Schlafli Symbol

Figure 9.3 Examples of network topologies along with their Schlafli symbols. The corresponding Wells symbols are (6,3), (4,82), (4,4) and (10,3)-a.

For multinodal nets the Schlafli Symbols are grouped according to the multiplicity of the node in the unit cell for example the net formed by the Si atoms in the silica form moganite (mog) is described as the 4-connected binodal net (4.64.8)2(42.62.82) because there are two crystallographic distinct nodes with multiplicity 2 1. 

A special type of point symbol is assigned to uniform nets, i.e. nets in which the shortest circuits at each angle are all equal in size. It follows that the Schlafli Symbol for each node in an n-connected uniform net with circuits of size A is n(n-i)/2, yxarnp cs indude the series of 3-connected 103 nets (srs, ths, bto, utp and others), the 4-connected 66 nets (dia, Ion, gsi, lcs and others) and also the binodal 3,4-connected (83)4(86)3 nets of C3N4 (ctn) and Pt304 (pto). 

Given this inadequacy other descriptors are needed. To differentiate nets with the same Schlafli Symbol a modified version of it, the Vertex Symbol (VS, or long symbol introduced by O Keeffe [1]) appears more useful. In a VS the size of the shortest rings at each angle is given with a subscript to denote the number of such rings. Solid state chemists define a ring as a circuit that has the property that there is no shorter path ( short cut ) between any two vertices on the circuit than the shortest one that is part of the circuit (see Fig. 1.3.4) [28]. 

Thus in the 3-connected srs 3D net there are five 10-rings at each angle and the VS is 105.105.105, and for the ths net, with two, four and four 10-rings at each angle, the VS is 102.104.104 (remember that for both nets the Schlafli Symbol is 103). 

Not all the angles of a 4-connected net necessarily possess rings. When there is not a ring an asterisk is inserted in the VS. Thus for the 4-connected CdS04 net (the net of the Cd and S atoms, with the -O- links considered as edges, cds, Schlafli Symbol 658) the Vertex Symbol is 6.6.6.6.62.. ... 

The Schlafli Symbols of these 6-connected nets are 48.66.8 (rob) for L = pyrazine, and 44.610.8 for L = l,4-bis(4-pyridyl)butadiene, respectively, so that the requirements for self-catenation are fulfilled, since the catenated 6-rings are the shortest rings [50], Perhaps one of the most fascinating examples is represented by [Ni(tpt)(N03)2] (tpt = tri-4-pyridyl-l,3,5-triazine) [51]. This contains a remarkable 3-connected chiral uniform net of 123 topology (twt, VS=124.12y. 127) (see Figure... 

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.

D Schlafli symbol Vertex symbol (VS) Vertex type Regular tilings Polyhedra and Uses shortest rings sql 4,4, hcb 6,3 rco 3,4, ... 

Delaney symbol or D-symbol 3D Schlafli symbol Vertex symbol (VS) Extended Schlafli symbol Extended Schlafli symbol 2- periodic nets All tilings Regular tilings 3- Periodic nets Uses shortest rings sql 4 , hcb 6, cem 3, 4 pcu 4,3,4 pts (4.4.82.82.83.881(4.4.87.87.87.87)... 

Point symbol (PS) long Schlafli symbol Schlafli symbol 3-periodic nets Uses shortest cycles pcu 4.4.4.4.4.4.4.4.4.4.4.T.. . pts 4. 8 ... 

Face symbol Delaney symbol of D-symbol Circuit symbol Vertex symbol Extended Schlafli symbol Polyhedra and cages (tiles) All tilings pcu 4. 6 rco [3. 4 ], pts [42.82]+ 8 ], pcu [4 ... 

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. 

Since this also is a unitiodal net, we only get one set of numbers. The Schlafli symbol is and the circuits arc shown in Figure 4.19. 

The Ivt-a net has vertex symbol 4-8-163 and genus 9, see Figure 6.19. Note the difference in tlie Schlafli symbol and the vertex symbol. The 10-gons in the Schlafli symbol arc circuits containing shortcuts while the 16-gons are the smallest fundamental rings. 

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