Other Approaches to Net Enumeration

Quotient graph enumeration

A complete enumeration of all 3–periodic networks can be achieved by systematically listing all possible quotient graphs . There are many technical difficulties associated with this approach and most of the networks generated this way have properties that are unphysical. Recent work has focussed on a priori filtering of the graphs for physically plausible structures.

Nets carried by 3D euclidean tilings

Many, but not all, “interesting” 3–periodic nets are carried by 3D euclidean tilings (meaning that the periodic net is the same as the framework formed by the edges and vertices of a tiling of space by generalised polyhedra). Combinatorial tiling theory has led to efficient algorithms for the systematic enumeration of 3D euclidean tilings, and hence a large proportion of periodic nets that are likely to be of interest chemically.

Homogenous sphere packings

Euclidean sphere packings are a relatively small class of three–periodic structures which nonetheless have significant physical and chemical relevance. Homogenous sphere packings have been enumerated via the systematic exploration of possible configurations in the 230 3D space groups. This approach is due to Werner Fischer, Elke Koch, and Heidrun Sowa.

All four approaches overlap, but access networks in a different order and from different classes. Our surface–covering approach to enumerating periodic networks retains the efficiency of the 3D tiling approach and generates many of the same nets. However, our approach also generates:

• networks that cannot be carried by 3D tilings,
• spatially distinct embeddings of the same network topology, e.g., self–catenated versions of a net, and
• multiple component interwoven networks.

History Of Our Approach

Stephen Hyde has been exploring structure via triply periodic minimal surfaces for many years, since he worked with Sten Andersson and Kåre Larsson in Lund on structures of microporous zeolites and related covalent atomic frameworks and self–assembly of lipid–water liquid crystals. Indeed, the genesis of this database can be traced to that time, in a very incomplete paper exploring nets on triply periodic minimal surfaces : ( S.T. Hyde and S. Andersson, “A systematic net description of saddle polyhedra and periodic minimal surfaces”, Z. Kristallogr., 168, 221–254, (1984). )

Further research by Stephen Hyde and Stuart Ramsden on molecular and atomic self–assembly, convinced them of the need for a fuller exploration the structural possibilities of 3D euclidean space, including polycontinuous spatial partitions (that generalise polyhedral cellular packings).

Exploration of structures and structural transitions via hyperbolic surface patterns remains — strangely enough — largely confined to solid state scientists. Our research into the hyperbolic crystallography of periodic minimal surfaces was stimulated by a paper by Sadoc and Charvolin ( Acta Cryst. A45:10 (1989) ). The hyperbolic subgroup computations required to enumerate surface tilings were carried out by Vanessa Robins, Stuart Ramsden and Stephen Hyde, building on this paper.

The recent mathematical advances in tiling theory due to Andreas Dress and his former students Daniel Huson and Olaf Delgado–Friedrichs have provided the necessary tools to exhaustively enumerate surface reticulations. Further references are given on the bibliography page.

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