Tetrahedron A polyhedron with

Tetrahedron A polyhedron with four equal-sized, equilateral triangular feces and four apices (corners). 

V A deltahedron is a polyhedron with all faces that arc equilateral triangles. The dcitahedra from n -4 to /t = 12 are tetrahedron (4), trigonal bipyramid (5). octahedron (6). pentagonal bipyramid (7). bisdisphenoid (dodecahedron) (8), tricapped trigonal prism (9). bicapped square aniiprism 110). octadccahedron (II), and icosahedron (12). Most of these arc illustrated in Chapters hand 12. Sec also Fig. 16.50. 

What is chirality The term chirality denotes the property of a molecule to be nonidentical to its mirror image. All coordination polyhedrons with identical ligands are achiral. When the ligands are different, the chiral compounds may be formed. Thus, a tetrahedron with four different ligands is chiral. A trigonal bipyramid and tetragonal pyramid may be (but not necessarily will be) chiral if three (or more) ligands are of different types, e.g. 

The molecule has the symmetry of the regular tetrahedron—four threefold rotation axes, three twofold rotation axes, and some symmetry planes. The four nitrogen atoms lie at the corners of a regular tetrahedron and the six carbon atoms lie at the corners of a regular octahedron. The polyhedron defined by the twelve hydrogen atoms is the truncated tetrahedron, a tetrahedron with each corner cut off by a plane parallel to the opposite face. 

Figure 5.29 Representations of tetrahedra found in crystal structure diagrams (a) a ball-and-stick diagram of a tetrahedron, with a central silicon atom surrounded by four oxygen atoms and (b) its representation as a polyhedron. The views in parts (c), and (d) are the equivalent to those in parts (a) and (b), along the direction A in part (b), in which one tetrahedral vertex is uppermost. The views in parts (e) and (f) are the equivalent to those in parts (a) and (b), along the direction B in part (b), in which one tetrahedral edge is towards the observer...

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. 

The optimized polyhedral model of p-belite without substitution of any foreign atom is illustrated in Figure 13.4. The crystal structural unit is made up of two kinds of CaO polyhedra (Ca(l)O = 7 Ca-0 bonds, Ca(2)0 = 8 Ca-0 bonds) and Si04 tetrahedron. The Ca(l)0 polyhedron with 7 Ca-0 bonds forms a distorted pentagonal bi-pyramid, and the Ca(2)0 polyhedron with 8 Ca-O bonds takes shape of a distorted anti-cube as show in Figure 13.5. 

Thus it is possible to define with respect to a (111) surface a polyhedron described by the 111 family. It is, in effect, a tetrahedron. However, closer inspection requires the definition of two tetrahedra relative to a direction normal to, say, the (111) plane, which of course would be the direction of indentation if a (111) surface or a (TIT) surface was indented. The slip plane tetrahedra are mirror images i.e., the triangular base of the tetrahedron lies on (111) or (TIT) and the apex is then below the surface, or of course the apex can be a (111) or (ITT) atom with the triangular base lying beneath the surface. These two glide polyhedra are sketched in Figure 3.31. 

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