Platonic icosahedron

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). 

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... 

The vertex-split Octahedron and the vertex-split Icosahedron are polycycles obtained from Octahedron and Icosahedron, respectively, by splitting a vertex into two vertices and the edges, incident to it, into two parts, accordingly. The vertex-split Octahedron is drawn on Figure 4.2,1 and both of them are given on Figure 8.3 they are the only, besides five Platonic r, q) — f, non-extensible finite (r, )-polycycles. 

Lemma 8.23 ([DSS06]) All finite elliptic non-extensible (r, q)-polycycles are two vertex-splittings (of Octahedron and Icosahedron see first two in Figure 8.3) and five Platonic r, q) (with a face deleted). 

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.

Figure B.2 shows polyhedra commonly encountered. The five Platonic (or regular) solids are shown at the top. Beside the octahedron and cube, the octahedron is shown inside a cube, oriented so the symmetry elements in common coincide. These solids are conjugates one formed by connecting the face centers of the other. The tetrahedron is its own conjugate, because connecting the face centers gives another tetrahedron. The icosahedron and pentagonal dodecahedron are conjugates. The square antiprism and trigonal...

Of course an icosahedron is not the only three dimensional design that can form a capsule. MacGillivray and Atwood proposed a structural classification for supramolecular assemblies based on the five Platonic and 13 Archimedean solids [21], The Platonic solids, illustrated in Fig. 3.10, are the tetrahedron, cube, octahedron, dodecahedron and icosahedron. 

Fig. 3.10 The Platonic solids (left to right) tetrahedron, cube, octahedron, dodecahedron and icosahedron...

The roots of molecular beauty can be traced back to the Platonic tradition. To Plato, the most beautiful bodies in the whole realm of bodies were the tiny polyhedra, now deemed the Platonic solids, which he proposed comprise the universe the four elements - earth (cube), fire (tetrahedron), air (octahedron), water (icosahedron) - and the ether (dodecahedron) (Fig. 1). Joachim Schummer, who has written [9] extensively on chemical aesthetics, writes ... 

The symmetry of many molecules and especially of crystals is immediately obvious. Benzene has a six-fold symmetry axis and is planar, buckminsterfullerene (or just fullerene or footballene) contains 60 carbon atoms, regularly arranged in six- and five-membered rings with the same symmetry (point group //,) as that of the Platonic bodies pentagon dodecahedron and icosahedron (Fig. 2.7-1). Most crystals exhibit macroscopically visible symmetry axes and planes. In order to utilize the symmetry of molecules and crystals for vibrational spectroscopy, the symmetry properties have to be defined conveniently. 

The five fundamental solids, the tetrahedron, the octahedron, the icosahedron and the dodecahedron were known to the Ancient Greeks. Constructions based on isosceles triangles are described for the first four by Plato in his Dialogue Timaeus, where he associated them with fire, earth, air, water and noted the existence of the fifth, the dodecahedron, standing for the Universe as a whole. These five objects are now known as the Platonic solids — defined as the convex polyhedra because they exhibit equivalent convex regular polygonal faces. 

Molecular frameworks with the shapes of each Platonic solid are known. In symmetry terms, the Platonic solids split into two families the tetrahedron, the cube and the octahedron, which have cubic symmetry, and the icosahedron and the dodecahedron, which have icosahedral symmetry. 

There are 6 five-fold rotational symmetry elements in an object of Ih point symmetry. Thus, in Figure 2.19b the 120 vertices of the great rhombicosidodecahedron are arranged in sets of 10 about the poles of these axes on the unit sphere. That construction emphasises that uniform contractions of these sets about these axes points will return the 12-vertex Platonic solid, the icosahedron, in which each vertex has Csv site symmetry. There are 10 three-fold rotational axes and, so, in Figure 2.19c the decoration pattern is arranged to divide the 120 vertices into sets of 6 about the 20 poles of these axes on the unit sphere. Again, uniform contraction of these subsets of vertices onto these positions on the unit sphere generates the fifth Platonic solid, the dodecahedron, and the site symmetry each vertex is Csy. 

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 ... 

FIGURE 15.23. Platonic figures found in virus structures, and their symmetries (see Ref. 191). (a) Tetrahedron, (b) octahedron, and (c) icosahedron. Rotation axes are indicated. 

Platonic solid Any one of five regular three-dimensional solids - the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. 

For nearly 30 years, the field of metal clusters has provided chemists a vast arena in which to work. Early results were often surprising. Crystallographic analysis revealed that compounds, such as Os3(CO)j2 and Rh6(CO),g, were often incorrectly formulated by traditional techniques. " It was quickly evident that a large number of ligand-stabilized metal clusters could be synthesized, and extensive exploratory research opened a rich, interesting field. Discovery of unprecedented structural features was the norm and remains common. The fact that each platonic solid (namely, the tetrahedron, octahedron, cube, icosahedron, and pentagonal dodecahedron) is now represented in transition metal cluster chemistry illustrates the structural variety present in this class of compounds. A short preview of particular metal clusters whose geometries approximately conform to these Platonic solids provides an introduction to some of the structural phenomena considered elsewhere in this chapter. 

Regular convex polyhedra are called Platonic solids because they play an important role in Plato s philosophy. Plato (ca. 428-347), however, was not the first to write about regular polyhedra - the mathematician Theatetus (ca. 380 B.C.), a friend of Plato s and pupil of Socrates (ca. 470-399), discovered the octahedron and the icosahedron and was also the first to write about the five regular polyhedra.256... 

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