Polyhedral model

Fig. 2.13 Structure of magnetite, a) Polyhedral model with alternating octahedra and tetrahedra-octahedra layers, b) Ball-and-stick model. Unit cell outlined, c) Ball-and-stick model ofthe arrangement of octahedra and tetrahedra. (a-c Stanjek, unpubl.)...

A similar outline drawing for the construction of buckminsterfullerene has been published.2 Several other polyhedral models constructed from simple materials have been described in a series of articles.3... 

Fig. 4. Polyhedral model of [Wi0O32]. The arrows show possible types of electron transfer with corresponding parameters.

Many models for the structure of liquid water have been proposed. One of the most useful is the polyhedral model. 

The polyhedral model is useful for understanding the properties of liquid water. 

These properties show that in liquid water there still exist a large number of intermolecular hydrogen bonds, thus lending support to the polyhedral model. 

Figure 7 Drawings of polyhedral models of (a), the M blocks BaFei20i9, and (b) the Y block, Ba2A/ 2Fei202 2 which stack along the hexagonal c-axis to form the barium ferrites (Reproduced by permission from Contemp. Phys., 1975, 16, 443)...

A polyhedral model of hectorite is shown in Figure 3 of [97]. Figure 1 of [98] shows ball-and-stick model of kaolinite. A combined polyhedral/ball-and-stick model of kaolinite is shown in Figure 1 of [99]. A ball-and-stick model of kaolinite is shown in Figure 1 of [100]. A ball-and-stick model of the 010 plane of montmorillonite is shown in Figure 1 of [101]. Ball-and-stick models of kaolinite, pyrophyllite, and illite are shown in Figure 7 of [60]. 

Actual foam contains bubbles whose shape is intermediate between spheres and polyhedra. Such foam is said to be cellular [214, 280]. The distinction between the cellular and polyhedral kinds of foam is rather conventional and is determined by very low moisture contents (of the order of some tenth of per cent). Nevertheless, the polyhedral model of foam cells is used rather frequently [38,125,244,438,480],... 

It was repeatedly proposed to use Kelvin s tetrakaidecahedron (that is, minimal truncated octahedron) [381, 407, 479] with eight hexagonal and six quadrangular faces as the polyhedral model of a foam cell and of a cell of any three-dimensional biological tissue. Note, however, that it was statistically shown [195] that Kelvin s tetrakaidecahedron is encountered in biological tissues among other tetrakaidecahedral cells only in 10% of the cases. 

Thus, in approximate modeling it is possible to use the following polyhedral model of the foam structure [214] (see Figure 7.1) ... 

Kepler s polyhedral model is beautifully illustrated and described on page 95 of the book by 1st-van and Magdolna Hargittai, Symmetry—Unifying Concept, Shelter, Bolinas, CA, 1994. This book also inspired my use of the polyhedra in Kepler s Harmonices Mundi. 

Favorable packing of the CO ligands about the cluster core was qualitatively and later (semi)quantitatively" investigated by Johnson et al. who, while searching for the best fit between the metal cluster and the optimum ligand polyhedron, eventually derived what is known as the Ligand Polyhedral Model (LPM). According to the LPM ... 

Lewis s cubical atom [3-72] deserves special mention. It was instrumental in shaping the concept of the shared electron pair. It also permitted a resolution of the apparent contradiction between the two distinctly different bonding types, viz., the shared electron pair and the ionic electron-transfer bond. In terms of Lewis s theory, the two bonding types could be looked at as mere limiting cases. Lewis s cubical atoms are illustrated in Figure 3-51. They are also noteworthy as an example of a certainly useful though not necessarily correct application of a polyhedral model. 

Photo 18 (right) Linus Pauling in the early 1950 s, studying a polyhedral model that may be related to structures of intermetallic compounds. 

FIGURE 9.10 Polyhedral models of some Mn minerals (a) bimessite with divalent (Mg) ions in the interlayer (b) bimessite with monovalent (Na) ions in the interlayer—note that Na+ ions are in the center, whereas ions in bimessite are closer to one of the sheets (c) lithioporite, with an (Al, Li)-OH sheet stacked in the z direetion (d) todorokite, showing the channels containing cations surrounded by water molecules. (Reprinted from Dixon, J. B. and White, G. N., Soil Mineralogy with Environmental Applications, Soil Science Society of America, Madison, WI, 367-388, 2002, with kind permission.)... 

FIGU RE 9.11 Polyhedral models of TiOj polymorphs (a) rutile, (b) anatase, and (c) brookite. (Reprinted with kind permission from Dr. K. Nomura, Intermediate Temperature Solid Conductors—Crystal Structure Gallery—AIST, http //staff.aist.go.jp/nomura-k/english/ itscgallary-e.htm.)... 

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