Читать книгу The Genius in my Basement - Alexander Masters, Alexander Masters - Страница 15

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Introducing

Just as a square can be rotated through four turns to get it back to where it was to begin with, and the results laid out in a Group Table, the same approach can be applied to every regular shape. The size of the table you need to draw depends on how many operations have to be performed before you’ve exhausted all the possibilities and ended up back where you started. An equal-sided triangle can be manhandled three times before it’s back on its feet:

As before, these represent the act of turning Triangle. The trick of the game is to find all the ways you can fiddle with Triangle and yet leave it looking just the same afterwards as it did before you began:

And (again as before, with Square) these turns combine in the most obvious way …

In words, turn Triangle once, then turn it again, and the result is two turns: one plus one equals two. It is easy to spin Triangle head over high-heels, if that’s what you want:

2 + 1 = 0

(two turns, followed by one turn, returns Triangle to its original position)

Remember, in Group Theory, turning a regular shape right round is taken to be the same as doing nothing at all. Full, completed turns don’t get totted up. It’s only the overall adjustment that matters:

2 + 2 = 1

The corresponding table (which, as with Square, looks like a pint-sized sudoku table) is therefore:

Once again, we’ve got through a mathematical section with a suspicious lack of awfulness, like someone who’s committed a crime in the woods.

Is that all there is to it? Was that really mathematics?

The mist in these woods is hushed. A distant leaf clatters among the branches like a falling pin.

Let it be whispered: a saucy chapter is approaching.