Application Bonding Schemes for Polyhedra

Leonhard Euler dominated the mathematics of the 18th century. One of his famous discoveries was the polyhedral theorem, which marks the beginning of topology. A polyhedron has three structural elements vertices, edges, and faces The numbers of these will be represented as i , e, and /, respectively. Then, for a polyhedron, the following theorem holds  

Theorem 15 In a convex polyhedron the alternating sum of the numbers of vertices, edges, and faces is always equal to 2. 

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 faces may be represented as closed chains of nodes, which are bordering a polyhedral face, (m, v,w. . The sequence forms a circulation around the face, in a particular sense (going from ( ) to w) over (u), etc.). The set of face rotations forms the basis for the face representation, which is denoted as /q(/). In a polyhedron the maximal site group of a face is C y, and in this site group the face rotation transforms as the rotation around the C axis, i.e. it is symmetric 

Theorem 16 The alternating sum of induced representations of the vertex nodes, edge arrows, and face rotations, is equal to the sum of the totally-symmetric representation, To, and the pseudo-scalar representation. Pc. The latter representation is symmetric under proper symmetry elements and antisymmetric under improper symmetry elements. 

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