Symmetry

Many kinds of symmetry occur in nature. The human figure is approximately symmetrical about a vertical line, which is one of the reasons why mirrors seem to invert right and left. This kind of symmetry is known as bilateral symmetry.

A shape may be symmetrical about several lines at once, or combine bilateral and rotational symmetry. A square is bilaterally symmetric about its diagonals and about lines through the centre parallel to a side.

An entirely different sort of symmetry is exhibited by wallpaper patterns, where the whole pattern can be displaced in various directions without looking any different.

The essence of symmetry is the way shapes can be moved around and still look the same. Individual points, however, need not stay in the same place. The important thing is not the position of the points, but the operation of moving them.

The fact that the product of any two symmetries is also a symmetry is usually expressed as: the set of symmetries is closed under the operation of multiplication. This set of symmetries, with its multiplication, is an example of a mathematical structure dignified by the title "group".

Every shape has a symmetry group. The human figure has two symmetries: the identity and reflection r about a vertical line. In general, to find the symmetry group of a figure we must: a) find all the symmetries; b) work out the multiplications. In every case you will find that the set is closed under multiplication.