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Constraints on the fundamental topological parameters of spatial tessellations
Author(s) -
Cowan Richard,
Weiss Viola
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300202
Subject(s) - facet (psychology) , adjacency list , mathematics , bounded function , regular polygon , space (punctuation) , topology (electrical circuits) , simple (philosophy) , combinatorics , computer science , geometry , mathematical analysis , psychology , social psychology , philosophy , personality , epistemology , big five personality traits , operating system
Tessellations of R 3 that use convex polyhedral cells to fill the space can be extremely complicated, especially if they are not “facet‐to‐facet”, that is, if the facets of a cell do not necessarily coincide with the facets of that cell's neighbours. In a recent paper [15][V. Weiss, 2011], we have developed a theory which covers these complicated cases, at least with respect to their combinatorial topology. The theory required seven parameters, three of which suffice for facet‐to‐facet cases; the remaining four parameters are needed for the awkward adjacency concepts that arise in the general case. This current paper establishes constraints that apply to these seven parameters and so defines a permissible region within their seven‐dimensional space, a region which we discover is not bounded. Our constraints in the relatively simple facet‐to‐facet case are also new.