John Milton, 'Paradise Lost', and General Relativity

John Milton (1608-1674) was a linguist, pamphleteer and poet, nowadays best remembered for having written Paradise Lost, first published in 1667.

England in 1667 had been experiencing decades of social upheaval, and an accelerating succession of crises. Oliver Cromwell's side had won the civil war, abolishing the monarchy and executing Charles I in 1649, but Cromwell had then died in 1658, and without Cromwell as a driving force, Parliament had decided to restore the Monarchy, with Charles II being appointed the new king in 1660. Milton had campaigned for religious reform, written campaign material for Cromwell and the Republic, and had held a post in the new regime before going blind. With the Restoration, Milton briefly became a wanted man, and his books were publicly burnt. As England was coming to terms with the abrupt political reversion, it got hit first by the Great Plague of 1665, and then by the Great Fire of London [*] [*] in 1666.

These were, as the saying goes, interesting times.

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Paradise Lost, Milton's masterwork, was an epic poem, originally in ten sections, that outlined the rebellion and fall of Lucifer and the subsequent fall of Adam and Eve, in the contemporary poetic equivalent of high-definition widescreen. Milton had something of a talent for visual imagery and a good turn of phrase, and some commentators later ruefully pointed out that Milton seemed to have been influenced by his experiences with the ill-fated English revolution, and not only made the rebellious Lucifer a sympathetic character, but given him some of the the best lines. "Better to reign in hell than serve in heaven", indeed!

The mind is its own place, and in itself
Can make a heaven of hell, a hell of heaven

There are some great phrases in the poem – when we talk about "the fabric" of spacetime, we're arguably borrowing from Milton – but to a physicist, one of the most surprising sections (along with his name-dropping of Galileo) is the bit where Milton writes :

... whether heavn move or earth
Imports not, it thou reckon right.

To a historian, those lines might be taken as an assertation of independence, and a rejection of the notion of centralised supreme power. There's no single authority that decides what is really "moving" and what isn't. No church, no Pope, no religious leader, no monarch.

But to a physicist, they set out the general principle of relativity. When the Earth spins on its axis and loops along its orbit around the Sun, it's convenient to think of the Earth as moving and the background starfield as fixed. But in reality, there's no "special" status accorded to those stars. They're just stars, each following their own line of least resistance as they drift in the wash of their own individual surrounding gravitational tides and currents. It doesn't matter whether we say that the Earth rotates beneath Heaven, or that Heaven rotates around the Earth. If you calculate properly, (said Milton), the answers should be the same in either case.

And if they're not, you've done it wrong.

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While Milton was getting his piece finalised and published in 1665-67, the plague had other consequences. Cambridge University sent its students home in the fall of '65, and one these, a previously unremarkable and undistinguished student, chose to make use of the two years of enforced comparative isolation back at his family's farm in the village of Woolsthorpe to work through some ideas on gravity, optics and mathematics. His name was Isaac Newton.

My Favourite 3D Fractal

It's this one:

It reduces to the Mandelbrot Set on one plane, and to the "Tricorn" or "Mandelbar" fractal on another. And there's a stream of self-similar solids emanating from the main pointy bit.

I don't only like this fractal because I don't know of anyone else having found it before me ... okay, perhaps that is a contributing factor ("It's MY fractal, Miiiiiine!"), but there are other fractal solids in the same series and I don't like those nearly so much. And according to my YouTube stats, neither do other people.

Now, if only I had access to a high-resolution 3D printer ...

Edible 3D Fractals

If you're into fractals, you've probably already heard of the Romanescu.

It's a relative of the cauliflower, and it produces a spiralling cone made up of smaller spiralling cones, which in turn are made of of smaller ... well, you get the idea. Imagine a pinecone, where each segment is a protruding mini-pinecone, composed of segments that are also mini-mini-pinecones, and you have some idea of what a romanscu looks like, up-close.
Oh, and they're a livid shade of green.

If M.C. Escher had designed a vegetable, this would be it.

You might now be tempted to track down lots of awfully pretty artsy photos of romanescu on the net. I'd advise you not to bother. Instead take a trip to your local chi-chi supermarket and buy some, for real.
I got a pair for about one pound fifty, at the local Waitrose. They're awfully cute, they're much more fun to stare at in real life than someone else's photo, and when you eventually get bored with them, you can eat them!

Steam them, or wet them, cover them and shove them in the microwave for a couple of minutes.
They taste a little like cauliflower. Cheese sauce is optional.

Summer Glau

Here's an odd thing. I added a "Summer Glau" head to the collection at relativitybook.com, and it struck me ... if anyone runs this software, their PC will be pretending to be an actress, who is pretending to be a robot, which is pretending to be a human.

You know that you've reached a certain level of technology when things start to get this complicated ...

Curvature is Important

Curvature allows us to comprehend views of reality that can't otherwise be seen, or appreciated without an understanding of a few basic principles. Curvature allows connections and interrelationships and juxtapositions that you may find it impossible to see if you don't have the necessary mind-set.

This doesn't just apply to theoretical physics, mathematics and abstract logical structures. it also applies to real life.

Let's suppose that we're signwriters, and we have a famous department store as a client. They'd like an impressive curved sign over their main entrance, proudly displaying their name. Wouldn't it be awful if we neglected to take into account how that curved set of letters looked from different angles, and accidentally built a sign that said a Very Rude Word?

At this point you're probably remembering the fictional Great Big Sign in Douglas Adams' "Hitchhiker" series ... the one built by Sirius Cybernetics that when collapsed to half its original size, spelt out the message "Go stick your head in a pig" ... you're probably thinking that I'm about to describe some tortured hypothetical example that would never really happen in real life ... some crazy laboured combination of store name and typeface and sign geometry that would be so improbable that it'd never ever happen.

And so, sceptical reader, I invite you to examine this real-life department store sign:

It's for a store called T J Hughes. Naturally, above the store's entrance we see the words T J HUGHES proudly displayed, in large red capitals. It's on a corner, and the letter sequence follows the curve.

If we turn the corner, cross the road, and look back at the sign, we still see the final “S” facing us, and to its right we see in white, slightly shrunken by perspective, the reversed white backsides of the letters H, J and T. Unfortunately, the letter J is very narrow, and the curved base of the letter is out of sight. So the J looks like an I, and although the H and T are seen reversed, they're symmetrical and still look like a perfectly normal H and T.

At this point you should be able to take a wild guess at the problem.

Here's the photo:

That's right. Seen from the right, the sign above their storefront really does say

Unfortunate, no?

This isn't a doctored photograph. The shop is real, and the sign has been there for an awfully long time. Here's the store's website, its location on Google Maps and their wikipedia entry. This really happened.

Like the title says, curvature is important. Ignore it at your peril.

Cool Circley Fractal!

While playing about with some Mandelbrottish code, I've managed to find a fractal that looks like it's entirely made out of circles:


Instead of having Mandelbrots that spawn "mini-Mandelbrots", it's got circles spawning "mini-circles".

Yaaay Me!

I may have to adopt a version of this as my new logo ...

Mathematics, Certainty, and the Mathematical Impossibility of the Saturn V

Back in the early Twentieth century, after Konstantin Tsiolkovsky had initially tried to get people interested in the idea of liquid-fuelled rocketry, the story goes that an elite mathematical society decided to study the problem, and decided that LFRs were impractical. They didn't just decide that LFRs were unlikely to to be useful to launching payloads into space, they proved mathematically that such a device wasn't workable. Or so they thought.

The argument was nice. It said, to launch a rocketship to any given height requires a certain amount of fuel, and with an LFR, a certain amount of additional infrastructure (fuel tanks, pumps, piping, the rocket engine itself, etc.). The more infrastructure you have, the more weight you have to carry, and the more “dead weight” you end up with at the end. If you want to get higher, you have to carry more fuel, which means that you start with more weight, which in turn means that you have to carry even more fuel to compensate.


Solid-fuelled rockets were simpler. They were basically big fireworks, and when you used solid fuel, the “fuel problem” wasn't intractable. If you calculated “weight of fuel required plus weight of payload” for a given height, it was solvable. It helped that the weight of the fuel being carried was constantly reducing as the flight progressed. With a solid rocket, you might start off with a monstrous amount of rocket propellant on the launchpad, but by the time you got into space you had something much more lightweight.

With a liquid-fuelled spaceship (it was argued) the equation was different. Once again, the higher you wanted to go, the greater the amount of fuel you had to carry ... but with a liquid-fuelled craft, all that propellant had to be carried in tanks, and fed through pipes. Although the tankwork became more efficient as you built larger tanks, the total weight of tankange and pipage was always greater for a larger craft, and since this weight had to be subtracted from the final payload weight, you found that beyond a certain height, your payload was effectively 100% infrastructure.
If you wanted to lift a person into space, with accompanying life-support, liquid rocketry obviously wasn't the way to go. It was an impractical idea, and funding liquid fuelled rocket research was clearly a waste of time.

A few decades later, when we landed on the moon, we got there using LFRs.

Where Things Went Wrong

When Apollo 11 went to the Moon, NASA didn't attempt to lift the entire Saturn V rocket into space. That would have been silly. They took a rocket that could lift a heavy payload pretty high, and instead of enlarging it, they then sat a second rocket on top of it. The payload for the first rocket was another rocket. Rocket #2 was smaller, was launched from the new height (with a good starting velocity), and didn't have to carry the dead weight of rocket #1, which simply fell back and crashed into the sea. The Saturn V had multiple stages, with only the last stage(s) leaving Earth with their final payload.

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The mathematicians would probably say that they didn't do anything wrong. Their calculations were exactly correct for the problem that they were given. The problem was that the question wasn't asked carefully enough ... or rather, that they'd asked a version of the question that was too careful, phrased in such a way as to be easily solvable and to give a definite answer.
And by doing that, they got an answer that was simple, straightforward, provably correct ... and physically wrong.

There are three main lessons here:

1. The "test particle" fallacy

Firstly, the physical behaviour of compound systems is sometimes quantatively or even qualitatively different to the results that we predict from simply studying individual components in isolation. The limits of a two-rocket system aren't the same those of a one-rocket system. If we take a “test particle” approach to physics, and derive a set of laws based on the idealised behaviour of single non-interacting idealised objects, and then turn those predictions into elegant mathematics and beautiful self-contained geometrical models, it doesn't mean that the answers given by those models are going to be correct. We can have beautiful, rigorously-derived geometry and mathematics that allows just one solution, and that provably has zero errors in any of its derivations, and creates breathtakingly elegant resonances across multiple fields of mathematical theory ... and physically, it it can still be quite wrong.

This isn't always appreciated.

2. Physics vs. mathematics

Secondly, mathematics is not physics. It might well be that "all physics is mathematics", but that can mean that physics is a subset of mathematics, rather than the thing itself ... physics is obliged to correspond to reality, while mathematics is not, so the two disciplines aren't automatically interchangeable. Modern physics is now so strongly math-based that researchers can spend their lives learning to manipulate the textbook mathematical machinery, without necessarily realising that the resulting answers aren't guaranteed to be physically meaningful. These guys are liable to tell you that "You can't argue with the math", but sometimes, if you're a physicist, your ability to argue with (and occasionally overturn) mathematically-proven results is what makes you worth your salary and your job title. There are situations where not understanding the importance of disputing math, regardless of the apparent strength of the proofs, means that perhaps you haven't really understood the idea of physics at all.

3. Proof vs Certainty

Thirdly, in physics, there's a sometimes a tradeoff between calculability and correctness. Sometimes the things that you do to a problem to make it well-defined and easily modellable destroy delicate-but-critical characteristics of the original problem. Instead of a "correctly vague" answer to an "indistinct" problem, you then end up with an unambiguous answer to a well-defined problem .. that doesn't actually correspond to the thing that you're trying to model.

In a worst-case scenario, a desire to be able to definitively solve a problem can, if your tools aren't up to the job, lead to a process of successive approximation that converges more and more definitely on an answer that's emphatically wrong. In everyday situations a physicist will use common sense to ignore answers that obviously aren't right, but when we're working at the edge of known theory, the selection process becomes more dangerous.

If we're not careful, all we end up doing is generating mathematically rigorous retrospective justifications for whatever it is that we already happen to believe. But what what we believe is based on our cumulative experiences to date. Our current belief system is a logically-perfect inverse projection of an imperfect dataset, designed to recreate the particular set of rules that we inherited from the generation before us.

And it's wrong

Putting Periscopes on Airplanes

One the subject of airplanes, a wacky idea I had about twenty years ago was designing aircraft cockpits to include periscopes.

A dumb idea? Not necessarily.

See, one of the problems with large aircraft is taxi-ing. Aircraft are designed to fly. They aren't really designed to be driven. So if you're in the cockpit of a 747, and you want to park it somewhere, you can't really see what you're doing. You're high up, your visibility sucks, and you have this nose-cone thing sticking out in front of you, guaranteeing that you have no chance of seeing what's just in front of your wheels. Consequently, expensive aircraft get trashed from time to time while they're still on the ground, while somebody is simply trying to find somewhere to park them.

What you need in this situation is a periscope – not to look up, but to see how things look from beneath the aircraft. You want to be able to pull a lever, and have a chunk of optics pop out of the bottom of the aircraft giving you a 180-degree or 270-degree view, relayed up to the cabin and perhaps projected onto a curved mirrored trim just below the windscreen. Worried about debris on the runway? Check the periscope. Trying to park? Check the periscope. Coming in to land, and not sure if your landing gear is deployed? Check the periscope. Unsure if one of your engines has just flamed out? ... you get the idea.

Consider the case of Concorde. One of the most difficult engineering tasks in designing Concorde was supposedly the design of the nose. Concorde has a looong nose, and it needs to be pointy and smoothly tapered for efficient supersonic flight. But when Concorde comes in to land, it glides in at an angle with its nose in the air, and the pilots can't see where they're landing. To get around this the engineers developed a "droop snoot" for the plane – an entire nose section that could swivel to point downwards when the plane landed, to give the pilots a fighting chance of seeing what they were doing. This was a difficult bit of engineering, with double windshields and so on.

Wouldn't a pop-down periscope system have been simpler?

Okay, so nowadays the idea's probably becoming a bit redundant. With recent aircraft, with their instrumentation displayed on LCD panels, it's probably easier to embed reliable cameras into key positions in the airframe and allow the copilot to switch one of their screens to camera view. The data networks are probably already in place, and the instrumentation panels are now flexible and modular. We're probably approaching the point where a pilot will trust a set of cameras more than a set of odd additional direct optics.

But for the last few decades, large planes probably really should have all had periscopes.