Platonic Structures

After successes in the syntheses of several bicyclophanes, Wennerstrom and Norinder [158] proposed a highly symmetric cage cyclophane with the tetrahedral arrangement of four benzene rings. Later, Vogtle et al. synthesized a tetrahedral analog with saturated bridges and named it spheriphane (118) [ 159]. 

Other molecules that contain four key aromatic units arranged in a tetrahedral array, do not necessarily have Td symmetry. Distortion creates two groups of lower symmetric structures, D2d and Qv Two examples of what stem from D2d symmetry are the adamantane-like macrotricycle (119, R=C6H13), synthesized by Moore and Wu [160], and Tani et al. s cross-oriented biphenylophane (120) [134] obtained with its parallel polycyclic isomer (99). 

Another group of distorted tetrahedra contains triangular pyramidal graphs with C3v symmetry. Vogtle and Wambach were the first to synthesize compound 

121 and several of its analogs in which the floor plate benzene is the bottom of the basket or the peak of the pyramid [161]. Still et al. designed and synthesized 

Alternatives to the square prisms are known from Cram s container molecules called carcerands [183,184]. Molecules, such as 132 [185,186], are made by quadruply linking two cavitand units, thus creating a void inside capable of incorporating a small organic molecule which is then called a carceplex. The encapsulated guests vary from simple solvent molecules to highly reactive 

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Norberg-Schultz, 1985. This curious affectation might be neo-platonic. Structure is being forced back under the surface of things, leaving only vestiges. 

FIGURE 2 Extended Schlafli space of the Platonic Structures. 

Thus in the diamond network, which corresponds to the Platonic (integer) topology of the Platonic polyhedra, one can readily trace the uniform 6-gon, puckered circuitry of the network connected together by all 4-con-nected, tetrahedral vertices. Diamond s topology classifies the network as a regular, Platonic structure-type. 

This limitation was already painfully obvious to the organic chemists in the 1880s these are statie struetures, whereas of eourse any moleeule at any temperature is a jelly-like pulsating, librating and vibrating entity. Only a terribly simplistic eye would see a molecule frozen into this Platonic archetype of the structural formula. 

Thus, it is herein that we now describe the results of this analysis which we regard as the development of a general strategy for the construction of spherical molecular hosts. [11] We will begin by presenting the idea of self-assembly in the context of spherical hosts and then, after summarizing the Platonic and Archimedean solids, we will provide examples of cubic symmetry-based hosts, from both the laboratory and nature, with structures that conform to these polyhedra. 

Thus, the Platonic and Archimedean solids not only provide a means for host design, but a way in which to maximize chemical information, allowing the chemist to simplify the structures of complex molecular frameworks and, in effect, engineer host-guest systems. 

Historically, most chemists have modeled the structure of molecules using a highly idealized platonic representation, where atoms are represented as vertices and bonds as paths between vertices. Chemoinformatics has very successfully adopted this representation and based many of its techniques around the metaphor of the connection table , i.e., a list of all atoms and bonds, which occur in the molecule. While this approach is quite successful for well defined chemical entities, it begins to break down for rapidly interconverting isomers, for example, and is completely inappropriate for polymers. In the majority of cases, the successful application of chemoinformatics to a given problem depends on the availability of a connection table. 

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 10.47 The 13 Archimedian semiregular solids, constructed from combinations of two or more different two-dimensional figures. Note that, in addition to the triangles, squares and pentagons allowed in the Platonic solids, hexagons, octagons and decagons also make an appearance, but always in conjunction with other shapes. In particular, structure (j) corresponds to buckminsterfullerene, C60, which requires pentagons to effect spatial closure, even though the structure consists predominantly of hexagons.

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. 

Examination of the vertical column entitled Platonic hydrocarbon in Fig. 6 illustrates a dimensional progression of exoskeleton structures beginning with tetrahedrane -> cubane -> dodecahedrane -> buckminsterfullerene. This progression shows how space may be incarcerated from the sub-nanoscopic to the nanoscopic level by geometric closure with exoskeleton structures. 

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