Platonic solid

Kepler s obsession with the Platonic solids probably relates to the close connection of these bodies with the golden ratio. 

The cube has a three-fold axis along each body diagonal. Grinding down the vertex of a solid cube along the three-fold axis exposes a new triangular face. Grinding away from all vertices until the triangles meet, an octahedron is produced. A similar, inverse operation performed on an octahedron produces a cube. The two polyhedra are said to be reciprocal. The dodecahedron and the icosahedron are reciprocal in the same sense. 

Because of their elements of five-fold symmetry the two large polyhedra have a close relationship with the golden ratio. For instance, three mutually perpendicular and interpenetrating golden rectangles define the vertices of an icosahedron. 

In the case of a cube it is noted that two perpendicular golden rectangles can be inscribed in a square such that each vertex divides a side of the square in the ratio I t. The edge of an inscribed tetrahedron AA) is divided by the rectangles in the ratios AC/AD = BD/AD = t. 

A circle of radius r/2, with respect to AD = 1, centred at F, intersects the edges in the points B and C, such that BBE defines a golden triangle and BCECB is a regular pentagon. 

Solid Vertices Edges Face Type Faces 

possesses 32 symmetry, and requires a minimum of 12 asymmetric units the cube and octahedron, which belong to the point group Oh, possess 432 symmetry, and require a minimum of 24 asymmetric units and the dodecahedron and icosahedron, which belong to the point group Ih, possess 532 symmetry, and require a minimum of 60 asymmetric units. The number of asymmetric units required to generate each shell doubles if mirror planes are present in these structures. 

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M. T. Pope and A. MbUer, eds., Toljoxometallates From Platonic Solids to A.nti-RetroviralA.ctivity Kluwer Academic Pubhshers, Dordrecht, the Nethedands, 1994. 

Pope, M. T. Muller, A. (eds). Polyoxometalates From Platonic Solids to Anti-Retroviral Activity . Kluwer, Dordrecht, 1994. 

You say that your nonlinear molecule has the high symmetiy of a regular polyhedron, such as a tetrahedron, cube, octahedron, dodecahedron, icosahedron,... sphere. If it is a sphere, it is monatomic. On the other hand, if it is not monatomic, it has the symmetry of one of the Platonic solids (see the introduction to Chapter 8). 

As stated, the Platonic solids constitute a family of five convex uniform polyhedra made up of the same regular polygons and possess either 32, 432, or 532 symmetry. As a result, the three coordinate directions within each solid are equivalent, which makes these polyhedra models for spheroid design. 

We will now illustrate four octahedral hosts related to the Platonic solids. Three are based upon the cube while one possesses features of both a cube and an octahedron. 

As a final artistic piece, consider Figure H.4 by Professor Carlo H. Sequin from the University of California, Berkeley. His representation is a projection of a 4-D 120-cell regular polytope (a 4-D analog of a polygon). This structure consists of twelve copies of the regular dodecahedron — one of the five Platonic solids that exist in 3-D space. This 4-D polytope also has 720 faces, 1200 edges, and 600 vertices, which are shared by two, three, and four adjacent dodecahedra, respectively. 

To implement the above plan we consider the regular polyhedra, sometimes called the Platonic solids, of which there are five. By a regular polyhedron we mean a polyhedron... 

Our first task is to show that the five Platonic solids do, in fact, represent all the possibilities. This is quite easy to do. 

TABLE 3.2 The Five Regular Poiyhedra or Platonic Solids... 

It is clear, therefore, that the five Platonic solids are the only regular poiyhedra possible. Let us now examine them to see what symmetry operations may be performed on each one. 

B3.9 Yet another dodecahedron, called apyritohedron is shown below. To what point group does it belong Why is it not one of the Platonic solids ... 

I. Croups with very high symmetry. These point groups may be defined by the large number of characteristic symmetry elements, but most readers will recognize them immediately as Platonic solids of high symmetry, a. Icosahedrd, Ik.—The icosahedron (Fig. 3.10a), typified by the B12H 2 ion (Fig. 3.10b), has six C3 axes, ten C3 axes, fifteen C2 axes, fifteen mirror... 

Platonic solids have been studied since antiquity and in a multiplicity of artistic and scientific contexts. More generally, polyhedral maps are ubiquitous in chemistry and crystallography. Their properties have been studied since Kepler. In the present book we are going to study classes of maps on the sphere or the torus and make a catalog of properties that would be helpful and useful to mathematicians and researchers in natural sciences. 

R. G. Finke in Polyoxometalates From Platonic Solids to Anti-Retroviral Activity, M. T. Pope and A. Muller, Eds., Kluwer, Dodrecht, 1994, pp. 267-280. (This monograph is Vol. 10 of Topics in Molecular Engineering, J. Maruani, Ed.)... 

Figure 2.13. The dodecahedron and the icosahedron are two of the five Platonic solids (regular polyhedra), the others being the tetrahedron, the cube, and the octahedron, (a) The dodecahedron has twelve regular pentagonal faces with three pentagonal faces meeting at a point, (b) The icosahedron has twenty equilateral triangular faces, with five of these meeting at a point.

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