Congruence and Symmetry

Congruence and Isometries

Two figures are said to be congruent if their points can be placed in one to one correspondence in such a way that pairs of corresponding points are the same distance apart. That is, for points X, Y, in the first configuration there are points X', Y', in the second configuration such that d(X, Y) = d(X', Y'). A transformation that preserves distances is termed an isometry.

The identity correspondence is a trivial type of isometry.

An isometry that leaves exactly one point invariant is called a rotation.

If an isometry leaves two distinct points invariant it must leave the whole line of points containing them invariant. An isometry that leaves exactly one line invariant is called a reflection.

An isometry that leaves no points invariant is either a translation, which moves all points a fixed distance in a fixed direction, or a transflection, which combines a translation with a reflection in a line parallel to the line of translation.

Symmetries

Self-correspondent isometries are termed symmetries. A figure that has only the identity self-correspondence is termed asymmetric, whereas a figure that has one or more non-identity symmetries is termed symmetric. A symmetric figure can be divided into two or more similarly placed congruent parts, known as its components. A figure with n components may be said to have n-ary symmetry (for particular values of n we have corresponding terms: 1 unary, 2 binary, 3 ternary, and so on; unary symmetry is an alternative name for asymmetry).

Instead of measuring the amount of symmetry in a figure by counting the number n of congruent components we could use d = log2 n (i.e. n = 2^d) which is the degree of symmetry. Figures with degrees of symmetry 0, 1, 2 and 3 may then be referred to as nully, singly (or simply), doubly and triply symmetric, corresponding to unary, binary, quaternary and octonary. In cases where n is not a power of 2, d becomes fractional. For example when n = 3, then d = 1.585 approximately.

Binary Symmetry

A line in which a figure may be reflected without alteration is called an axis of symmetry (plural, axes). A point about which a figure may be rotated (other than a multiple of 360) without alteration is called a centre of symmetry; if one exists it is unique. A figure can have both an axis and a centre of symmetry; in fact any figure that has two axes is necessarily symmetric by rotation about the point of intersection of the axes, the minimum angle of rotation being twice that between the two axes.

Translational Symmetry

Figures can be constructed that can be repeated at regular intervals so as to cover an area of any size. This type of design is said to exhibit translational symmetry of the type seen in wallpaper patterns. The simplest such patterns are those consisting solely of equally spaced parallel straight lines.