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 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 E_{2}. Then the
Euclidean geometry of the plane is the study of the properties of subsets of
the plane which are invariant under motions of E_{2}. 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 E_{2} 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.
E_{3} 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 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 G consists of rotations about its
center through 0_{A}^{°}, 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 S_{3}, 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 G
of the configuration _{A}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
The symmetry group of any polygon with 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 C, generated by a rotation
through an angle of 2p/_{n}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 D.
_{n}
In discussing the symmetry groups of polyhedra, we will, as indicated above,
always assume that the geometry is given by the group S consisting of the even
permutations of _{n}n symbols; it is of index 2 in S, that is,
its order is half that of _{n}S, or _{n}n!/2. Thus the
order of A_{4} is 12. The cube and the regular octahedron have the
same symmetry group, namely S_{4}. 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 A_{5}. 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 S_{4}.
We are now in a position to give at least one precise meaning to the
statement "Figure A strictly contains the
symmetry group G of _{B}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 G. Thus we have, in fact, two notions whereby we may compare
symmetry - and they have the merit of being consistent. Indeed, if _{B}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 - " |