Topology polyhedra

For a convex polyhedron, topologically like a sphere, with F faces, V vertices and E edges, Euler s law states that V—E+F = 2. A sphere has genus zero and if another more complex polyhedron can be deformed to take the shape of a sphere with N handles, then it has the genusN. This is a useful, but not a complete, characterization... 

Figure C2.12.6. Framework topology of ZSM-5. The 5-ring polyhedron is connected into chains which fonn the ZSM-5 stmcture with the 10-membered openings of the linear channels.

Buckminsterfullerene, an allotrope of carbon, is topologically equivalent to a truncated icosahedron, an Archimedean solid that possesses 12 pentagons and 20 hexagons (Fig. 9-16). [17] Each carbon atom of this fullerene corresponds to a vertex of the polyhedron. As a result, C6o is held together by 90 covalent bonds, the number of edges of the solid. 

Every polyhedron has a density. A polyhedron could be defined as the union of a finite number of convex polyhedra. A convex polyhedron is the intersection of a finite number of half-spaces. It may be bounded or unbounded. The family of polyhedra is closed with respect to union, intersection and subtraction of sets. For our goals, polyhedra form sufficiently rich class. It is important that in definition of polyhedron finite intersections and unions are used. If one uses countable unions, he gets too many sets including all open sets, because open convex polyhedra (or just cubes with rational vertices) form a basis of standard topology. 

Figure 5. Polyhedron model of locally minimum covering of the sphere by 32 equal circles. (Reprinted from Tarnai Wenninger (1990) with permission of Structural Topology (Montreal) and Dr M. J. Wenniner.)...

Here is a fixed integer, the Euler characteristic that marks the particular topology of the surface on which the polyhedron is embedded. However, in order to describe the topology completely, one also has to specify the orientability of the surface [3]. A surface is orientable if there is no walk on the surface that would take you from the outside to the inside. Such is the case of a sphere with handles. Otherwise, it is non-orientable. This is the case of a sphere with crosscaps. Based on this orientability, the infinite class of surfaces can be divided into two subclasses ... 

Scalar counting relations for sets of structural components can seen as expressions for characters under the identity operation of more general relations between representations of those sets. For example, the Euler relation in topology can be generalised to connect not only the numbers of edges, vertices and faces of a polyhedron, but also various symmetries associated with the structural features. The well-known Euler theorem... 

Assuming a fixed band structure (the rigid band model), a decrease in the density of states is predicted for an increase in the electron/atom ratio for a Fermi surface that contacts the zone boundary. It will be recalled that electrons are diffracted at a zone boundary into the next zone. This means that A vectors cannot terminate on a zone boundary because the associated energy value is forbidden, that is, the first BZ is a polyhedron whose faces satisfy the Laue condition for diffraction in reciprocal space. Actually, when a k vector terminates very near a BZ boundary the Fermi surface topology is perturbed by NFE effects. For k values just below a face on a zone boundary, the electron energy is lowered so that the Fermi sphere necks outwards towards the face. This happens in monovalent FCC copper, where the Fermi surface necks towards the L-point on the first BZ boundary (Fig. 4.3f ). For k values just above the zone boundary, the electron energy is increased and the Fermi surface necks down towards the face. 

Euler proved another topological theorem (1750, published in 1758) named after him for every convex polyhedron homeomorphic to a sphere,... 

The topology of a polyhedron can be described by a graph, called the 1-skeleton of the polyhedron. The vertices and edges of the 1-skeleton correspond to the vertices and edges, respectively, of the underlying polyhedron. Of fundamental importance are relationships between possible numbers and... 

The most simple molecular topology of such systems reported so far is a tetrahedral supermolecule obtained by reacting tetrakis(dimethylsiloxy)-silane with alkenyloxy-cyanobiphenyls (Fig. 22), as discussed previously. Such tetramers exhibit smectic A liquid crystal phases [179]. For such end-on materials, microsegregation at the molecular level favors the formation of the smectic A phases in preference to the nematic phase exhibited by the mesogenic monomers themselves. The use of different polyhedral silox-ane systems (Fig. 24) or the Ceo polyhedron as the template for multi- and polypedal hexakis(methano)fullerenes (Fig. 70) substituted with a large number of terminally attached mesogenic groups confirm the same tendency to the formation of smectic A phases (vide supra). 

Euler s formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy [Cauchy, 1813] and [L Huillier, 1861] and is at the origin of topology. 

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