Professor Stewart's Hoard of Mathematical Treasures (44 page)
Read Professor Stewart's Hoard of Mathematical Treasures Online
Authors: Ian Stewart
Tags: #Mathematics, #General
BOOK: Professor Stewart's Hoard of Mathematical Treasures
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. . . or reflect it . . .
... so it has no symmetries. No, that’s a lie: it has one symmetry: leave it alone. This is the trivial symmetry, and all shapes have it.
A cat with two tails looks the same when you reflect it, so it has an axis of reflectional symmetry (grey line).
The cat’s body has two axes of reflectional symmetry, and it also looks the same when you rotate it through 180°.
Four cats sitting in a square are symmetric under rotations of 0° (trivial), 90°, 180° and 270°. This is 4-fold rotational symmetry.
The same goes when you throw away the cats . . .
... but now there are four new axes of reflectional symmetry. So a square has eight different symmetries.
A cube has 48 symmetries . . .
. . . and a dodecahedron has 120.
A circle has infinitely many rotational symmetries (any angle) and infinitely many reflectional symmetries (any diameter as axis).
If this line of cats went on infinitely far, it would have translational symmetries: slide the cats an integer number of spaces right or left.
A cat crystal has translational symmetries in two different directions.
Symmetries need not be motions. Shuffling a pack of cards is a transformation . . .
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