Network topology notation

Figure 4.29 Common three-dimensional network topologies (a) a-Po (or NaCI), uninodal, with 6-connecting octahedral nodes (b) diamond lattice with tetrahedral nodes (c) rutile, binodal with 3- and 6-connecting nodes. The networks (d) 10,3a and (e) 10,3b (ThSi2) have the same Wells notation and Schlafli symbol (10 ) (the shortest route is shown with white nodes) but a different topology.

The use of networks to describe and design crystal structures inevitably leads to the possibility of interpenetration of multiple networks. In such cases it is essential that the topology of interpenetration, as well as the individual network topology, is analysed and understood in detail. For ID and 2D networks, interpenetration can occur either in a parallel or inclined fashion, leading to overall entanglements of the same or higher dimensionality. Such systems can be described in terms of a mD(/nD) pD parallel/inclined interpenetration notation. Even highly symmetrical 3D nets such as diamond or a-Po can show, on rare occasions, different modes of interpenetration. 

It is also true that Wells [3, Chapter 1], perfectly well introduced a systematic and rigorous coding of the topology of tessellations and networks he worked with, which is now called the Wells point symbol notation, and that this was a simple coding scheme over the eireuitry and valences, about the vertices, in the unit of pattern of the tessellations and networks. The Wells point symbol notation was, however, nonetheless an important development for the rigorous mathematieal basis it put the tessellations and networks on, formally, as quasi-solutions (n, p) for the Schlafli relation shown as Eq. (2). 

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