Ultimate Symmetry

The Dodecahedron has twelve pentagonal faces, twenty vertices, and thirty edges. A regular Dodecahedron consists of twelve regular pentagons, with three of those pentagons meeting at each vertex. The pyritohedron, a common crystal form in pyrite, is an irregular pentagonal Dodecahedron, having the same topology as the regular one but pyritohedral symmetry while the tetartoid has tetrahedral symmetry.

Figure IV.8: The Dodecahedron has twelve pentagonal faces, twenty vertices, and thirty edges, and its dual is the Icosahedron.

With its twelve pentagonal faces, Plato assigned the Dodecahedron to the Heavens with its twelve zodiac constellations. This agrees very well with the complex-time structure, but the Heavens is the element Fire  and the not the Quintessence that corresponds to the Tetrahedron as we mentioned above. As we also said, the dual of the Dodecahedron is the Icosahedron which corresponds to the Earth state .

If the five regular polyhedra are built with same volume, the regular Dodecahedron has the shortest edges, and the distance between the vertices on the same face not connected by an edge equals the edge length multiplied the golden ration .

The symmetry, structural integrity, and beauty of these solids have inspired architects, artists, and artisans from ancient Egypt to the present. Euclid devoted the last book of the Elements to the regular polyhedra, where he provided the first known proof that exactly five regular polyhedra exist. It is natural to wonder why there should be exactly five regular polyhedra, and whether there might conceivably be one that simply hasn t been discovered yet. However, it is not difficult to show that there must be five, and that there cannot be more than five. Thus Plato concluded that they must be the fundamental building blocks of nature.

The main characteristics of the regular polyhedra are summarized in the table 4.

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Table 4: The main properties of the Five Regular Polyhedra.

The regularity of the regular polyhedra means that they are all highly symmetrical. For each Platonic solid, it is possible to construct a circumscribed sphere or circumsphere that completely encloses the Platonic solid, and for which all of the vertices of the Platonic solid lie on the surface of the sphere, a mid-sphere that is tangent to each of the edges of the Platonic solid, and an inscribed sphere or in-sphere that is completely enclosed by the Platonic solid, and that is tangent to each of its faces. For each of the regular polyhedra, these three spheres are concentric. The radii of the spheres are called the circum-radius, the mid-radius, and the in-radius respectively.

The symmetry groups of the regular polyhedra are a special class of three-dimensional point groups known as polyhedral groups. The high degree of symmetry of the regular polyhedra can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and face-uniform. There are only three symmetry groups associated with the regular polyhedra rather than five, since the symmetry group of any polyhedron coincides with that of its dual. This is easily seen by examining the construction of the dual polyhedron. Any symmetry of the original must be a symmetry of the dual and vice versa. The three polyhedral groups are: the tetrahedral group, the octahedral group, which is also the symmetry group of the cube, and the icosahedral group, which is also the symmetry group of the Dodecahedron. The orders of the proper rotation groups are 12, 24, and 60 respectively; precisely twice the number of edges in the respective polyhedra. The orders of the full symmetry groups are twice as much again (24, 48, and 120). All regular polyhedra except the Tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin.