Polyhedron, regular

In accord with and in addition to Plato s view of convex polyhedra, there is only a limited number of polyhedra, regular or irregular, mostly more or less distorted, that are observed as cluster polyhedra, mostly deltahedra. 

Table D.2. Five platonic regular polyhedrons Regular polyhedron Description Volume Surface area No. of No. of No. of...

These local stmctural rules make it impossible to constmct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fiU space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal stmcture that can serve as a convenient mathematical idealization of polyhedral foam stmcture. Lord Kelvin considered this problem, and his minimal tetrakaidecahedron is considered the periodic stmcture of polyhedra that most nearly satisfies the mechanical constraints. 

Oxygen chelates such as those of edta and polyphosphates are of importance in analytical chemistry and in removing Ca ions from hard water. There is no unique. sequence of stabilities since these depend sensitively on a variety of factors where geometrical considerations are not important the smaller ions tend to form the stronger complexes but in polydentate macrocycles steric factors can be crucial. Thus dicyclohexyl-18-crown-6 (p. 96) forms much stronger complexes with Sr and Ba than with Ca (or the alkali metals) as shown in Fig. 5.6. Structural data are also available and an example of a solvated 8-coordinate Ca complex [(benzo-l5-crown-5)-Ca(NCS)2-MeOH] is shown in Fig. 5.7. The coordination polyhedron is not regular Ca lies above the mean plane of the 5 ether oxygens... 

A set of principles governing the structure of complex ionic crystals, based upon the assumption of a coordinated arrangement of anions about each cation at the comers of an approximately regular polyhedron, is formulated with the aid of considerations based upon the crystal energy. Included in the set is a new electrostatic principle which is of wide application and considerable power. 

You say that your nonlinear molecule has the high symmetiy of a regular polyhedron, such as a tetrahedron, cube, octahedron, dodecahedron, icosahedron,... sphere. If it is a sphere, it is monatomic. On the other hand, if it is not monatomic, it has the symmetry of one of the Platonic solids (see the introduction to Chapter 8). 

You have replied that your molecule, that is not a regular polyhedron, does not have a proper rotation axis of order greater than one. If its only symmetry element is a plane, it belongs to the group 6Jih a... 

Five points can be arranged on the surface of a sphere such that they are all equivalent, only in a planar pentagonal arrangement, which does not maximize the distance between the points. In other words, there is no regular polyhedron with five equivalent vertices. There are two... 

If we have N hard spheres (of radius rs) forming a close-packed polyhedron, another sphere (of smaller radius rc) can fit neatly into the central hole of the polyhedron if the radius ratio has a well-defined value (see also 3.8.1.1). The ideal radius ratio (rc/rs) for a perfect fit is 0.225.. (in a regular tetrahedron, CN 4), 0.414.. (regular octahedron CN 6), 0.528.. (Archimedean trigonal prism CN 6), 0.645... (Archimedean square antiprism CN 8), 0.732.. (cube CN 8), 0.902... (regular icosahedron CN 12), 1 (cuboctahedron and twinned cuboctahedron CN 12). 

Figure 13.15 shows a design based on a regular polyhedron, the pentagon [Himmelblau (1970)]. The experimental design matrix is... 

Although the connected coordination polyhedron approach is useful in crystal chemistry, it has certain implicit rigidity which makes it difficult to extend to less regular and more complex structures. Several alternative approaches to the description... 

The electron-counting scheme which allows to understand this situation and which also works for the simpler clusters was developed from the MO theory for boron hydride polyhedra 389-393). It rests on the fact that a regular triangular faced polyhedron of n boron atoms in the molecule requires n + 1 bonding... 

A by-product of the preparation of U(Et2NC02)4 from the tetrachloride and the amine is a product of composition U402(Et2NC02)i2 this is a tetramer in which there are two non-equivalent uranium(lV) sites. In one of them the coordination geometry is a distorted tricapped trigonal prism, and the geometry of the other does not fit any type of regular polyhedron.1 5... 

Proteins with cyclic or dihedral symmetry are particularly common. More complex rotational symmetries are possible, but only a few are regularly encountered. One example is icosahedral symmetry. An icosahedron is a regular 12-cornered polyhedron having 20 equilateral triangular faces (Fig. 4-24c). Each face can... 

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