Intuitive ideas about symmetry start with the simple concept of a mirror: a line in the plane, or a plane in space, in which points, lines, and other geometric forms are reflected. This article will begin with the symmetries of the plane; we’ll then see how the essence of plane symmetry can be captured using polygons and the algebraic structures called groups that describe them; turning these polygons into networks, or graphs as mathematicians like to call them, we’ll extend the idea of symmetry into more complex and intriguing forms. Finally, a return to geometry will explore the kinds of “exotic planes” these new symmetries define.
Let’s begin by giving the mirror concept a little formalism. Sticking to the two-dimensional plane, a pair of points, P and Q, are symmetric about a mirror line M if M reflects P exactly onto Q, and vice versa; more generally, a set of points, S = {P1, P2, P3,…}, is symmetric in M if the reflected image of S is exactly the same set of points as S: M(S) = S, with perhaps M(P1) = P2, and so on.
Elementary plane symmetry requires just one more concept, which is obvious once you think of it as a form of symmetry: rotation. A rotation R, around a centre-point C, creates the possibility of symmetry when it is an exact fraction of a full turn: for example, quarter-turns, half-turns, two-thirds-turns, one-fifth-turns, and so on. For these special rotations, repeating the same number of times as the fraction’s denominator brings you back to your starting point. So you can say, for example, that S has five-fold rotational symmetry if there is a one-fifth-rotation R with S = R(S).
For example, a pentagon has five-fold rotational symmetry about its centre: every one-fifth rotation “maps” the pentagon onto itself.
In fact, the pentagon also has some mirror, or reflection, symmetries:
Notice how the numbering of the pentagon’s vertices reveals and describes each symmetry operation. The key idea of this article is to describe symmetries entirely in terms of strings, or permutations, of integers …
The idea of a group as a special set of permutations; symmetry groups for the regular polygons
View a polygon as a graph (network of vertices and edges): the shape need not be perfect; leads to the symmetry group of a graph, for example trees, complete graphs, …
The Petersen graph and its symmetries; higher-order “Petersenesque” graphs
Geometric settings for the new symmetry groups
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