Simple City  

Two things (one interesting, one very silly) that I have recently learnt about the sine function:

1. The sine rule, as most school-students know, says that in any triangle, is constant, whichever side A and opposite angle a you pick.

Less well-known is that this quantity actually has a geometric meaning: it gives the diameter of the triangle’s circumscribing circle. The Math Less Travelled has pictures.

2.

This may seem an innocuous enough fact. But prepare to be amazed! If you convert into degrees, you get 666^o, which is of course the the Number of the Beast. And you might spot the ubiquitous Golden Ratio lurking on the other side of the equation. So now our formula becomes:

The standard numerological-satanic interpretation of this fact, I am reliably informed, is that the Devil is the opposite of God (hence the minus sign), and only half as powerful. Where exactly the trigonometry fits in, I’m not sure…

Everyone is familiar with George Bernard Shaw’s line: “He who can, does. He who cannot, teaches.”

Apparently it isn’t universally popular in educational circles.

But what did Shaw actually mean? I’d always taken it in the same way as everyone else: as a nasty swipe at the teaching profession. But my contact in the Shaw Society suggested an alternative explanation.

The quote comes from Shaw’s slightly odd 1903 work Maxims for Revolutionists: just a categorised list of aphorisms, which includes others of his most famous lines^[1].

Interpreted as such a maxim, the quote takes on another meaning altogether: it’s a description of how revolutionary societies should organise themselves. Everyone who can should get involved in the fighting, cooking, carrying, building, etc: doing. And those who cannot (on account of being too old, wounded, or whatever) should teach the others.

So, I believed this interpretation for a little while. But now I’m not so, erm, sure.

If you look at Maxims for Revolutionists, it’s quite short on practical advice for organising uprisings, and despite its title it does seem like a depository for his thoughts on various topics. Advice on “How to Beat Children”, for example, strikes me as being of limited use to people actively preparing for revolution. In particular the section on Education (line 31) does contain general snarking at teachers, or at least teachers of certain types: “When a man teaches something he does not know to somebody else who has no aptitude for it, and gives him a certificate of proficiency, the latter has completed the education of a gentleman.”

So now I don’t know. At any rate it doesn’t seem plausible that Shaw would have failed to notice the more obvious reading, and that interpretation doesn’t exactly run contrary to his attitude to the education system of his time (I’m told he described his own school in Dublin as a “futile boy prison” where he learnt “dishonourable submission to tyranny”). So it’s difficult to conclude that he didn’t intend it, at least as an overtone.

In any case, we can perhaps agree that teaching, as it should happen, was better summed up by Aristotle: “Those that know, do. Those that understand, teach.”

Sorry for the feeble pun in the title, by the way, but it’s a sort of unwritten rule when discussing Shaw.

[1] Though not my favourite: “I like flowers, I also like children, but I do not chop their heads and keep them in bowls of water around the house.”

The best-known techniques for telling knots apart are with knot invariants. These are algebraic objects (e.g polynomials) associated to knots. If two knots have different invariants, then you know they really are different knots rather than different configurations of the same knot.

But there is an alternative algorithmic approach to this question.

In 1970, the German-born mathematician Wolfgang Haken, at the University of Illinois, tackled the question of telling when two knots are the same: his tactic was to turn the whole problem inside out. Instead of comparing two knots floating in space, he looked at the knots’ complements: the 3-dimensional shapes that are left when you remove the knots from the surrounding matter, leaving knot-shaped holes. (Imagine setting the loosely knotted loops in blocks of glass, and then removing the strings.) If he could tell when these two objects could be deformed one into the other, then the same would go for the knots.

He set to work on a method which would take the two knot-complements, dissect them in stages, and eventually decide whether or knot they were the same. He made a great deal of progress, but Haken’s algorithm was left with holes in it when he moved onto other concerns (most notably in 1976 with Kenneth Appel he proved the famous Four Colour Theorem). However other people picked up the algorithm, notably Sergei Matveev at Chelyabinsk State University in Russia who filled in the final gap in 2003.

So in theory, mathematicians now have the foolproof^[1] method for distinguishing knots that they longed for. A tremendous achievement though this is, it may be too cumbersome ever to be fully implemented on a computer in the real-world. So other, less powerful but more practical algorithms are used in current knot tabulation efforts.

Another reason for mathematicians’ reticence is that the Haken algorithm doesn’t leave any fingerprints. In theory, it can provide a yes/no answer to the question of whether two specific knots are the same. But it can’t identify or describe individual knots: algebraic invariants are needed for that.

For the specific problem of recognising configurations of the unkot, more manageable algorithms have been found. But with these too, it is an open question whether they can be made to run fast enough (i.e in Polynomial Time) to be of widespread practical use in the real world: this is the so-called unkotting problem.

[1] perhaps “watertight” would be more accurate here – not a lot of mathematics is foolproof

While the large hadron collider is out of action, fans of scientific enormousness will be pleased to hear that not one but two new large prime numbers have been discovered. They are:

2^37,156,667 – 1
and
2^43,112,609 – 1

This second one is now the largest known prime, at 12,978,189 digits long. You can see the entire thing written out here, if you can’t wait for the poster to come out.

As you can see, both are Mersenne primes: prime numbers of the form 2ⁿ-1, for some n. The largest was discovered by Edson Smith at UCLA, and the other by Hans-Michael Elvenich in Langenfeld, Germany, both as part of the collaborative programme GIMPS (the Great Internet Mersenne Prime Search). Smith’s put together an FAQ about his discovery.

If you want to get involved in GIMPS yourself, there’s serious money at stake courtesy of an anonymous, loaded, prime-enthusiast.

The Guardian has won its legal battle against Matthias Rath – a vitamin-magnate who told desperate South Africans that his pills could cure AIDS, while “so-called anti-retroviral… drugs severely damage all cells in the body – including white blood cells – thereby not improving but rather worsening immune deficiencies and expanding the AIDS epidemic.” Ben Glodacre of Bad Science – who wrote the piece in the Graun that attracted Rath’s unsuccessful lawsuit – is justifiably pleased and proud. Meanwhile Rath is doubtless spitting mad, and hopefully, shortly, bust.