## 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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