Platonic bodies

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. 

Balasubramanian67 reported all irreducible representations of the octahedral (cubic) symmetry for vertex-, face- and edge-colorings of a pair of closely related Platonic bodies, the octahedron and its dual the cube using multinomial combinatorics. Combinatorial tables for all irreducible representations and all multinomial partitions were constructed and visualized by the Young tableaux. 

The ancient Greek philosophers searched harmony in the universe as well as in the human arts. Certain geometric constmctions are addressed as platonic bodies. The school named after Pythagoras tried to measure harmony in numbers. As simpler a ration of numbers that are characterizing a system, more harmonic the system is itself. The starting point was problems of music, because here harmony is very easy to demonstrate. 

But making new molecules is also a creative endeavor of the synthetic chemist. I shall mention the beautiful cage molecule dodecahedrane, (CH)2o and the race to make this molecular mimic of one of the classical Platonic bodies. Our myths of beauty and symmetry come to life in chemistry because of its creative element. 

The tetrahedron is one of the perfect Platonic bodies, which are shown in Figure 4.3 and are known by the names tetrahedron, octahedron, cube, dodecahedron, and icosahedron. These bodies were thought by the Greeks to symbolize the fom elements and a fifth one called Quintessence. The tetrahedron symbolized Fire, the octahedron Air, the cube Earth, and the icosahedron Water. According to this philosophy, the... 

FIGURE 4.4 The abstract dodecahedron body (a Platonic body) in (a) and the 3D representations of the dodecahedrane molecule, (CH)2q, in (b) and (c). The representation in (b) shows all the C—C and C—H bonds as sticks, while the one in (c) shows the packing of the atomic spheres, C (blue) and H (gray). (The dodecahedron image on the left-hand side of the figure was obtained from http //commons.wikimedia.0rg/wiki/File Zeroth stellation of dodecahedron.png.)... 

Figure 10.2 The Platonic bodies with their corresponding (n,p) designation. The icosahedron was once thought to be impossible to find in inorganic structures, but, as pointed out by Coexter in the second edition of his book, is in fact the structure of the B 2 molecule [5,6].

This is the same equation as 10.5, giving the five Platonic bodies as the solutions for integer values of n and p. For an infinite number of vertices the denominator should be zero, thus yielding equation 10.4 again. In this formulation it is not evident that these solutions should give 2D-nets- However, an enclosed volume covered by an infinite number of faces also has infinite values of n and p. Since 10.4 gives finite values of n and p these solutions have to correspond to a 2D-net,... 

As can be seen in Figure 10.12 we arc now talking about very different polyhedra from the five Platonic bodies. Each type of polyhedra (there can be several types of polyhedra used in the tiling) will have a certain number of faces defined by the rings (r), a certain number of edges (q) and a certain number of vertices, (p). The number of different types of polyhedra is s. 

FIGURE 13 Platonic body-centered cubic (b.c.c.) structure as the structure of CsCl, this network can be represented by the Wells point symbol and lies in space group (Im-3m). 

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 ... 

Fig. 1 The Platonic solids are the most beautiful bodies according to Plato. Like molecules, these imperceptibly small objects were thought to compose the physical world...

There is a large body of evidence that describes the behaviour of compounds containing pentacoordinate phosphorus, and these types of species are certainly involved as intermediates in many reactions. Holmes (1980b) has presented a very complete picture of the nature of such species. Thus the corner of the three-dimensional energy diagram contains the five-coordinate phosphorus as a platonic ideal whose existence has been demonstrated as a reality in many instances. Concerted reactions may avoid this corner, but it is certain that stepwise substitution processes involving these species as intermediates do exist (Kluger and Thatcher, 1986). 

He was familiar with the five regular Platonic solids. Like Pythagoras he accepted a fundamental relationship between physical reality and number, manifesting itself in constructs such as the tetraktys and the golden ratio, shown below. He surmised that the distances of the heavenly bodies from... 

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

Clarke s reference to eye-beams relies on an understanding of an active eye that emits pneuma to unite with particles projected by the object. This way of seeing also resembles the neo-platonic theory of love - a concept that popular texts, such as The Problems of Aristotle, rendered in elementary form the louer sendeth co[nt]inuall beames of the eie towards that which he loueth. And those beames are like vnto arrowes, because the louer doth dart them into the bodie (sig. Lyv). The evil eye depends on the same principles -vapours are emitted from one person s eye to penetrate the victim s eyes. 

The alchemists of this period who made useful contributions to the development of chemistry were not totally distracted by the attempt to make gold artificially. Most of these workers were iatrochemists, and Paracelsus has already been mentioned (Chapter 2). He too was influenced by the renewed interest in neoplatonism, and he applied the neo-Platonic doctrine of the microcosm and macrocosm to medicine. He saw the human body as a microcosm of all that existed in the universe (the macrocosm), and thus the organs of the body were the equivalents of the stars. Such mystical ideas resulted in Paracelsus concentrating... 

Humans are twofold in nature mortal in body, but immortal in essence. Even though we are immortal, as long as we are bound in bodies we are subject to fate as the author of Poimandres makes clear, although humans are above the cosmic framework, they became slaves within it Poim. 15). This fundamental dualism—the direct heritage of Platonic thought—means that the body is fundamentally connected with sexual intercourse and the cycle of physical birth and death ... 

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