Space-filling arrangements of polyhedra

The polyhedra in the space-fillings of Andreini do not include the regular or semi-regular polyhedra with 5-fold symmetry. It is not possible to fill space with regular dodecahedra or icosahedra or the Archimedean solids derived from them (or with combinations of these polyhedra) owing to the unsuitable values of the 

Space-filling arrangements of regular and semi-regular polyhedra. 

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In certain 3D nets there are well-defined polyhedral cavities, and the links of the net may alternatively be described as the edges of a space-filling assembly of polyhedra. At least four links must meet at every point of such a net, and the most important nets of this kind are, in fact, 4-connected nets. Space-filling arrangements of polyhedra leading to such nets are therefore described after we have dealt with the simpler 4-connected nets. 

These structures provide an elegant example of the interrelations of nets, open packings of polyhedra, space-filling arrangements of polyhedra, and the closest packing of equal spheres. 

It may often be convenient to describe the crystal structure in terms of the domains of the atoms [40], The domain is the polyhedron enclosed by planes drawn midway between the atom and each neighbor, these planes being perpendicular to the lines connecting the atoms. The number of faces of the polyhedral domain is the coordination number of the atom and the whole structure is a space-filling arrangement of such polyhedra. 

Since the water framework is formed by a space-filling arrangement of face-sharing polyhedra, 12 (i.e., 24/2) of the 44 protons in the [20H2O+4OH]+ form disordered bonds with the adjacent polyhedra. 

Class (b). In this class the (4-connected) networks are the edges of space-filling arrangements of pentagonal dodecahedra and one or more of the related polyhedra /s = 12,/e = 2, 3,4 (Table 15.3). 

Silanation reaction of silane (SiELi) with OH groups Sorption uptake of liquid or gas by a microporous material Space-filling spatial arrangement of polyhedra such that each polyhedron shares all its faces with other polyhedra Stacking fault misalignment of layer(s) arising from a fault plane... 

All the polyhedra intermediate between the dodecahedron and truncated icosahedron can be realized except /s = 12,/ = 1, and some in more than one form (different arrangements of the 5-gon and 6-gon faces). A number of these polyhedra are of interest in connection with the structures of clathrate hydrates (p. 543), because certain combinations of these solids with dodecahedra form space-filling assemblies in which four edges meet at every vertex. Two of these polyhedra, a tetrakaidecahedron and a hexakaidecahedron, are illustrated in Fig. 3.4. 

Such a view of crystal structures leads to a simplistic but nonetheless very useful concept of a silicate mineral, that is, a mineral is an arrangement of boxes in space (the coordination polyhedra), and we construct such a mineral by filling the boxes with appropriate cations. In a simple structiue there might be only two kinds of boxes, representing tetrahedra and octahedra, appropriately linked together. Thus, we can change the chemical composition of a mineral by replacing all or part of the cations in one type of box by... 

The way of the best choice to model PS s structure on both molecular and supramolecular levels begins with allocation of primary building units (PBUs), which without gaps and overlaps would fill a 3D space occupied by a PS. An universal method for allocation of such PBUs in both ordered and randomly arranged PSs, formed of packings of convex particles (or pores), is based on the construction of the assembles of Voronoi polyhedra (V-polyhedra) and Delaunay simplexes (or D-poly-hedra), which form Voronoi-Delaunay tessellation [100],... 

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