The regular solids

The Greeks had a considerable knowledge of polyhedra, but only during the past two hundred years or so has a systematic study been made of their properties, following the publication in 1758 of Euler s Elementa doctrinae solidorum. From Euler s relation between the numbers of vertices (A o) edges (A i), and faces N ) of a simple convex polyhedron 

I In addition to the five convex regular solids there are four stellated bodies, produced by extending outwards the faces of the convex solids until they meet. 

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Equilibrium in any reaction is determined by a compromise between tendency toward minimum energy f golf balls roll downhill ) and tendency toward maximum randomness. Reaction (29) and reaction (30) both involve increase in randomness since the regular solid lattice dissolves or melts to become part of a disordered liquid state. Both reactions produce ions. But reaction (29) proceeds readily at 25°Q whereas reaction (30) does not... 

Figure 2-54 shows Kepler and his planetary model based on the regular solids [84], According to this model the greatest distance of one planet from the sun stands in a fixed ratio to the least distance of the next outer planet from the sun. There are five ratios describing the distances of the six planets which were known to Kepler. A regular solid can be interposed between two adjacent planets so that the inner planet, when at its greatest distance from the sun, lays on the inscribed sphere of the solid, while the outer planet, when at its least distance, lays on the circumscribed sphere. 

Figure 2-54. Johannes Kepler on Hungarian memorial stamp and his Planetary Model based on the regular solids [87],...

The simplest semi-regular polyhedra are obtained by symmetrically shaving off the comers of the regular solids. They are the truncated regular polyhedra and are marked with the superscript a in Table 2-4. One of them is the truncated icosahedron, the shape of... 

FIG. 3.3. Polyhedra (a) the regular solids, (b) some Archimedean semi-regular solids, (c) some Catalan semi-regular solids. (The numbers are the numbers of vertices.)... 

Polyhedra related to the pentagonal dodecahedron and icosahedron In equation (1) for 3-connected polyhedra (p. 62) the coefficient of is zero, suggesting that polyhedra might be formed from simpler 3-connected polyhedra by adding any arbitrary number of 6-gon faces. Although such polyhedra would be consistent with equation (1) it does not follow that it is possible to construct them. The fact that a set of faces is consistent with one of the equations derived from Euler s relation does not necessarily mean that the corresponding convex polyhedron can be made. Three of the Archimedean solids are related in this way to three of the regular solids ... 

J. Sci.. in press). Indeed, in the case of a solid-solution with a small difference in the size of the substituting ions (relative to the size of the non-substituting ion), the first parameter, ao, is usually sufficient (8). Equations 5 and 6 then become identical to those of the "regular" solid-solution model of Hildebrand (9). For the case where both ao and ai parameters are needed, equations 5 and 6 become equivalent to those of the "subregular" solid-solution model of Thompson and Waldbaum (10). a model much used in high-temperature work. Equations 5 and 6 can also be shown equivalent to Margules activity coefficient series (11). 

The platonic solids (the regular solids, regular polyhedra) are the only convex polyhedra with equivalent convex regular polygons as faces. They are the building blocks of the universe in Plato s theory of five elements (in the Timaeus), and are examined in Euclid s f/ewewte. 

External electric fields transform as the regular solid spherical harmonics, (p), for which the Laplacian vanishes, V r Yemi, v) = 0. There is therefore no spin-free correction of 0(c ) to the property operator, and the operator can be written as... 

The angular dependence of the coefficients C R, < a. < b) can be expressed in a closed form. The relevant formulae are obtained by asymptotic expansion of the polarization series truncated at some finite order. In practice such an asymptotic expansion is best performed by evaluating the polarization energies (as given by equations 9, 18, and 21) using the multipole expansion of the electrostatic potential l/ ri — r2. The latter expansion can be written in terms of either the Cartesian or the spherical tensors. The spherical formulation appears to be more popular because it leads much more easily to closed formulae and only this formulation will be considered in this article. Denoting by (r) the regular solid harmonic r Cim(0,), where Cim 0,) is the spherical harmonic in the Racah normalization and with the Condon and Shortley phase, one can write ... 

In a consideration of the foregoing results, it should be remembered that the molar volume of a melt is almost invariably larger than that of the regular solid, 17% so in the case of KCl. Although this might seem contrary to the observed decrease in the closest interionic separations, a parallel decrease in coordination number of nearest neighbors and particu-... 

The next step in the derivation of the multipole expansion of the two-electron integrals is to separate the electronic coordinates according to (9.13.4). Invoking the addition theorem for the regular solid harmonics [24]... 

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