Cube, snub

Fig. 2. Parameter plot for NaZn13, showing the line corresponding to a regular icosahedron, the point corresponding to a regular snub cube, and the parameters actually obtained in the previous and present investigations.

A regular snub cube (i.e. one with all edges equal) requires the unique values... 

In Fig. 2 conditions (11) and (12) are plotted on a parameter map together with the parameter values reported for NaZn13 by Zintl Hauke, those reported for KCd13 by Ketelaar, and those determined for NaZn13 in the present investigation. The uncertainties are indicated by the radii of circles drawn around the points determined by the parameter values since Zintl Hauke reported no uncertainty value, the uncertainty reported by Ketelaar (0-003) was assumed. It is seen that the parameter values obtained in the present work lie between those of Zintl Hauke and those of Ketelaar, and that they differ considerably from the values required either by a regular icosahedron or by a regular snub cube. 

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. 

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),...

We have recently demonstrated the ability of six resorcin[4]arenes and eight water molecules to assemble in apolar media to form a spherical molecular assembly which conforms to a snub cube (Fig. 9.3). [10] The shell consists of 24 asymmetric units - each resorcin[4]arene lies on a four-fold rotation axis and each H2O molecule on a three-fold axis - in which the vertices of the square faces of the polyhedron correspond to the corners of the resorcin[4]arenes and the centroids of the eight triangles that adjoin three squares correspond to the water molecules. The assembly, which exhibits an external diameter of 2.4 nm, possesses an internal volume of about 1.4 A3 and is held together by 60 O-H O hydrogen bonds. 

Using information obtained from X-ray crystallography, we have described the structure of a chiral, spherical molecular assembly held together by 60 hydrogen bonds. [10] The host, which conforms to the structure of a snub cube, self-assembles in apolar media and encapsulates guest species within a cavity that possesses an internal volume of approximately 1.4 nm3. 

HaS196] R. H. Hardin and N. J. A. Sloane, McLaren s improved snub cube and other new spherical designs in three dimensions, Discrete and Computational Geometry 15 (1996) 429-441. 

Atwood et al. have gone on to prepare a truly enormous molecular capsule along these principles, which conforms to the geometry of a snub cube (structure (1) in Figure 10.47). Reaction of six equivalents of the octol [4] resorcarene (with either methyl or undecyl feet ) with eight water molecules results in... 

Figure 10.48 (a) Space filling view of the structure of the snub cube (f) formed from six... 

Figure 11.1. Calix[4]resorcinarene, which forms a snub cube with 60 hydrogen bonds.

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 above sections examples of the snub cube, the great rhombicuboctahedron, and the small rhombicuboctahedron have been presented. The guests are badly disordered for all of the capsules except for that made from p-sulfonatocalix [4]arene anions, pyridine V-oxide, and lanthanide ions. 

Figure 2-61. Two artistic representations of semi-regular polyhedra (photographs by the authors), (a) Snub cube fountain in Pasadena, California [98] (b) Cubocta-hedron on top of a garden lantern in the Shugakuin Imperial Villa in Kyoto [99],...

W. P. Schaefer, The Snub Cube in the Glanville Courtyard of the Beckman Institute at the California Institute of Technology. Chemical Intelligencer 1996, 2(4), 48-50. 

Four other polyhedra based on 24-vertex cages exhibiting cubic symmetry can be formed from the regular orbit structure of the previous section. Two of these are chiral pairs, the dextro snub cube and its chiral partner, the laevo snub cube the third is the regular orbit of Td point symmetry, while the fourth is the regular orbit of Th symmetry. 

In Figure 2.14a, the 48-vertex structure of the great rhombicuboctahedron is divided into two sets of 24 points, coloured to distinguish two sets related by the inversion operation. Each set of 24 vertices now exhibit O symmetry and are examples of the chiral polyhedra based on the snub cube structure, displayed as the ri-isomer as a projection in Figure 2.14b and as a perspective drawing in Figure 2.14b. 

The various rotational axes can be identified by examination of the snub cube structure. Figure 2.14c, which spans the regular orbit of O. As the lower orbits, Oe, Os and O12, Figures 2.14d-f, are all intrinsically achiral, any object of O symmetry must contain at least one copy of the chiral regular orbit, it is allowed that the lower order orbits Oe, O12 and Os of Oh symmetry can be formed by coalescing appropriate local sets of vertices of the regular orbit onto the poles of the rotational axes as shown in Figures 2.15d-f. 

Figure 2.14 Division of the regular 48-vertex orbit of 0[, symmetry (a) into the two 24-point sets of the chiral snub cube structure. The dextro projection (b) is also drawn in perspective (c). Coalescing sets of vertices onto the rotational poles leads to the other orbits O, O12 and Os of the O symmetry group. These are achiral they are also orbits of 0[,.

In the XYj/ and XYj3 structures the X atoms occupy holes in which they are surrounded by 22 and 24 Y atoms respectively. For a given Y atom the smaller X metals form the XYx 1 phase and the larger ones the XYj 3 phase, while if X is too large even the XYj 3 phase is not formed (Table 29.8). This XY13 structure is of particular interest on account of the coordination polyhedra. The X atom has 24 Y neighbours at the vertices of a nearly regular snub cube, which is one of the less familiar Archimedean solids and has 6 square faces parallel to those of a cube and... 

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