Dual polyhedra

The numbers of vertices and edges in a pair of dual polyhedra P and P satisfy the relationships v = f,... 

Dual polyhedra have the same symmetry elements and thus belong to the same symmetry point group. 

If we replace the vertices of an octahedron with faces and the faces with vertices, we obtain a cube. Starting from a cube, we can obtain an octahedron in the same way. The technical term for this relationship is dual, thus the cube and the octahedron are dual polyhedra. In the same way, the dodecahedron and the icosahedron are dual polyhedra, whereas the tetrahedron is its own dual. It thus follows that these five regular polyhedra represent three types of symmetry, tetrahedral, octahedral and icosahedral (Table 2.7). The icosahedral groups contain fivefold axes and are, hence, not crystallographic. 

Dual polyhedra have the same symmetry elements and thus belong to the same symmetry point group. Thus in the example above both the octahedron and the cube have the Oh symmetry point group. Also note in general that the dualization of a prism gives the corresponding bipyramid and vice versa. 

Figure 7. The equilibrium of the space frame modelling C28. (a) The primal curved-edged polyhedron formed from bars. (6) The dual polyhedron formed from moments. 

By the way, do you know what a dual polyhedron is If you take a boron octahedron and I put a carbon in the center of each of the faces and throw away the borons, then I get cubane. Now, here is a proposal I made in 1978, before buckminsterfullerene was discovered it has 32 borons and 60 faces. If I put a carbon in the center of each of its faces and throw away the borons, then I get buckminsterfullerene. This B32H32 is the dual polyhedron of buckminsterfullerene. When the discovery of buckminsterfullerene happened in 1985, I remembered that I d done something along those lines. Then later Lou Massa and I looked at these analogues in general so we could go on to propose other examples, all related by the Descartes-Euler formula, which can be found in Coxeter s book. 

Sah has described a yet more general way to extend the leapfrog transformation to a whole sequence of possibilities. These transformations may be represented within our algebra as DT( where a and b are integers such that a > 0 and a > > 0. The particularities of the new transformation are conveniently described in terms of the triangular lattice acts on the dual polyhedron to replace each triangular face by... 

The numbers of vertices and edges in a pair of dual polyhedrafP and fP satisfy the relationships v =/, e = e,f = v, in which the starred variables refer to the dual polyhedron P Thus, in the case of the trigonal bipyramid (fP)/trigonal prism (IP ) dual pair depicted above, = f= 6, e = e = 9, J = v = 5. 

A given polyhedron P can be converted into its dual P by locating the centers of the faces of P at the vertices of P and the vertices of P above the centers of the faces of P. Two vertices in the dual P are connected by an edge when the corresponding faces in P share an edge. 

Dualization of the dual of a polyhedron leads to the original polyhedron. 

The degrees of the vertices of a polyhedron correspond to the number of edges in the corresponding face polygons in its dual. 

Crystal structures may be described in terms of the coordination polyhedra MX of the atoms or in terms of their duals, that is, the polyhedra enclosed by planes drawn perpendicular to the lines M-X joining each atom to each of its neighbours at the mid-points of these lines. Each atom in the structure is then represented as a polyhedron (polyhedral domain), and the whole structure as a space-filling assembly of polyhedra of one or more kinds. We can visualize these domains as the shapes the atoms (ions) would assume if the structure were uniformly compressed. For example, h.c.p. and c.c.p. spheres would become the polyhedra shown in Fig. 4.29. These polyhedra are the duals of the coordination polyhedra illustrated in Fig. 4.5. These domains provide an alternative way of representing relatively simple c.p. structures (particularly of binary compounds) because the vertices of the domain are the positions of the interstices. The (8) vertices at which three edges meet are the tetrahedral interstices, and those (6) at which four edges meet are the octahedral interstices. Table 4.9 shows the octahedral positions occupied in some simple structures c.p. structures in which tetrahedral or tetrahedral and octahedral sites are occupied may be represented in a similar way. (For examples see JSSC 1970 1 279.)... 

Taking the Dual To take the dual of a polyhedron is to replace vertices by faces and vice-versa, as was already mentioned in Sect. 3.7 in relation to the Platonic solids. The dual has the same number of edges as the original, but every edge is rotated 90°. Hence the relations between v, e, for the dual and v, e, f for the original are ... 

Trivalent Polyhedra The dual of a deltahedron is a trivalent polyhedron, meaning that every vertex is connected to three nearest neighbours. The fullerene networks of carbon are usually trivalent polyhedra. This reflects the sp hybridization of carbon, which can form three specialized forms of the Euler symmetry theorem can be formulated. We may start from Eq. (6.138) and replace vertices by faces. The edge terms remain the same since they are totally symmetric under the local symmetries of the edges. Rotations of the edges by 90° will thus not affect these terms. 

The duals of the Archimedean solids are the Catalan polyhedra. They are named after the French-Belgian mathematician Eugene Charles Catalan, who first described them in 1862. Given the characteristics of their duals, discussed above, Catalan polyhedra have more than one type of vertices but all their faces are equivalent, noimegular polygons. Furthermore, each Catalan polyhedron has the same nnm-ber of edges and the same symmetry as its Archimedean dual. The names of the Catalan solids and their duals are given in Table 4. 

The number of edges and symmetry of each Catalan polyhedron can be found in the entry of the corresponding Archimedean dual in Table 3. 

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