Cutting up Doughnuts

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.

Cyclamate Sweeteners Shrivel Your Testicles

Sometimes I notice details that don't quite seem to connect, and it bugs the hell out of me until I eventually find out why.

Some years back, I saw an episode of a detective show (perhaps "Columbo"?), whose resolution hinged on the idea that there was a particular artificial sweetener used in soft drinks in the 1960's that then got withdrawn. That was a bit before my time, but it puzzled me that the writers seemed to assume that the viewer knew about this, but somehow I hadn't heard of it. So I thought I'd look the thing up to see if it was true.
Apparently it was – according to the literature, cyclamate sweeteners appeared in mainstream products before being banned by the FDA in 1969. But a substance doesn't just get banned from food production without kicking up a bit of media discussion, and I was arrogant enough to figure that if I hadn't come across any such discussion then ... there was probably more to this subject than met the eye. My antennae started twitching. Something smelled wrong.

So I looked deeper. The listed reason why cyclamates were withdrawn, according to the first wave of usual sources, was a possible "slightly elevated cancer risk". Again, this didn't smell right – I'd heard a lot over the years about the similar alleged (small, supposed) risk of bladder cancer associated with saccharine, which wasn't banned in most countries, so where had the corresponding discussion about cyclamates gone? There was an cultural anomaly here – the public discussions that ought to have happened seemed to be missing, and I now badly wanted to find out why.

This was long before the days of Google, so in about 1990 I found myself at an outpost of the British Library, crouched over an old green-screen text terminal that had basic online subscription access to a medical research database, and there I found the obvious answer to why industry people didn't discuss cyclamate side-effects. A distinctive three-word medical term, that once heard and explained, you don't tend to forget.

Cyclamates are associated with ITA.

ITA stands for Irreversable Testicular Atrophy.

=:0

Yes, you read that correctly. It seems that if you're a primate, and male, cyclamate sweeteners can cause your balls to progressively shrivel and wither away. Permanently. Researchers don't know why.

So that was it. That was why nobody told the public what they'd discovered, and why the product was taken off the market, using the iffy "cancer" argument as an excuse. It wasn't in the interests of the drinks companies to mention that they'd probably been chemically castrating some of the guys who drank their cola, it wasn't in the interests of the FDA to admit that they'd approved a substance for mainstream use that had been permanently shrivelling the Male American Public's manly bits. It probably also wasn't in the interests of the research community to go around telling outsiders about their part in a major public health boo-boo that many of those outsiders would find difficult to forgive, if they knew about it. But if you bypassed the commentaries and looked up the original research papers, there it was, in stark black-and-white. ITA.

Cyclamate Returns

The story's moved on since then. Some companies really liked using cyclamate as an ingredient – it was cheap, people preferred the taste to saccharine, and it avoided the later nasty public-relations mess associated with Nutrasweet and product safety. And the biochemical/food industry did some digging of their own, and realised that actually, organisations like the FDA might not be allowed to ban cyclamates.

See, the FDA's remit is public health, and the industry lobbyists started arguing that that having permanently shrunken testicles doesn't actually count as a health problem (no matter how angry the owners of those testicles might be if they found out why). It isn't traditionally associated with work disability (unless you're a porn star), and there's no obvious associated reduction in life expectancy, unless you start to guess at factors like depression and impotence-related suicides.

A product that damages the testicles of male customers doesn't obviously kill anybody. In fact (the industry argued), the associated reduction in testosterone in guys with withered testes might even be associated with a statistical reduction in the frequency of certain testosterone-related cancer deaths (for instance, high testosterone is supposed to exacerbate prostate cancer). Since castrati were generally reckoned to have a greater life expectancy than "intact" blokes, there was an argument that the use of their sweetener product might actually extend average male lifespans rather than reduce them. So (the industry argued) not only were the FDA not allowed to use the ITA argument as a reason for banning the product, they couldn't use the weaker "bladder cancer" argument either, unless they could show that the hypothetical life-expectancy reduction due to increased incidence of bladder cancer was expected to be greater than the corresponding LE extension due to reduced male hormone levels. Cyclamates probably weren't "harmful" if you judged "harmfulness" by a number based on medical life expectancy.
A further argument was that if the potential sexual-function-impairment aspects of cyclamates were outside the FDA's remit, then the FDA was not supposed to tell anyone about them in their reports, because it wasn't the job of a government department to use state funding to "bad-mouth" a product by mentioning negative aspects of that product that weren't anything to do with them. The FDA were supposed to shut up.

So now cyclamates are finding their way back into food, and for a few years now, industry groups have been more bullish about lobbying for the FDA ban to be removed. As a page on the National Cancer Institute website delicately puts it:

" A food additive petition is currently filed with FDA for the reapproval of cyclamate. The FDA's concerns about cyclamate are not cancer related. "

---==--

One last point: if you're reading this and feeling smug because you're female ... well, don't.
See, we don't yet know the mechanism by which cyclamates appear to damage and kill off seminal frond tissue, and it may well be that cyclamates might be doing do something similar to the female reproductive system, unnoticed. Atrophied ovarian tissue is going to be more difficult to spot in the lab than atrophied testicles, simply because its not as easy to do a "before and after" weighing comparison with internal organs. We also don't know whether cyclamate's attack on the germ cells might be associated with heritable genetic or epigenetic damage to the reproductive cells that survive. If your guy's taking cyclamate sweeteners, is it damaging his sperm, and affecting any kids that you might have with him, from that sperm? And what happens if you're pregnant with a male child while ingesting a lot of cyclamate? Does the reproductive-tissue-shrinkage effect in adult males have a counterpart that affects the growth and development of that tissue in the fetus?

There are some genuinely nasty possibilities here.

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.