shuttle's main engines. Given a solar farm and enough time, it'd be nice to have a local fuel production plant on the Moon, making rocket fuel simply from local materials. We'd probably need a robotic refueller to pick up H2 + O2 from the plant, fly back to Earth orbit, find the satellite and fill up its tanks (or swap a standardised empty satellite launch tank with a nice pre-refilled one).
H2 + O2 + fuel cell = mobile power
Fuel cells have a capacity that's only limited by the amount of hydrogen and oxygen you have to feed them. If you're building a water-splitting plant anyway, you might want to send along a spare set of empty fuel cells.
Water+ electricity + rock + atmosphere = food
Sure, we can set up a hydroponics lab to grow our own veggies in space, recycle biomass, and use the plants help remove CO2 and other nasties from the air ... and in theory we can get pretty damned close to a sealed self-perpetuating system. But in practice, you need topups, and safety margins, and an awful lot of water to get the thing started (as the name "hydroponics" kinda suggests). If you're going to be growing algae or fungus or plants to eat, there's a lot of water locked up in the system while they're going through their cycle. Industrial biological reactors usually need whole tanks of the stuff, and water's actually pretty heavy. If water's costing you thousands of dollars per kilo to ship from Earth, it's not cheap stuff. It's probably not quite as expensive as gold, but with current shuttle per-kilo launch costs, it's in the ball-park.

With water, the moon becomes a solar-powered robotically-constructed and remotely-operated gas station and hydroponics plant, remote-controllable from the Earth, with a mild gravity penalty. It can have its own fleet of little refuelling craft, powered by locally-produced lunar rocket fuel.

Without water, its just a big chunk of rock with some handy boulders to hide behind when there's a bad solar storm.

Anyone whose job involves thinking a decade or two ahead about future lunar, manned or deep space payload missions will be watching this story very carefully.

Black Holes, Coordinate Reversals, and r=3M

2009-09-18T09:11:00.005+01:00

Coordinate projections sometimes have a habit of going wierd when you try to project them past a gravitational horizon. Sometimes you can do it, sometimes you can't, and sometimes the attempt turns various things inside out.
A cool physical inversion that happens outside the horizon was used as the March 1993 cover story for Scientific American: Black Holes and the Centrifugal Force Paradox (by Marek Artur Abramowicz).

The effect isn't really paradoxical, but it's counter-intuitive until you think it through. Normally, if you orbit a body, you can break free of that body by firing up your spaceship's engines and going faster – too fast to be able to orbit at your current distance.
What the BHCFP says is that if you're skimming too close to a black hole event horizon, and you fire up your engines, then the faster you try to circle, the more that your trajectory is deflected inwards, towards the hole. The centrifugal forces that would normally throw you away from the body, now seem to be inverted, pointing inwards rather than outwards.

The critical threshold beyond which this effect appears is the distance r=3M, exactly one-and-a-half times the radius of the horizon surface (which is at r=2M).

It turns out that the r=3M radius is the photon orbit. It's the critical distance at which light aimed at 90 degrees to the mass will be deflected enough by gravity to perform a complete orbit and end up at its starting-point. The SciAm article has some nice computer graphics showing what a circular self-supporting scaffolding tube constructed around the hole at r=3M would look like to an observer standing inside it ... it'd appear to be straight, and if the observer pulled out a telescope and looked far enough along the tube, they'd expect to see the back of their own head.

So r=3M is special. From the perspective of the observer at r=3M who's hovering with the aid of rocket engines, or standing in our circular tube up above the hole, the universe seems to be divided into two regions. On one side they see the black hole and its immediate surroundings, and on the other, they see the starfield that represents the outside universe. Topologically, both regions can be thought of as solid spheres, with their external parallel surfaces meeting at r=3M. Both regions are trying to impose their will on the observer's local geometry, but at r=3M, a stationary observer feels the geometrical competition between the effect of the two spheres as being in balance (although in order to maintain their position hovering above the hole, they're feeling rather a strong gravitational pull!). Spin either one of the two spheres, and the observer will be pulled towards it – spin both at exactly the same rotational rate – the effect that we'd see if we passed along the tube at high speed – and the radial gravitomagnetic effects of both spheres cancel.

So if you built an electric train to run around the interior of the tube, it'd feel the black hole's conventional gravitational attraction pulling it against one side of the tube ... but that pull would seem to be exactly the same no matter how quickly it circled the hole.

The author's moral is that if you're in a spaceship close to a black hole, and you want to escape, don't just throttle up your engines, actually point your ship away from the damned thing, or you're liable to get a nasty crashy surprise.

"Observerspace" Description:

When we think about the optics of the situation, though, perhaps the hypothetical spaceship captain wouldn't be all that surprised:

See, if we imagine standing on a suspended non-orbiting platform at r=3m, we find ourselves looking along the r=3M surface in any (perpendicular) direction. The surface appears to us to be a flat plane cutting through our location. And because our view along r=3M circles around the hole indefinitely, our view along this apparent plane repeats indefinitely, too – the plane appears extend indefinitely far in all directions, showing us older and older views of the surface at greater distances, right back to the time that the black hole originally formed. So logically, anything that we see to one side of the plane corresponds to the interior of the r=3M sphere, and everything we see to the other corresponds to the contents of the "rest-of-the-universe" sphere.
The outside universe only seems to exist on one side of this plane. On the other, gravitational lensing effects make the black hole's r=2M surface beneath us appear to be opened out into a second indefinitely-repeating surface, at some distance below the 3M plane.

Once we're at the 3M surface, there are two ways that we can go.
If we slowly winch ourselves upwards away from the hole, then we see the flat 3M boundary of the outside universe curving itself back into a more normal-looking inward-facing enclosing sphere. But if we allow ourselves to be lowered further towards the black hole, to less than r=3M, then the 3M surface continues to distort past being a flat plane, to becoming a concave surface that curves above us, away from the hole. Instead of the universe surrounding the black hole, it now seems to us that the black hole (and the r=3M surface) is surrounding the universe!
The region that we know ought to be just above the 2M surface appears visually to us to be part of a concave shell, apparently wrapped around a ball representing the remaining universe. The abstract, "topological" idea that our location can affect the choice of which sphere is "really" on the inside or outside now appears to us, visually, to be concrete reality!

The further we descend (slowly) towards 2M, the more pronounced the effect becomes, the more sharply the 2M surface appears to be curved around us, and the more that the outside starfield above appears to shrink to something that looks like a little bright ball suspended somewhere above, in the enveloping black-holey gloom directly above us, like a tiny planet or star.

So if we're hovering too close to r=2M, (or flying past in a spaceship) we shouldn't really be surprised if increasing our forward speed results in our colliding with part of the hole, because that's exactly what our forward view tells us is directly in front of us (and on every side, and directly behind us). If we want to escape from the hole's influence and get back to normal space, then we have to aim our spaceship at the little shrunken blob of compacted blueshifted starfield directly above us. All other directions point at the black hole.

So the rule-of-thumb for navigating within r=3M would seem to be: forget about your ship's fancy gyroscopic navigation systems, just look out of your window and make sure that the ship's nose appears to be pointed approximately at the part of the universe that you want to go to. But don't take your eyes off the forward view, because the harder your engines fire on your way out, the the stronger those distortion effects are going to become.

My Chocolate Tree is Unhappy

2009-09-16T16:33:00.006+01:00

I keep chocolate trees. They're not too difficult to grow (if you set up an incubator), but keeping the things alive as houseplants without a controlled environment can be tricky. They generally do okay until you have One Bad Day with light levels that are too bright, or too dim, or the humidity's too low, or the temperature is too hot or too cold, and the things panic and drop all their leaves and turn into ugly bare sticks. And when that happens, it seems to take about eight months to coax the things into producing more proper leaves, and get back into the swing of things. Maybe it's a way of outliving predators - if any beasties have eaten the last set of leaves, the tree waits until they and their offspring have all starved to death before growing any more. Dunno.

I had two gorgeous bushy indoor trees last year, sitting by the back window, and moved them to the front of the house where the light levels were slightly lower. One day later, all the leaves had gone sickly. A day or so later they all fell off. A couple of earlier trees got trashed by a few hours of unusually harsh UV light on one clear winter's morning.

After a number of house-moves, I'm now down to just one small tree, which is only about a year old. It had a nice cluster of healthy dark-green leaves. But after just one hour's car journey (on a fairly hot day), the thing had virtually turned albino. The leaves went almost white, apart from the veins, and it's been struggling ever since. Once a leaf loses its "green", it's one short step away from dying completely, and going brown and falling off, and when all the leaves fall off, you're in trouble.

So what I have to do now is coddle the thing so that the existing leaves hopefully last until the plant has decided to try cautiously growing some new ones.
Maybe I should switch to growing something less challenging. If I used a set of mirrors to catch and redirect daylight around the room, indoor climbing roses would be nice ...

Dark Stars and Hawking Radiation

2009-09-11T02:15:00.001+01:00

Some people have trouble getting used to the idea of Hawking radiation outside the context of strict quantum mechanics. For those people, I'd suggest that they consider the mechanics of a crusty old Nineteenth-Century “Dark Star” model.

The Dark Star was the predecessor to the modern black hole, and the basic properties of the object were worked up and published by John Michell back in 1784. Michell worked out many of the “modern” Twentieth-Century black hole properties from Newtonian principles, including the r=2M event horizon radius, gravitational spectral shifts, and a method of calculating the number of these “invisible” gravitationally-cloaked objects by finding the proportion of unseen “companion stars” in binary star systems, and then using statistics to extrapolate that proportion to the larger stellar population.

The main difference between an old “dark star” and John Archibald Wheeler's 1950's-era “black hole” was that dark stars could emit faint traces of indirect radiation. In theory, signals and particles could still migrate upstream out of the dark star's gravitational trap by using local objects as accelerational stepping-stones, whereas under GR1915, this mechanism couldn't exist – objects smaller than their r=2M event horizon radius weren't just incredibly dark, but totally black. Their signals and radiation-pressure signature weren't just absurdly faint, but entirely missing. The thing really was, as Wheeler memorably described it, a truly black "hole" in the surrounding landscape.

From the perspective of the Twenty-First Century, we can describe the difference in another way: dark stars emit classical Hawking radiation and GR1915 black holes don't.
Some people will take issue with that statement. They'll say that a hypothetical dark star's radiation-pattern is about acceleration effects rather than QM, and that Hawking radiation is all about particle-pair-production, a completely different mechanism.

So here's the sanity-check exercise. Suppose that the GR1915 description of horizon behaviour was wrong, and that a more "dark-starry" description was right … but that we still believed in GR1915. More general approaches (like statistical mechanics) would have to insist that the radiation effect was real, even though GR1915 disagreed. So how would we explain the reappearance of our naughty radiation effect?

There are number of stages we'd have to go through:

In a thought-experiment, catch an escaped particle and measure its trajectory.
Extrapolate that trajectory back to the originating body as a smooth ballistic trajectory. In our "dark star" scenario, this extrapolated trajectory is wrong – the particle only escaped by being "bumped" out of the gravitational pit by interactions with other bodies or radiation – but in our GR1915 description there's no self-supporting atmosphere outside the black hole to allow this sort of acceleration mechanism, so we have to (wrongly) assume an unaccelerated path.
Notice that the earliest part of this (fictional!) escape-path is superluminal. In order to escape along a ballistic trajectory, a particle would have to have started out travelling at more than the speed of light (!).
Apply coordinate systems. Using a distant stationary observer's coordinates, we break the fictitious trajectory into two parts, an initial superluminal section, and the later, legal, sub-lightspeed part of the calculated path. The first section appears to be off-limits in our coordinate system, and an orderly transition between the two, as the particle supposedly jumps down through the lightspeed barrier seems impossible, but …
… then we then notice that in a very idealised description of a superluminally-approaching particle, the particle ends up described as time-reversed ("tachyonic" behaviour). If an (over-idealised) particle approaches at more than the speed of its own light (which shouldn't normally happen, but ...), we'd end up describing it as being seen to arrive before it was seen to set out. Our artificial coordinate system approach then describes the particle as being seen to originate at the nearest part of its path, and to be apparently moving away from us at sub-light speeds, as its earlier signals eventually arrive at our location in reverse order.
Time-reversal counts as a reversal of one dimension, which flips a left-handed object into its right-handed twin, and vice versa (chiral reversal). So if our particle was an electron, this artificial approach would describe the earlier part of its supposed path as belonging to a positron, instead.
Our final description would then say that a particle and its antiparticle both appeared to pop into existence together outside the horizon (from nowhere) and moved in opposite directions, with the "matter" particle escaping and being captured by our detector, and its "antimatter" twin moving towards the black hole to be swallowed.

And this is, essentially, the 1970's QM description of Hawking radiation.

The Moon, considered as a Flat Disc

2009-09-06T23:36:00.007+01:00

Mathematics doesn't always translate directly to physics.
That statement might sound odd to a mathematician, but consider this: even if you believe that physics is nothing but mathematics, that makes physics a subset of mathematics ... which means that there'll be other mathematics that lies outside that subset, that doesn't correspond cleanly to real-world physical theory. The key (for a physicist) is to know which is which.

That's not to say that "beauty equals truth" isn't a good working assumption in mathematical physics – it is – the problem is that the aesthetics of the two subjects are different, and mathematical beauty doesn't necessarily correspond well to physical truth. The physicist's concept of beauty is often different to that of the mathematician.

The "beauty equals truth" idea is often used as an argument for special relativity. SR uses the Lorentz relationships, and to a mathematician, it can sometimes seem that these are such beautiful equations that a system of physics that incorporates them has to be correct.

But the Lorentz relationships can also appear in bad theories, as a consequence of rotten initial starting assumptions:
Our Moon is tidally locked to the rotation of the Earth, so that it always shows the same face to us, and we always see the same circular image, with the same mappable features. Now suppose that a 1600's mathematician has a funny turn and decides that it's so outrageously statistically improbable that the moon would just coincidentally just happen to have an orbit that results in it presenting the same face to us at all times, that something else is going on. Our hypothetical "crazy mathematician" might decide that since we always see the same disc-image of the Moon, that perhaps, (mis)applying Occam's Razor, it really IS a flat disc.

Our mathematician could start examining the features on the Moon's surface, and discover a trend whereby circular craters appear progressively more squashed towards the disc's perimeter. We'd say that this shows that we're looking at one half of a sphere, but our mathematician could analyse the shapes and come up with another explanation. It turns out that, in "disc-world" the distortion corresponds to an apparent radial coordinate-system contraction within the disc surface. For any feature placed at a distance r from the disc centre, where R is the disc radius, this radial contraction comes out as a ratio of 1 : SQRT[1 - rr/RR ] .

In other words, by treating the Moon as a flat disc, we'd have derived the equivalent of the Lorentz factor as a ruler-contraction effect! :)
Our crazy mathematician could then go on and use that Lorentz relationship as the basis of a slew of good results in group theory and so on. They could argue that local physics works the same way at all points on the disc surface, because the disc's inhabitants can't "see" their own contraction, because their own local reference-rulers are contracted, too. Our mathematician could arguably have advanced faster and made better progress by starting with a bad theory! So "bad physics" sometimes generates "good" math, and sometimes the worse the physics is, the prettier the results.

The reason for this is that, sometimes, real physics is a bit ... boring. If we screw physics up, the dancing pattern of recursive error corrections sometimes generates more fascinating structures than the more mundane results that we'd have gotten if we simply got the physics right in the first place.

Sometimes these errors are self-correcting and sometimes they aren't.
If we considered the Earth as flat, then, because it's possible to map a flat surface onto a sphere (the Riemann projection), it'd still be theoretically possible to come up with a complete description of physics that worked correctly in the context of an infinite rescaled Flat Earth. We'd lose the inverse square law for gravity, but we'd gain some truly beautiful results, that would allow, say, a lightbeam aimed parallel to one part of the surface to appear to veer away. We'd end up with a more subtle, more sophisticated concept of gravitation than we'd tend to get using more "sane" approaches, and all of those new insights would have to be correct. In fact, studying flat-Earth gravity might be a good idea! We'd eventually end up deriving a mathematical description that was functionally identical to the physics that we'd get by assuming a sphericial(ish) Earth ... it'd just take us longer. Once our description was sufficiently advanced, the decision whether to treat the Earth as "really" flat or "really" spherical would simply be a matter of convenience.

But with the "moon-disc" exercise, we don't have a 1:1 relationship between the physics and the dataset that we're working with, and as a result, although the moon-disc description gets a number of things exactly right, the model fails when we try to extend it, and we have to start applying additional layers of externally-derived theory to bring things back on track.
For instance, the "disc" description breaks down at (and towards) the Moon's apparent horizon. For the disc, the surface stops at a distance R from the centre, and there's a causal cutoff. Events beyond R can't affect the physics of the disk, because there's no more space for those events to happen in. The horizon represents an apparent causal limit to surface physics. But in real life, if the Moon was a busier place, we'd see things happening in the visible region that were the result of events beyond the horizon, and observers wandering about near our horizon would see things that occur outside our map. So if we were to use statistical mechanics to model Moon activity, and were to say that the event-density and event-pressure have to be uniform (after normalisation) at all parts of the surface, then statistical mechanics would force us to put back the missing trans-horizon signals by giving us "virtual" events whose density increased towards the horizon, and whose mathematical purpose was to restore the original event-density equilibrium. In disc-world, we'd have to say that the near-edge observer sees events in all directions, not because information was passing through (or around) the horizon, but because of the disc-world equivalent of Hawking radiation.

So in the disc description, the telltale sign that we're dealing with a bad model is that it generates over-idealised horizon behaviour that can't describe trans-horizon effects, and which needs an additional layer of statistical theory to make things right again. In the "moon-disc" model, we don't have a default agreement with statistical mechanics, and we have to assume that SM is correct, divide physics artificially into "classical" and "quantum" systems, and retrofit the difference between the two predictions back onto the bad classical model – as a separate QM effect, as the result of particle pair-production somewhere in front of the horizon limit – to explain how information seems to appear "from nowhere" just inside the visible edge of the disc.

Clearly, in the Moon-disc exercise this extreme level of retrofitting ought to tell our hypothetical crazy mathematician that things have gone too far, and suggest that the starting assumption of a flat surface was simply bad ...
... but in our physics, based on the early assumption of flat spacetime, and generating the same basic mathematical patterns, we ran into a version of exactly the same problem: Special relativity avoided the subject of signal transfer across velocity-horizons by arguing that the amount of velocity-space within the horizon was effectively infinite (you could never reach v=c), but when we added gravitational and cosmological layers to the theory, the "incompleteness problem" with SR-based physics showed up again. GR1915 horizons were too sharp and clean, and didn't allow outward flow of information, so to force the physics to obey more general rules, we had to reinvent an observable counterpart to old-fashioned transhorizon radiation as a separate quantum-mechanical effect.

So the result of this sanity-check exercise is a little humbling. We can demonstrate to our hypothetical 1600's "crazy mathematician" that the Moon is NOT flat, no matter how much pretty Lorentz math that generates, and we can use the horizon exercise to show them that their approach is incomplete. By assuming that their model is wrong, we correctly anticipate the corrections that they'd have to make from other theories in order to fix things up. That ability to predict where a theory fails and needs outside help is the mark of a superior system, and shows that the "Flat-Moon" exercise isn't just incomplete, it generates results that are physically wrong, and that don't self-correct. It's faulty physics.

But the same characteristic failure-pattern also shows up in our own system, based on special relativity. So have we made a similar mistake?

On Catching Rainbows

2009-09-02T23:06:00.005+01:00

I saw a nice rainbow yesterday.

I was out to do some shopping but took a random detour, following my feet. The detour just happened to take me to a suitable road junction, at exactly the right time. By rights, I shouldn't have been there to take the picture.

But "lucky catches" aren't just about accidentally being in the right place at the right time by nothing but dumb good luck, or about preserving a certain random element in your approach (although that certainly helps). If you want to be able to catch something that other people miss, you have to expect to spend at least some of your time in places where they aren't, and looking at things that don't always seem to be immediately necessary to the job in hand.
You also have to be prepared for the possibility of success (I try to keep a camera with me, and it had just enough juice left in the batteries to fire off a few shots for the critical sixty or seventy seconds), you have to be able to recognise the preliminary signs of something interesting (I saw a faint 'bow forming, realised what was coming, and was able to fish the camera out and find something to shield it from the rain, in time) and you also have to be prepared to look stupid (standing in the rain with a plastic folder over your camera, taking photos of the sky, at an angle where most of the people who can see you have no idea what you're doing).

But the main thing is to have your eyes open. If you're absolutely sure that nothing interesting is going to happen, then on the occasions when it does happen, you're liable to miss it.

The same thing goes for theoretical physics. If you want to catch things that have eluded other people (whether it's math, or theory, or experimental research), you don't always have to be so much smarter than everyone else, or to have better equipment. Sometimes it's enough just to be prepared for the possibility of being surprised. If you're too rigid about what you're trying to find, you miss out. In my case, I was popping out for a plank of wood for some shelving, and I came back with a plank of wood and a bloggable photograph of a rainbow. If I'd been more singleminded in my shopping, I'd have only come back with the bit of wood.

M.C. Escher's "Relativity", Intransitivity, and the Pussycat Dolls

2009-08-29T15:16:00.001+01:00

There's a nice example of intransitive geometry in the latest Pussycat Dolls video ("Hush hush").
No, really, there is. It's the bit where the girls are on four staircases attached to the sides of a cube, that each have a different local direction of "down". The "stairwell" section of the video starts at about 58 seconds in and goes on until about a minute thirty. While you're waiting for it to start you'll have to put up with the sight of Nicole Scherzinger nekked in a bathtub making "ooo, yeah" noises for nearly a minute, though. Sometimes doing research for this blog is really tough.

The video seems to be inspired by the famous "Relativity" lithograph by M. C. Escher, which had three intersecting sets of stairs and platforms set into three perpendicular walls, as a piece of "impossible" architecture (physically you could build it, but you wouldn't be able to walk on all the surfaces as the people do in the illustration).Escher's illustration was incredibly influential, and as well as the Pussycat Dolls video (!), there are some more literal tributes online, including Andrew Lipson's recreation of the scene using Lego, part of the 1986 movie Labyrinth, and a funny short video called Relativity 2.0, that has people trapped in a nightmarish Escherian shopping mall.

If you know of any other especially good ones, please add them to the end of this post as a comment!

Next, we need a Beyonce video illustrating the event horizon behavour of acoustic metrics ...

Special Relativity is an Average

2009-08-22T23:44:00.003+01:00

Textbooks tend to present special relativity's physical predictions as if they're somehow "out on a limb", and totally distinct from the predictions of earlier models, but SR's numerical predictions aren't as different to those of Nineteenth-Century models as you might think.

One of the little nuggets of wisdom that the books usually forget to mention is that most of special relativity's raw predictions aren't just qualitatively not particularly novel, they're actually a type of mathematical average (more exactly, the geometric mean) of two earlier major sets of predictions. So, in the diagram above, if the yellow box on the left represents the set of predictions associated with the speed of light being fixed in the observer's frame (fixed, stationary aether), and the red box on the right represents the set of physical predictions for Newtonian optics (traditionally associated with ballistic emission theory), then the box in the middle represents the corresponding (intermediate) set of predictions for special relativity.

If we know the physical predictions for a simple "linear" quantity (visible frequency, apparent length, distance, time, wavelength and so on) in the two "side" boxes, then all we normally have to do to find the corresponding central "SR" prediction is to multiply the two original "flanking" predictions together and square root the result. This can be a really useful method if you're doing SR calculations and you want an independent method of double-checking your results.

This usually works with equations as well as with individual values.
F'rinstance, if the "linear" parameter that we were working with was observed frequency, and we assumed that the speed of light was fixed in our own frame ("yellow" box), we'd normally predict a recession Doppler shift due to simple propagation effects on an object of

frequency(seen) / frequency(emitted) = c / (c+v)

, whereas if we instead believed that lightspeed was fixed with reference to the emitter's frame, we'd get the "red box" result, of

frequency(seen) / frequency(emitted) = (c-v) / c

If there was really an absolute frame for the propagation of light, we could then tell how fast we were moving with respect to it by measuring these frequency-shifts.

The "geometric mean" approach eliminated this difference by replacing the two starting predictions with a single "merged" prediction that we could get by multiplying the two "parent" results together and square-rooting. This gave

frequency(seen) / frequency(emitted) = SQRT[ (c-v) / (c+v) ]

, which is what turned up in Einstein's 1905 electrodynamics paper.

The averaging technique gave us a way of generating a new prediction that "missed" both propagation-based predictions by the same ratio. Since the numbers in the "red" and "yellow" blocks already disagreed by the ratio 1: (1- vv/cc), the new intermediate, "relativised" theory diverged from both of these by the square root of that difference, SQRT[ 1 - vv/cc ]. And that's where the Fitzgerald-Lorentz factor originally came from.

---==---

Why is it important to know this?

Well, apart from the fact that it's useful to be able to calculate the same results in different ways, the "geometric mean" approach also has important implications for how we go about testing special relativity.
Our usual approach to testing SR is to compare just the the "yellow" and "orange" predictions, identify the difference, say that the resulting differential Lorentz redshift/contraction component is something unique to SR and totally separate from any propagation effects, and then set out to measure the strength of this relative redshift/contraction component, in the range "zero-to-Lorentz". Having convinced ourselves that these effects are unique to SR, we usually don't then bother to check whether the data might actually make a better match to a point somewhere to the right of the diagram.
Since the "yellow box" predictions are so awful, special relativity comes out of this comparison pretty well.

But once you know the averaging method, you'll understand that this is only half the story -- these "derivative" effects that appear under SR but not "Classical Theory" ("orange" but not "yellow") must have counterparts under Newtonian optics ("red"), and these are usually stronger than the SR versions. So any experimental procedure or calculation that appears to support the idea of time dilation or length-contraction in an object with simple constant-velocity motion under SR would also generate an apparent positive result for those effects if SR was wrong and the older "Newtonian optics" relationships were the correct set (or if some other intermediate set of relationships was in play). We can say that special relativity's concept of velocity-based time dilation didn't exist under NO, but hardware doesn't care about concepts or interpretations, only results ... and the result of performing an SR-designed test in an "NO universe" would be that the test would throw up a "false positive" result apparently supporting SR (with an overshoot that'd then have to be calibrated out).

And, actually, the situation is worse than this.
... Since the "yellow" and "red" blocks represent the two extremal predictions for theories that allow linkage between the velocity of a light-signal and the motion of a body ("yellow" = zero dependency, "red" = full dependency), they also seem to represent the cutoff-limits for a whole slew of old Nineteenth-Century "dragged aether" models, all of which would be expected to produce similar physical effects to special relativity, differing only in their scaling and strength. So typical test procedures designed to isolate the "new" SR effects should be able to generate "false positive" results with almost all of these old theories and models.

While some of special relativity's concepts might have been new, its testable numerical predictions lie right in the middle of a pre-existing range. Any time you see a claimed experimental verification of SR that forgets to take this into account, treat it with caution.

Fibonacci Kitchenware (well, almost)

2009-08-17T11:39:00.005+01:00

I popped into Habitat yesterday, and they're selling a range of five pseudo-Fibonacci nesting trays (four smaller trays plus a bigger one to hold them). It's just a shame that they chose such and awful selection of colours for them (who the heck decided on yellow, brown and navy blue??!?).