Apologies to my hordes of devoted followers for my recent absence…I’ll make up for it by including some (shamelessly copied and pasted) pictures today.
I’m going to talk a bit about topology, which is a kind of generalisation of geometry; if we imagine every object to be made of some kind of incredibly pliable substance which enables it to be morphed into any other vaguely similar object, then we can study general properties of space rather than being constricted by such trifling matters as shape and form. The basic rule is: if you can stretch, squeeze or twist (no cutting or glueing allowed!) a shape into another one, then they have the same topological properties. The classic – and rather over-used – saying is that a topologist can’t tell his coffee cup from his doughnut (they both have one hole).
Topologists spend a lot of their time considering holes, and trying to work out things like how many different ways there are to wrap spheres of different dimensions around each other, and which knots are actually unknots (not knots!). It is a very visually appealing branch of mathematics (see the link on the right to “sketches of topology” for some nice images) and there are some wonderfully odd constructions with special properties that topologists like to study.
For example: take a strip of paper, twist it once and glue the ends together. You will end up with a shape with only one edge (trace it around with your finger if you don’t believe me) called the Möbius band. This has various odd properties…for example if you cut it in half down the centre of the strip you will end up with not two pieces, but one intact twisted strip.
There is a 3-dimensional analogue of this called a Klein bottle, which has only one surface…this means that its inside is its outside. It is hard to say how much water Klein bottles would hold, as they live in 4-dimensional space, and contain themselves! This hasn’t stopped people trying to embed them in 3 dimensions however, and there are some beautifully-made representations on display at the London Science Museum.
Mathematics is rarely in the news at all, but there was a brief burst of publicity for topology a few years ago when a Russian mathematician called Grigori Perelman solved a very famous problem known as the Poincaré conjecture (and subsequently turned down the $1m prize and stopped doing mathematics altogether). The conjecture – now a theorem – says (in very inexact and non-technical terms) that the surface of any 4-dimensional object is topologically equivalent to the surface of a 4-dimensional sphere. While this sounds pretty obscure, topology can actually be highly applicable, and is becoming ever more so as physics increases in strangeness. It is interesting to note that, just as we experience the surface of the 3-dimensional earth that we walk on as being a 2-d shape, the shape of the universe at a given point in time is modelled by cosmologists as a 3-d “slice” of 4-d space-time, exactly the sort of object the Poincaré conjecture concerns.
It is fair to say that topology is a rather inaccessible subject to non-experts nowadays, but there are some interesting results which anyone can understand. The Hairy Ball Theorem, for example, asserts that there is no non-vanishing continuous vector field on the sphere. A continuous vector field on a sphere can be pictured as follows: for every point on the outside of the sphere, assign an arrow facing in a direction tangential to the sphere, so that the directions and lengths of the arrows vary continuously (I mentioned continuity in my last post) from point to point. The theorem says that there is no way to do this in a way such that at least one of the arrows has zero length.
The reason this is known as the Hairy Ball Theorem is of course that instead of arrows we can think of the sphere as being covered in hairs; a bit like a coconut, only with more hair. Then the theorem says that there is no way to comb the hairs on the coconut without creating a “cowlick” – a point where there is nowhere for the hair to be combed apart from straight up. So far, so mildly diverting…but ultimately not very useful. The good thing about having such a general theorem, however, is that there are many ways in which we can specialise it. For example, the earth is (almost) a sphere, and the wind on the earth is certainly a continuously varying vector field (at any point the wind is moving roughly tangentially to the surface, has magnitude and direction, and does not change abruptly from one point to the next). So now our theorem says: at any one time, there is always at least one point on the surface of the earth with zero wind. In reality these points will be the centre of a cyclone, or anticyclone.
I’ll briefly discuss one more important theorem with interesting applications to the real world: the Brouwer Fixed Point Theorem. The statement of this goes, roughly: letbe a continuous function taking a certain type of space to itself (a continuous function is just a smooth mapping of each point in the space to another point). Then there is some pointsuch thatThat is, there is some point which is fixed by the function. This theorem has various interesting applications. For example, the action of stirring a cup of tea is just a continuous function mapping the tea to itself. So the Brouwer fixed point theorem says that no matter how hard you stir (or even shake!) the tea, there will always be one point of it which will remain in the same place it was in the teacup before you stirred it. Similarly, if you take a piece of paper from a pile, crumple it up, and lay it back on top of the pile, there will always be some point on the paper which ends up directly above where it was originally (the action of crumpling a piece of paper is a continuous function from the space of points of the paper to itself).
Now it is unlikely that knowing the hair on a coconut cannot be effectively combed is going to advance mankind very much. But topology does have many very important applications to physics, astronomy, and computer science…I suppose my conclusion is, once again: pure mathematics is not useless! (And some of the theorems have funny names).