Schlafli notation

Fig. 3.22 shows a network of atoms which can be considered as a triangular net, T net that is the 36 net. We may incidentally notice that the Schlafli Notation PN... 

Figure 4.28 Common two-dimensional sheet networks (a) a (4,4)-net (or 4". 6 in Schlafli notation) (b and c) (6,3)-nets (Schlafli, 6 ) using differently shaped nodes.

The extended Schlafli notation has been put forward by O Keeffe [5,9] and is, as the name indicates, an expansion of the Schlafli notation. This notation is also called vertex symbol or long notation. Now we will not only note the size of the smallest rings of a link pair, but we will also count how many of these there are. 

Thus, all possible ways to unite a link pair using smallest rings are counted and then added as a subscript index to the ring size as M>i-Ny2 0y,... The connectivity p is now deduced from the number of link pairs, each of these having a Mji symbol assigned to it. Thus summing the number of link pairs is equal to / (/ -1 )/2 (equation 4.1). For example for a three-connected net there will be three link pairs (Myi-Nyj-Oys) and for a four-connected net there will be six (My -Ny2-Oy3 Py4-Qy5-Ry[Pg.63]

The order of the link pairs are the same as for the short (Schlafli) notation when p-3 and p>4. However, for nets where the nodes have p-4 a different order is used. The link pairs are paired opposite to each other and ordered with the lowest numbers first. In these pairs, each link is used once. Designating the links as a, b, c and d, the order w ill be (ab,cd)(ac,bd)(ad,bc). 

In this section we will use two nets to show how to count the rings using the Schlafli and the vertex symbol (extended Schlafli) notation described earlier. 

Starting with the Schlafli notation where only one circuit is counted for each link pair wc will find a total of six circuit (one cadi for AB, AC, AD, BC, BD and CD). The resulting symbol is 6 and the circuits are shown in Figure 4.16. 

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.

Although the Schlafli symbol gives somewhat more information, both types of notation result in there being more than one net with the same set of numbers. The added letter in Wells notation does of course take care of this, but new nets are hard to designate since there is no central committee handing out extra letters. 

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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