Symmetry

I

Introduction

Symmetry, orderly, mutually corresponding arrangement of various parts of a body, or of a geometric shape. When such a body possesses a high degree of symmetry, a balanced, pleasingly proportioned form often results. Symmetries are of the highest importance in mathematics, physics, and chemistry, and also feature prominently in biology and in the arts.

II

Types of Symmetry

The classification of different types of symmetry is part of geometry. The main classes of symmetry are identified as rotational symmetry, reflection symmetry, and translational symmetry.

A

Rotational Symmetry

An object (the word “object” will be used to include geometric shapes) is said to have rotational (or radial) symmetry if it coincides with itself more than once when it is rotated through some fraction of a complete turn about some point. (Every object coincides with itself when it is rotated through 360°.) The point about which it is rotated is called the centre of rotational symmetry. For example, if an equilateral triangle is rotated about its centre, it will coincide with itself three times in one complete rotation. It is said to have rotational symmetry of order three.

B

Reflection Symmetry

An object is said to have reflection symmetry if it can be divided, in imagination, into two halves that are mirror images of one another. For example, in most typefaces the capital letters A, H, and W show reflection symmetry: in each case the left half of the letter is the mirror image of the right half. The line dividing the halves—in these cases, the vertical line through the middle of each letter—is called the axis of symmetry. The letter H has another axis of symmetry as well: the horizontal line through the middle, since the upper and lower halves of the letter are mirror images of each other.

When an object shows reflection symmetry, as a whole it is identical with its own mirror image. An object like a glove or a shoe, by contrast, is distinct from its own mirror image.