Ranganathan, S and Lord, EA (2008) Parallelohedra and topological transitions in cellular structures. In: Philosophical Magazine Letters, 88 (9-10). pp. 703-713.
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In 1891, Fedorov showed that tilings of three-dimensional space by congruent convex polyhedra, in which all the tiles are in the same orientation, belong to just five topological classes. In 1953, Cyril Stanley Smith generalized Fedorov's result by dispensing with the convexity requirement. In this and the later work of Ferro and Fortes, solutions were obtained by applying topological transitions, of the kind that occur during grain growth, to the well-known space-filling by Kelvin 14-hedra. We demonstrate an anomalous solution to the generalized Fedorov problem that is not derivable by this method, and which provides a counter example to some conjectures suggested by O'Keeffe. Finally, a further generalization is proposed, that has relevance in the study of periodic networks. We conclude with a few examples to indicate some interesting directions for possible future developments of the idea that Fedorov and Smith had initiated.
|Item Type:||Journal Article|
|Additional Information:||Copyright of this article belongs to Taylor & Francis.|
|Keywords:||3D tiling;cellular structure;Fedorov;networks;parallelohedron;polyhedron;topology.|
|Department/Centre:||Division of Mechanical Sciences > Materials Engineering (formerly Metallurgy)|
|Date Deposited:||06 Oct 2008 09:13|
|Last Modified:||19 Sep 2010 04:50|
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