Dodecahedrons

The truncated octahedron and the rhombic dodecahedron provide periodic cells that are approximately spherical and so may be more appropriate for simulations of spherical molecules. The distance between adjacent cells in the truncated octahedron or the rhombic df)decahedron is larger than the conventional cube for a system with a given number of particles and so a simulation using one of the spherical cells will require fewer particles than a comparable simulation using a cubic cell. Of the two approximately spherical cells, the truncated octahedron is often preferred as it is somewhat easier to program. The hexagonal prism can be used to simulate molecules with a cylindrical shape such as DNA. 

The most important stmctural variables are again polymer composition, density, and ceU size and shape. Stmctural foams have relatively high densities (typically >300 kg/m ) and ceU stmctures similar to those in Figure 2d which are primarily comprised of holes in contrast to a pentagonal dodecahedron type of ceU stmcture in low density plastic foams. Since stmctural foams are generally not uniform in ceU stmcture, they exhibit considerable variation in properties with particle geometry (103). 

Fig. 11. Clathrate hydrates (a) basic structural component (H4QO2Q pentagonal dodecahedron) (b) type I host stmcture (two face-sharing 14-hedra are...

Figure 16.2 The icosahedron (top) and dodecahedron (bottom) have identical symmetries but different shapes. Protein subunits of spherical viruses form a coat around the nucleic acid with the same symmetry arrangement as these geometrical objects. Electron micrographs of these viruses have shown that their shapes are often well represented by icosahedra. One each of the twofold, threefold, and fivefold symmetry axes is indicated by an ellipse, triangle, and pentagon, respectively.

Tellurium nitride was first obtained by the reaction of TeBt4 with liquid ammonia more than 100 years ago. The empirical formula TeN was assigned to this yellow, highly insoluble and explosive substance. However, subsequent analytical data indicated the composition is Tc3N4 which, in contrast to 5.6a and 5.6b, would involve tetravalent tellurium. This conclusion is supported by the recent preparation and structural determination of Te6N8(TeCl4)4 from tellurium tetrachloride and tris(trimethylsilyl)amine (Eq. 5.5). The TceNs molecule (5.12), which is a dimer of Tc3N4, forms a rhombic dodecahedron in which the... 

The most symmetrical structure possible is the cube Oh but, except in extended ionic lattices such as those of CsCl and CaF2, it appears that inter-ligand repulsions are nearly always (but see p. 1275) reduced by distorting the cube, the two most important resultant structures being the square antiprism D4h and the dodecahedron Did (Fig. 19.10). 

Figure 19.10 (a) Conversion of cube to square antiprism by rotation of one face through 45° (b) Conversion of cube into dodecahedron. 

Eight O atoms form a dodecahedron around the Ti and the 4 N atoms form a flattened tetrahedron. 

Granat-baum, m. pomegranate tree, -dode-Icaeder, n. rhombic dodecahedron. 

Zwdlfflachner, m. dodecahedron. zwdUseitig, a. twelve-sided, dodecahedral do-decagonal, zwblfte, a. twelfth. 

In one of the cages within which gas molecules are trapped in methane hydrate, water molecules form a pentagonal dodecahedron, a three-dimensional figure in which each of the 12 sides is a regular pentagon. 

Dodecahedron of oxygen atoms surrounded by four out of a total of twelve tetrakaidecahedra. 

Fig. 1. The structure of gas hydrates containing a hydrogen-bonded framework of 46 water molecules. Twenty molecules, arranged at the comers of a pentagonal dodecahedron, form a hydrogen-bonded complex about the comers of the unit cube, and another 20 form a similar complex, differently oriented, about the centre of the cube. In addition there are six hydrogen-bonded water molecules, one of which is shown in the bottom face of the cube. In the proposed structure for water additional water molecules, not forming hydrogen bonds, occupy the centres of the dodecahedra, and...

We have found that the proposed structure of water, based upon the centred pentagonal dodecahedron, accounts in a reasonably satisfactory way for several properties of water, including the dispersion of dielectric constant and the radial distribution curve as determined by x-ray diffraction. A detailed description of this work will be published later. 

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

Replacement of each water molecule by an MnAli2 icosahedron with shared faces leads to an infinite framework with 136 Mn and 816 Al atoms in the unit cube. This framework is similar to the framework of covalently bonded carbon atoms in a diamond crystal, with one body diagonal of each pentagonal dodecahedron in place of each C—C covalent bond. 

There is an interesting tendency of the ring strains of the large cage molecules. Persiladodecahedrane Sij Hj (SE = 32.3 kcal mol ) is less strained than the carbon congener, dodecahedron (SE = 43.6 kcal mol ) [76, 77]. Low strain of silicon congener is more apparent for persilafulleran (SE =114 kcal... 

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