Platonic solid geometry

Typically, synthetic capsules have a contained volume in the range 200-350 A3. In 1997, we discovered a spherical assembly consisting of [(C-methylresorcin[4]arene)6 (H20)8], 1. This assembly, Figure 1, with an enclosed volume of 1375 A3, was characterized by a single-crystal X-ray diffraction study and was found to be stable in nonpolar solvents [25]. Supramolecular assembly 1 ultimately led to the discovery of the link between 1 and the solid geometry principles of Plato and Archimedes [26], Before this discussion progresses, it is useful to examine briefly Platonic and Archimedean solids. 

When I was 8 or 9, one of my favorite books was on geometry, and it described how to build the Platonic solids. I remember making these polyhedra. This interest has remained with me ever since. I kept making polyhedra from time to time, and some very elaborate ones too. 

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. 

Different representations of a tetrahedral electron geometry (a) the Platonic solid having four faces composed of equilateral triangles, (b) as the shape inscribed by connecting opposite corners of a cube, and (c) in the conventional representation used to draw tetrahedral molecules, [(a) E. Generalic,... 

The transition-metal clusters often display highly symmetrical metal frameworks, normally with symmetries derived from the platonic or archimedian solids or variations thereof. The post-transition element clusters, on the other hand, are not necessarily aflfected by such confinements. However, as will be explained in Sec. 1.29.4, relationships between the structure and the electron count do exist for the naked clusters, and regular, closed geometries are found for a rather large group of these species. For instance, the trigonal prismatic symmetry of the Rhf, framework in Fig. lA is also found in the naked tellurium cluster... 

Metal triangles, tetrahedra, and octahedra form the basic building blocks of transition metal carbonyl clusters. The smaller clusters with between three and six metal atoms often adopt these pseudo-spherical deltahedral geometries, but as the nucle-arity of the cluster increases, condensed structures, built up from the smaller poly-hedra by vertex-, edge-, or face-sharing, tend to be favored in preference to the larger spherical deltahedra based on the Platonic and Archimedean solids. In general, this is in contrast to the structures foimd in borane chemistry. 

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