Dodecahedron, pentagonal

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...

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. 

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. 

Another contribution is represented by an investigation of a cubic thallium cluster phase of the Bergmann type Na13(TlA.Cdi A.)27 (0.24 < x <0.33) (Li and Corbett 2004). For this phase too the body centred cubic structure (space group Im 3, a = 1587-1599 pm) may be described in terms of multiple endo-hedral concentric shells of atoms around the cell positions 0, 0, 0, and 14,14,14. The subsequent shells in every unit are an icosahedron (formed by mixed Cd-Tl atoms), a pentagonal dodecahedron (20 Na atoms), a larger icosahedron (12 Cd atoms) these are surrounded by a truncated icosahedron (60 mixed Cd-Tl atoms) and then by a 24 vertices Na polyhedron. Every atom in the last two shells is shared with those of like shells in adjacent units. A view of the unit cell is shown in Fig. 4.38. According to Li and Corbett (2004), it may be described as an electron-poor Zintl phase. A systematic description of condensed metal clusters was reported by Simon (1981). 

Support for this postulation came from work done on the shape of the ideal foam cell [32-40]. Ross and co-worker [34,35] proposed three minimal geometric structures, i.e. those which will subdivide space with minimum parti-tional area. These were the pentagonal dodecahedron, the minimal tet-rakaidecahedron, originally suggested by Thomson (Lord Kelvin), and the P-tetrakaidecahedron (Fig. 4). 

The pentagonal dodecahedron, however, is not entirely space-filling, i.e. a close-packed array of such figures has a number of interstitial voids. On the other hand, Kelvin s tetrakaidecahedron and the P-tetrakaidecahedron are. The latter requires 4% more surface area, so a system of such figures would spontaneously rearrange to the more stable array of Kelvin cells. Thus, it would seem that Kelvin s tetrakaidecahedron is the ideal candidate nevertheless, this is not observed in real systems Pentagonal faces are shown on foam cells. These... 

Fig. 4. Regular pentagonal dodecahedron (RPD) (a), Kelvin s minimal tetrakaidecahedron (Kelvin s cell) (b) and /J-tetrakaidecahedron (c)...

The PDH complex contains three enzymes—pyruvate dehydrogenase (EJ, dihydrolipoyl transacetylase (E2), and dihydrolipoyl dehydrogenase (E3)—each present in multiple copies. The number of copies of each enzyme and therefore the size of the complex varies among species. The PDH complex isolated from mammals is about 50 nm in diameter—more than five times the size of an entire ribosome and big enough to be visualized with the electron microscope (Fig. 16-5a). In the bovine enzyme, 60 identical copies of E2 form a pentagonal dodecahedron (the core) with a diameter of about 25 nm (Fig. 16-5b). (The core of the Escherichia coli enzyme contains 24 copies of E2.) E2 is the point of... 

The structure is based on a body-centered lattice. At each lattice point there is a small atom (Zn, Al). It is surrounded by an icosahedron of twelve atoms (Fig. 11-14). This group is then surrounded by 20 atoms, at the corners of a pentagonal dodecahedron, each atom lying directly out from the center of one of the 20 faces of the icosahedron. The next 12 atoms lie out from the centers of the pentagonal faces of the dodecahedron this gives a complex of 45 atoms, the outer 32 of which lie at the corners of a rhombic triacontahedron. The next shell consists of 60 atoms, each directly above the center of a... 

Finally, we turn to the pentagonal dodecahedron and the icosahedron. These two polyhedra have the same symmetry. They are related to each other as the cube and octahedron are related. The symmetry elements and operations are as follows. 

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