Figures and Configurations


A geometrical figure is any simple pattern, picture or diagram that can be shown as an array of black marks on a white background. In the case of a drawing made on paper such a figure may be drawn by using a pencil, pen or brush to apply graphite, ink or paint to the white surface of paper or canvas. In the case of a diagram on a computer screen such a figure will be made up of tiny pixels, which are small squares or rectangles. A geometrical configuration is a more complicated diagram made by superposing simpler figures. The component parts of figures may be labelled, that is given names, so that we can study them verbally as well as visually.


Possibly the simplest type of figure is a point, which is anything that can for some purpose be represented by a dot. The thing represented doesn't actually have to be small. In working out the orbits of the planets in the solar system it is often convenient to treat the sun, planets and other astronomical bodies as points. In cosmology we may even represent whole groups of galaxies as points, though this may be stretching the analogy as far as it will go. On smaller scales molecules, atoms, electrons and smaller particles of matter are often treated as points, though here once again we are stretching the analogy rather a lot. Subatomic structures are now often described in terms of other analogies, such as waves, strings or glue. For ease of visualisation we will think of a point as a carefully made small mark on our drawing paper, but it should be borne in mind that this is only one context. For instance a point on a distant advertising hoarding may turn out to be a very large spot when seen close up.

Lines and Curves

Another type of figure is a line or curve formed by moving the tip of a drawing instrument across the drawing surface. It may represent the path of a moving point.


Two figures are said to correspond (to each other) if their points and lines and other defining features can be placed in (one-to-one) correspondence. That is for each feature X in the first there is a uniquely determined corresponding feature Y in the second, and further for each feature Y in the second there is a uniquely determined feature X in the first corresponding to it. A correspondence can be regarded as a type of relation between the figures, or it can be regarded as an operation acting from one to the other of them, in either direction. Thus one figure may be regarded as being produced from the other, i.e. as being a transformation of it, or one may be regarded as being used to describe the other, i.e. as being a representation of it. These are aspects of the same relationship, seen from different viewpoints. Denoting the transformation by a prime (') the element corresponding to X is X'.

It may happen that the points X and X' coincide, that is X' = X, in which case X is called an invariant element of the correspondence and belongs to both figures. If all the points of the second figure are points of the first figure we have a self-correspondence. If all the points are invariant we have the identity self-correspondence. If X' = Y and Y' = X for all X, where X and Y are distinct, then we have a binary self-correspondence.