Dodecahedron, regular

Figure 7.9b. Different perspective views of the regular dodecahedron and icosahedron.

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. 

Figure 6-6 In the gas phase, H3Or can be tightly surrounded by 20 molecules of H20 in a regular dodecahedron held together by 30 hydrogen bonds. Small dark and white atoms are H. Dark H atoms are hydrogen bonded. [From S. Wei. Z. Shi. and A. VV Castleman. Jr..

What is the shape-factor for a regular dodecahedron, one of whose faces is a-units in size ... 

In the Preface to the Third Edition of his Regular Polytopes [20], the great geometer H. S. M. Coxeter calls attention to the icosahe-dral structure of a boron compound in which twelve boron atoms are arranged like the vertices of an icosahedron. It had been widely believed that there would be no inanimate occurrence of an icosahedron, or of a regular dodecahedron either. 

In 1982, the synthesis and properties of a new polycyclic C2oH2o hydrocarbon, dodecahedrane, was reported [21], The twenty carbon atoms of this molecule are arranged like the vertices of a regular dodecahedron. When, in the early 1960s, H. P. Schultz discussed the topology of the polyhedrane and prismane molecules (vide infra) [22], at that time it was in terms of a geometrical diversion rather than true-life chemistry. Since then it has become real chemistry. 

A solid figure with 12 faces A regular dodecahedron is a regular polyhedron with 12 faces. Each face is a rgular pentagon. 

The involvement of qrmmetry in chemistry has a long history in 640 B.O. the Society of Pythagoras held that earth had been produced from the reguliu hexahedron or cube, fire from the r ular tetrahedron, air fiom the regular octahedron, water from the regular icosahedron, and the heavenly sphere from the regular dodecahedron. Today, the chemist intuitively uses symmetry every time he recognizes which atoms in a molecule are equivalent, for example in pyrene it is easy... 

Stretching of the substantially dodecahedral cells is schematically illustrated in Fig. 10 (a regular dodecahedron is projected on the two planes perpendicular to Fig. 8 as a hexagon and on that of Fig. 8 as a decagon). 

These are groups which contain more than one threefold or higher axis. We will limit our consideration to the symmetry groups which describe the Platonic solids Td for the regular tetrahedron, Oh for the cube and regular octahedron, I/, for the regular dodecahedron and icosahedron, and JCh for the sphere. Some molecules in the cubic groups are shown below ... 

The phenomena just mentioned lead to formation of a polyhedral foam the shape of the air cells approximates polyhedra. For cells of equal volume, the shape would be about that of a regular dodecahedron (a body bounded by 12 regular pentagons), and the edge q then equals about 0.8/-, where r is the radius of a sphere of equal volume. Actually, the structure is less regular, because of polydispersity. Moreover, close packing of true dodecahedrons is not possible. In a two-dimensional foam, say, one layer of bubbles... 

The operation of some of the other improper rotation axes can be illustrated with respect to the five Platonic solids, the regular tetrahedron, regular octahedron, regular icosahedron, regular cube and regular dodecahedron. These polyhedra have regular faces and vertices, and each has... 

Figure 4.5 The five Platonic solids (a) regular tetrahedron (b) regular octahedron (c) regular cube (d) regular icosahedron (e) regular dodecahedron. The point group symbol for each solid is given below each diagram...

Fig. 2.25. Tetrahedrally interconnected atoms in eclipsed configuration, a) Pentagonal ring (the tetrahedral angle of 109° 28 is exaggerated in the drawing) b) regular dodecahedron (amorphon).

Alternatively, analogous to the T 0 (e +12) problem discussed above, the dual of the icosahedron, namely the regular dodecahedron, can be used for a top-rep of the T h problem (Figure 1.20b). In this case the roles of the vertices and face midpoints are reversed from those of the icosahedron top-rep (Figure 1.20a). 

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