We want now to take up the more mathematical aspects of symmetry. Indeed, at this stage, we lack a precise definition of symmetry - we cannot even give a meaning, in general, to the statement that one geometrical figure is more symmetric than another. Of course, a square is more symmetric than an arbitrary rectangle, and a rectangle is more symmetric than an arbitrary quadrilateral. But can we, for example, always compare the symmetries of regular polygons?
We are guided in our definitions by the approach of the great German mathematician Felix Klein (1849 - 1925) to understanding the nature of geometry. Consider, for example, the usual plane Euclidean geometry, in which we study the properties of planar figures which are invariant under certain Euclidean motions.These motions certainly include translation and rotation, but it is a matter of choice whether they include reflexion. For example the FAT 7-gon (Figure 4(b)) is invariant under rotations through 2p/7 about its center, but not under reflexion in its plane. Thus, to define our geometry, we must decide whether we allow Euclidean motions which reverse orientation. Of course, it we allow certain Euclidean motions, we must also allow compositions and inverses of such motions, so we postulate a certain group G of allowed motions. If A is a planar figure, then, for any g Î G, Ag is again a planar figure and, in the G-geometry of A, we study the properties of the figure A which it shares with all the figures Ag as g varies over G; such properties are called the G-invariants of A, abbreviated to invariants if the group G may be understood.
Example 3.1 Let G be the group of motions of the plane generated by translations, rotations and reflexions (in a line); we call this the Euclidean group in 2 dimensions and may write it E2. Then the Euclidean geometry of the plane is the study of the properties of subsets of the plane which are invariant under motions of E2. For example, the property of being a polygon is a Euclidean property; the number of vertices and sides of a polygon is a Euclidean invariant. On the other hand, as we have hinted, orientation is not invariant with respect to this group, though it would be if we disallowed reflexions. Thus, by means of a motion in E2 the triangle ABC may be turned over (flipped) to form the triangle A'B'C' as shown in Figure 7. But the orientation of the triangle ABC is anti clockwise, while the orientation of the triangle A'B'C' is clockwise.
Example 3.2 We may 'step up a dimension', passing to the group E3 of Euclidean motions in 3-dimensional space. Notice that it is natural to think of reflexions in a line (of a planar figure) as a 'motion' since it can be achieved by a rotation in some suitable ambient 3-dimensional space containing the plane figure. However, it requires a greater intellectual effort to think of reflexion in a plane (of a spatial figure) as a motion in some ambient 4-dimensional space! Who would think of turning the golden dodecahedron (see Figure 8 of [Rec]) inside out? Thus it is common not to include such reflexions in defining 3-dimensional geometry. This preference is, however, a consequence of our experience of living in a 3-dimensional world and has no mathematical basis. However, since, in this article, and its companion article, we are highlighting the construction of actual physical models of geometrical configurations, it is entirely reasonable to omit 'motions' to which the models themselves cannot be subjected.
We now introduce the key idea in the precise definition of symmetry. Let a geometry be defined on the ambient space of a geometric configuration A by means of the group of motions G. Then the symmetry group of A, relative to the geometry defined by G, is the subgroup GA of G consisting of those motions g Î G such that Ag = A, that is, those motions which map A onto itself, or, as we say, under which A is invariant. Thus, for example, if our geometry is defined by rotations and translations in the plane, and if A is an equilateral triangle, then its symmetry group GA consists of rotations about its center through 0°, 120°, and 240°; if, in our geometry, we also allow reflexions, then the symmetry group has 6 elements instead of 3, and is, in fact, the very well-known group S3, called the symmetric group on 3 symbols - the symbols may be thought of as the vertices of the triangle. We must repeat for emphasis that the symmetry group GA of the configuration A is a relative notion, depending on the choice of 'geometry' G.
It is plain that no compact (bounded) configuration can possibly be invariant under a translation. Thus when we are considering the symmetry group of such a figure we may suppose G to be generated by rotations and, perhaps, reflexions. Moreover, any such motion in the plane is determined by its effect on 3 independent points and any such motion in 3-dimensional space is determined by its effect on 4 independent points. Since a (plane) polygon has at least 3 vertices and a polyhedron has at least 4 vertices, and since any element of the symmetry group of a polygon or a polyhedron must map vertices to vertices, it follows that the symmetry group of a polygon or a polyhedron is finite (compare the symmetry groups of a circle or a sphere).
The symmetry group of any polygon with n sides is, by the argument above, a subgroup of Sn, the group of permutations of n symbols, also called the symmetric group on n symbols. If G is generated by rotations alone, and the polygon is regular, this group is the cyclic group of order n, often written Cn, generated by a rotation through an angle of 2p/n radians about the center of the polygonal region. If G also includes reflexions, this group has 2n elements and includes n reflexions; this group is called a dihedral group and is often written Dn.
In discussing the symmetry groups of polyhedra, we will, as indicated above, always assume that the geometry is given by the group G generated by rotations in 3-dimensional space. Then the symmetry group of the regular tetrahedron is the so-called alternating group A4. In general, An is the subgroup of Sn consisting of the even permutations of n symbols; it is of index 2 in Sn, that is, its order is half that of Sn, or n!/2. Thus the order of A4 is 12. The cube and the regular octahedron have the same symmetry group, namely S4. It is easy to see why the symmetry groups are the same; for the centers of the faces of a cube are the vertices of a regular octahedron, and the centers of the faces of a regular octahedron are the vertices of a cube. Likewise, and for the same reason, the regular dodecahedron and the regular isocahedron have the same symmetry group, which is A5. It is a matter of great interest and relevance here that the symmetries of the Diagonal Cube and the special braided octahedron of Figure 7 and Figure 16, respectively (of [Rec]) each permute the four braided strips from which the models are made. This provides a beautiful explanation of why their symmetry group is the symmetry group S4.
We are now in a position to give at least one precise meaning to the statement "Figure A is more symmetric than Figure B". If it happens that the symmetry group GA of A strictly contains the symmetry group GB of B, then we are surely entitled to say that A is more symmetric than B. Notice that the situation described may, in fact, occur because B is obtained from A by adding features which destroy some of the symmetry of A. For example, the coloring of the strips used to construct the braided Platonic solids of Figure 6 of [Rec] will reduce the symmetry in all cases but that of the cube.
However, the definition above is really too restrictive. For we would like to be able to say that the regular n-gon becomes more symmetric as n increases. We are thus led to a weaker notion which will be useful provided we are dealing with figures with finite symmetry groups (e.g., polygons and polyhedra). We could then say - and do say - that A is more symmetric than B if GA has more elements than GB. Thus we have, in fact, two notions whereby we may compare symmetry - and they have the merit of being consistent. Indeed, if A is more symmetric than B in the first sense, it is more symmetric than B in the second sense - but not conversely.
Notice that we deliberately avoid the statement - often to be found in popular writing - "A is a symmetric figure". We regard this statement as having no precise meaning!