persuading the world that mathematics can be fun 2: cohomology and local triviality

I’m feeling particularly brave today, and with this level of bravery comes a degree of foolhardiness that strikes me as reckless.  But I have set myself a quest, and part of that quest is to bring mathematical ideas out into the light.

So let’s do this.   Consider an impossible object.  The particular I want to consider is the Penrose triangle or impossible tribar, thought the same sort of discussion may well hold for some of the other objects on the list.

If we restrict our attention to just one of the corners of the Penrose triangle, we see nothing out of the ordinary.  Just the corner of a triangular object that might have been built out of pieces of wood or steel.  Not much to see here.

But it is when we look at the Penrose triangle as a whole, not one small piece at a time, that things go terribly, horribly wrong.  It is not possible to build a Penrose triangle in normal space, regardless of the materials used.

Lots of people find the Penrose triangle and other similar impossible objects interesting for all sorts of reasons.  For me, there is one particular aspect that I find interesting, one among many, and this is that the Penrose triangle illustrates a phenomenon that mathematicians find fascinating.

This is the phenomenon of getting something interesting globally, about an object as a whole, even when it’s not all that interesting locally, in its small pieces.  The how of getting this something interesting is where the mathemagic happens and so I don’t want to go into detail here.

But I feel that the general idea of taking two objects that are made up of the same looking small pieces, but being able to distinguish one from the other, is I think one of the great ideas in mathematics.

And it’s a remarkable slippery and subtle idea.  To convince you of this, let me end with an example.

Take a line, an ordinary straight line.  If we look at very small pieces, say intervals of length ε (a symbol that mathematicians use to refer to the very very very small), then we can build up the line by overlapping lots (and by lots, here we need to mean infinitely many) of these small intervals.

If on the other hand we look at a circle of radius 1 000 000 000, we can also build the circle by overlapping lots (though by lots, here we mean finitely many even though the number needed will be huge) of these small intervals.

The line and the circle are very different, and yet we can build them out of the same small pieces, by using enough of these small pieces.

The line and the circle are very different; we can keep going around and around and around a circle, something we can’t do for a line.  We keep going the same direction on a circle and we keep passing the same point, which we can’t do on a line.

So, different objects, built from the same small pieces.  Or impossible objects, built from possible small pieces.  Both reflections of the same basic idea, that even when things looks simple on the small scale, they get interesting when viewed on the large scale.

~ by Jim Anderson on 5 November 2017.

One Response to “persuading the world that mathematics can be fun 2: cohomology and local triviality”

  1. Oh, that’s really interesting! I need to think about that. But first must write fiction. As a reward I am allowed an evening of playing about with tiny tiny bits of lines…

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