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Introducing

To understand Simon’s particular genius – how it developed and why for a few years he led the braying pack of mathematicians hunting down the Monster – the reader needs to know about squares.

On the face of it, the study of symmetries is a subject for children. A square has symmetry: you can rotate it, and the result looks just as if you’d done nothing at all:

The same goes for an equal-sided triangle:

A circle, cube, sphere, and a host of other shapes with names like dodecadodecahedron (twenty-four faces) and icosidodecadodecahedron (forty-four faces) each has similar symmetrical properties.

In order to develop mathematics out of such simple stuff, we have to keep a diary of these symmetries.

For example, to keep track of these four moves, we can represent them like this:

Note that there’s a sense of self-containment about this set of operations. A square has four sides and therefore only four distinct ways of rotating. After that, you’ve exhausted all the possibilities. No amount of rotating will paint it green or puff it up to twice its original size. Other operations are needed to perform that sort of thing.

If we rotate a square in any of the above four ways, it still looks to the outsider just like the square we started with:

But, privately, we know we’ve been fiddling. For example, if we rotate a square through two turns (i.e. flip it head over heels), we can represent this:

In other words,

signifies the act of swivelling a square through two 90-degree turns, without anybody noticing.

Naturally, if you turn a square by one turn through 90 degrees, then do it again, that’s the equivalent of two 90-degree turns overall:

1 + 1 = 2

Similarly, rotate a square once, followed by two more turns, and the result is equivalent to three turns. You’ve almost gone the whole way round:

1 + 2 = 3

And so on. Rotate a square by two turns, then do nothing, go off and play with somebody else’s crayons, and no one’s going to be fooled – it’s still just two turns:

2 + 0 = 2

A square looks just the same after any combination of these operations, or all of them:

The figures with arms and legs are simply diary entries to keep track of the secret things we’ve been doing to the square in the playpen.

What happens if we turn a square, say, five times? That’s the equivalent of spinning it through a full cycle, then throwing in an extra single turn for good measure:

Group Theory isn’t interested in recording such clever-clogs stuff. Turn a square round five times and you might as well just have turned it once. It’s the final outcome only that matters, so it’s put down as an ordinary single turn:

So, although it seems possible that:

3 + 2 = 5

because the first four turns make a complete rotation, head over heels, back exactly to where we started, we ignore them as wasted effort, and just focus on the one left-over turn, which got us somewhere:

3 + 2 = 1

In this respect, rotating a square is the same as rotating the hour hand on an ordinary clockface. If it’s two o’clock and we add twelve hours, we don’t say it’s fourteen o’clock (unless we’re being tiresome). We say it’s two o’clock again.

All these combinations of turns can now be written down as a chart. These tables are the lifeblood of Group Theory. Every mystery of this secretive, sly subject is contained in them. The one that applies to turns of a square is:

It’s read in the same way as the distance chart at the front of a road atlas. It’s not a calculating device; it’s a secretarial way of keeping track of information. If you’re six years old and want to remind yourself what happens when you turn a square once, then turn it again, i.e.:

take the row corresponding to 1, run your chocolatey finger along until you come to the column corresponding to 1, and there’s the answer – it’s equivalent to two turns:

What’s baffling is that there can be anything complicated enough about this ‘study’ of symmetries to bring it out of the playroom in the first place.