Polyhedral Topology

Before considering coordination polyhedra it is first useful to consider the static topology of polyhedra in general. Of fundamental importance are relationships between possible numbers and types of vertices (v), edges (e), and faces (f) of polyhedra. In this connection the following elementary relationships are particularly significant  

This arises from the properties of ordinary 3D space. 2. Relationship between the edges and faces  

In equation (2) f is the number of faces with i edges (e.g.,/3 is the number of triangular faces, is the number of quadrilateral faces). This relationship arises from the fact that each edge of the polyhedron is shared by exactly two faces. Since no face can have fewer edges than the three of a triangle, the following inequality must hold in all cases  

In general, if a face withedges is capped, the following relationships will be satisfied V2 = V + 1 2 = +fk, /2 =/i +/r - 1 An example of such a capping process converts a square antiprism into a capped square antiprism, i.e.. 

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Polyhedral isomerizations may be studied using either a microscopic or macroscopic approach. The microscopic approach uses details of polyhedral topology to elucidate possible single polyhedral isomerization steps, namely which types of isomerization steps are possible. Such isomerization steps consist most commonly of so-called diamond square-diamond processes or portions thereof. The microscopic approach to polyhedral isomerizations is relevant to understanding fluxional processes in borane and metallaborane polyhedra. 

Another process of interest in polyhedral topology is the dualization of poly-hedra. A given polyhedron V can be converted into its dual V by locating the centers of the faces of V at the vertices of V and the vertices of V above the centers of the faces of V. Two vertices in the dual V are connected by an edge when the corresponding faces in V share an edge. 

Are there any other possible uses for the construction of complex topological species One possible application is in the mass production of DNA polyhedral catenanes by biological means, such as the polymerase chain reaction (PCR) (Saiki et al. 1986) or by production in vivo. Figure 21 illustrates that semi-conservative replication (the mechanism used by DNA polymerases) cannot reproduce a stable branch. The DNA with different sequences in the two arms of the branch (cartooned as dashed and solid lines) leads to two heterologous duplex DNA molecules, rather than a second branched molecule. 

The secondary structure unit in zeolites A. X, and V is the truncated octahedron. These polyhedral units are linked in three-dimensional space through the four- or six-membered rings, The former linkage produces the zeolite A structure, and the latter the topology of zeolites X and Y and of the mineral faujasite. 

The main properties of a polyhedral surface are undoubtedly related to the topology of the underlying manifold that it decorates. This topology can be completely described by two properties the Euler characteristic and the orientability of the surface. The former can easily be calculated from the celebrated Euler theorem [9] which states that the number of vertices, edges, and faces, denoted as V, E, and F respectively, obey the following mle ... 

Zwijnenburg, M. A., Bromley, S. T., Jansen, J. C. and Maschmeyer, T. Toward understanding extra-large-pore zeolite energetics and topology a polyhedral approach, Chem. Mater., 2004, 16, 12-20. 

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). 

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