Archimedean solid

Rhombitruncated cuboctahedron (8), Truncated dodecahedron (9), Truncated icosahedron (10), Rhombicosidodecahedron (11), Snub dodecahedron (12), Rhombitruncated icosidodecahedron (13) (see also Table 9.2). 

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Fig. 9.2 (a) X-ray crystal structure of 1 and (b) the snub cube, one of the 13 Archimedean solids. The square faces of the snub cube correspond to the resorcin[4]arenes the eight shaded triangles that adjoin three squares correspond to the water molecules of 1. 

The discovery that members of the resorcin[4]arene family self-assemble to form 1, owing to its classification as an Archimedean solid, prompted us to examine the topologies of related spherical hosts with a view to understanding their structures on the basis of symmetry. In addition to providing grounds for classification, we anticipated that such an approach would allow us to identify similarities at the structural level, which, at the chemical level, may not seem obvious and may be used to design large, spherical host assemblies similar to 1. 

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. 

Fig. 9.9 The 13 Archimedean solids, in order of increasing number of vertices. Truncated tetrahedron (1), Cuboctahedron (2), Truncated cube (3), Truncated octahedron (4), Rhombicubocta-hedron (5), Snub cube (6), Icosidodecahedron (7),...

As stated, the Archimedean solids constitute a family of 13 convex uniform polyhe-dra made up of two or more regular polygons and, like the Platonic solids, possess either 32, 432, or 532 symmetry. As a result, the three coordinate directions within each solid are equivalent, making these polyhedra, in addition to the Platonic solids, models for spheroid design. 

Buckminsterfullerene, an allotrope of carbon, is topologically equivalent to a truncated icosahedron, an Archimedean solid that possesses 12 pentagons and 20 hexagons (Fig. 9-16). [17] Each carbon atom of this fullerene corresponds to a vertex of the polyhedron. As a result, C6o is held together by 90 covalent bonds, the number of edges of the solid. 

Fig. 9.17 The 13 Archimedean duals derived from corresponding Archimedean solids (see Fig. 9.9). Triakis tetrahedron (1), Rhombic dodecahedron (2), Triakis octahedron (3), Tetrakis hexahedron (4), Deltoidal icositetrahedron (5), Pentagonal icositetrahedron (6), Rhombic tri-... 

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. 

Indeed, we anticipate that the Platonic and Archimedean solids may be used for the construction of hosts which conform to those solids not yet realized and additional members of each family, where supramolecular synthesis, via self-assembly, will play a major role in their design, ushering in an era of spherical host-guest chemistry. 

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. 

Each face of a particular solid is the same shape tetrahedra, octahedra and icosa-hedra are composed of equilateral triangles cubes have square faces dodecahedra are composed of regular pentagons. The Archimedean solids are composed of at... 

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. 

In addition to the Platonic solids, there exists a family of 13 convex uniform polyhedra known as the Archimedean solids. Each member of this family is made up of at least two different regular polygons and may be derived from at least one Platonic solid through either truncation or twisting of faces (Figure 3, Table 2). In the case of the latter, two chiral members, the snub cube and the snub dodecahedron, are realized. The remaining Archimedean solids are achiral. 

We were also able to link the spherical assembly to the Archimedean solid known as the snub cube, Table 2. In a recent review, we have set forth structural classifications and general principles for the design of spherical molecular hosts based, in part, on the solid geometry ideas of Plato and Archimedes [27]. Indeed,... 

In the following discussion, the advantages of the use of the Platonic and Archimedean solids as models for supramolecular assemblies are clearly pointed out. However, the limitations in the use of these models in complex supramolecular assemblies are also revealed. Both the power and the limitations of this approach are made clear with regard to assembly of / -su 1 lbnatocal ix[4]arene anions, 4, into the large spherical twelve-p-sulfonatocalix[4]arene entity 4 (Figure 4). 

A further set of semi-regular polyhedra, many of which are also important in Chemistry, follows on relaxation of the requirement for equivalent polygonal faces. The thirteen Archimedean polyhedra have equivalent vertex figures on all vertices and all faces remain planar and equilateral, but are of two or three distinct kinds. In orbit terms, the Platonic and Archimedean polyhedra all have single orbits of vertices, but, whereas, the face centres of Platonic solids also fall into single orbits, those of Archimedean solids span either two or three. Again, the Archimedean solids fall into cubic and icosahedral families. 

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