Regular cubes spheres

Irregular-shaped particles exhibit greater surface area than regular-shapea cubes and spheres, the amount of this increase being possibly 25 percent. The effect of particle size and size distribution on effective surface, in a shaft employed for calcination of limestone, is shown in... 

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). 

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). 

Relatively little appears to be known about the influence of shape on the behaviour of particulate solids and it is notoriously difficult to measure. Whilst a sphere may be characterised uniquely by its diameter and a cube by the length of a side, few natural or manufactured food particles are truly spherical or cubic. For irregular particles, or for regular but non-spherical particles, an equivalent spherical diameter de can be defined as the diameter of a sphere with the same volume V as the original particle. Thus... 

Figure 16.1 All face-regular ( 4,7, 3)-spheres that are 7Ri, besides Cube...

The involvement of qrmmetry in chemistry has a long history in 640 B.O. the Society of Pythagoras held that earth had been produced from the reguliu hexahedron or cube, fire from the r ular tetrahedron, air fiom the regular octahedron, water from the regular icosahedron, and the heavenly sphere from the regular dodecahedron. Today, the chemist intuitively uses symmetry every time he recognizes which atoms in a molecule are equivalent, for example in pyrene it is easy... 

Figures 27.15 through 27.18 show Hoemer s data (Ref 18) for five regular geometries, a rod (side-on), a sphere, a cube (face-on and comer-on), a cylinder (face-on), and a disc (face-on). These geometries represent the range of idealized fragment shapes most commonly found.

The examination of coordinate transformations as local contractions and expansions of decorations about the poles of the principal rotational axes on the unit sphere for objects of Oh symmetry leads to intermediate geometries corresponding to particular Archimedean polyhedra related to the cube. In a similar manner, partial contractions and expansions of the decorations of the regular orbit of Ih point symmetry, i.e. the vertices of the great rhombicosidodecahedron, leads to the remaining polyhedra within the icosahedral family of Archimedean structures and orbits of Ih. 

These are groups which contain more than one threefold or higher axis. We will limit our consideration to the symmetry groups which describe the Platonic solids Td for the regular tetrahedron, Oh for the cube and regular octahedron, I/, for the regular dodecahedron and icosahedron, and JCh for the sphere. Some molecules in the cubic groups are shown below ... 

Observation A regular tetrahedron is formed by the spheres, the coordination number is 12. It is possible to completely fit the basic cube into the tetrahedron packing with the use of the central hole for one corner of the cube. 

Individual solid particles are characterized by their size, shape, and density. Particles of homogeneous solids have the same density as the bulk material. Particles obtained by breaking up a composite solid, such as a metal-bearing ore, have various densities, usually different from the density of the bulk material. Size and shape are easily specified for regular particles, such as spheres and cubes, but for irregular particles (such as sand grains or mica flakes) the terms size and shape are not so clear and must be arbitrarily defined. 

Such bodies are said to be spherically isotropic. Examples are spheres and the five regular polyhedrons—the tetrahedron, hexahedron (cube), octahedron. 

As an example, in a cube one has v = 8,e = 12, f = 6, and hence 8 — 12-1-6 = 2. The 2 in the right-hand side of Eq. (6.128) is called the Euler invariant. It is a topological characteristic. Topology draws attention to properties of surfaces, which are not affected when surfaces are stretched or deformed, as one can do with objects made of rubber or clay. Topology is thus not concerned with regular shapes, and in this sense seems to be completely outside our subject of symmetry yet, as we intend to show in this section, there is in fact a deep connection, which also carries over to molecular properties. The surface to which the 2 in the theorem refers is the surface of a sphere. A convex polyhedron is indeed a polyhedron which can be embedded or mapped on the surface of a sphere. Group theory, and in particular the induction of representations, provides the tools to understand this invariant. To this end, each of the terms in the Euler equation is replaced by an induced representation, which is based on the particular nature of the corresponding structural element. In Fig. 6.8 we illustrate the results for the case of the tetrahedron. 

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