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Enumeration of \(4\)-connected \(3\)-dimensional nets and classification of framework silicates. \(3D\) nets based on insertion of \(2\)-connected vertices into \(3\)-connected plane nets. (English) Zbl 1239.05094

Summary: \(4\)-connected \(3\)-dimensional nets containing zigzag and saw chains may be derived from stacks of parallel congruent \(3\)-connected \(2\)-dimensional nets by linking them together with zigzag or saw chains. Such derivation is governed by the following rules: (1) Every vertex must lie on an infinite h (horizontal) path in the plane of the original \(2\)-dimensional net; (2) the number of z (zigzag) edges in any (projected) polygonal circuit in the original \(2\)-dimensional net must be even; as a consequence of (1) and (2), we have: (3) z edges must connect infinite h paths of different heights. Using these rules, all possible translationally and radially symmetric nets derivable from \(6^3\), \(3.12^2\), \(4.8^2\), \(4.6.12\) and \((5^2.8)_2(5.8^2)_1\) are considered. Of particular interest are the radially symmetric nets, which consist of mirror-related sectors within which there is translational symmetry; such nets can describe sector-twinned crystals.

MSC:

05C30 Enumeration in graph theory
82D25 Statistical mechanics of crystals
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