Archimedean

The presence of scroll helicity replaces a set of concentric cylinders by a single sheet rolled upon itself (Fig. 1). Assuming that the distance between the successive rolls of the scroll is constant, its cross-section can be conveniently represented in polar coordinates by the Archimedean spiral ... 

Structural characteristics of compounds with X Me = 8 are collected in Table 17. Na3NbF8 and Na3TaF8 compounds that form similar crystal structure [77], The structure of Na3TaF8 was determined by Hoard et al. [136], by means of X-ray diffraction of a single crystal. Na3TaF8 is composed of sodium cations and isolated complex ions TaF83, in an Archimedean antiprism configuration, as shown in Fig. 23. 

Fig. 3.4.5 Schematic description of the two-dimensional SPIRAL-SPRITE technique. Cx and Gy are the phase encode magnetic field gradients that are amplitude cycled to traverse /(-space along an Archimedean Spiral. A single data point is acquired from the FID at a fixed encoding time tp after an rf excitation pulse. TR is the time between rf pulses.

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

A scroll vacuum pump uses two interleaved Archimedean spiral-shaped scrolls to pump or compress gases (see Fig. 1.9). One of the scrolls is fixed, while the other orbits eccentrically without rotating, thereby trapping and compressing gases between the scrolls and moving it towards the outlet. 

The intensely yellow compounds 10 and 11 crystallize in the tetragonal space group PA/nnc. The oval Ei0Lii6 cluster skeleton (E = P, As) is topologically best described as a twofold capped Archimedean antiprism formed by the 10 E atoms, of which each of the 16 triangular faces are occupied by a lithium center (Fig. 12) (21). 

Fig. 12. (Left) Molecular structure of the mixed-valent cluster 10 and 11 in the crystal. (Right) Representation of the E10-dicapped-Archimedean antiprism as anionic partial structure in 10 and 11 (E = P, As). For clarity the organic groups of the silyl groups have been omitted.

If we have N hard spheres (of radius rs) forming a close-packed polyhedron, another sphere (of smaller radius rc) can fit neatly into the central hole of the polyhedron if the radius ratio has a well-defined value (see also 3.8.1.1). The ideal radius ratio (rc/rs) for a perfect fit is 0.225.. (in a regular tetrahedron, CN 4), 0.414.. (regular octahedron CN 6), 0.528.. (Archimedean trigonal prism CN 6), 0.645... (Archimedean square antiprism CN 8), 0.732.. (cube CN 8), 0.902... (regular icosahedron CN 12), 1 (cuboctahedron and twinned cuboctahedron CN 12). 

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. 

To the best of our knowledge, there is one host which conforms to the structure of an Archimedean dual. Harrison was the first to point out that the quaternary structure of ferritin, a major iron storage protein in animals, bacteria, and plants, corresponds to the structure of a rhombic dodecahedron. [45] This protein, which is approximately 12.5 nm in diameter, consists of 24 identical polypeptide subunits (Fig. 9.18), and holds up to 4500 iron atoms in the form of hydrated ferric oxide with... 

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. 

Smith, J.V. and Bennett, J.M. (1981) Enumeration of 4-connected 3-dimensional nets and classification of framework silicates the infinite set of ABC-6 nets the Archimedean and. sigma.-related nets. Am. Mineral., 66 (7-8), 777-788. 

Fig. 33. Skeletons of the gold clusters (Au(AuL)8] and [Au(AuL)io] compared with the icosahedron and the bicapped Archimedean antiprism. Central atoms are shaded peripheral atoms are unshaded.

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