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Is there a survey anywhere of space-filling polyhedra? MathWorld's article, space-filling polyhedron, mentions about 400 being seen in pre-1981 books and papers. Wikipedia mentions 28 convex uniform honeycombs, and the article honeycomb.

Is there a modern count anywhere for how many space-filling hexahedra or icosahedra exist? Can the 3D coordinates be downloaded?

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From your MathWorld link: "In the period 1974-1980, Michael Goldberg attempted to exhaustively catalog space-filling polyhedra. According to Goldberg, there are 27 distinct space-filling hexahedra..." Are you looking for any possible updates? –  amWhy Jul 6 '11 at 21:22
    
Yep. Is that number 27 still reliable? From the later papers, it seems Goldberg missed a lot. Is there data solid enough to support an integer sequence for OEIS? –  Silas Pike Jul 6 '11 at 21:30
    
What sort of integer sequence did you have in mind? Tesselations by dimension, or some restricted-to-3D concept? –  hardmath Jul 6 '11 at 23:21
    
In 2D, for n-gons 3-6, there are 1,1,14,3 families of tiling polygons, according to Grunbaum. In 3D, 3-12, there are 5,?,27,56,49,?,?,40,16 types of space-filling polyhedra, according to MathWorld. For example, there are 5 spacefilling tetrahedra. –  Silas Pike Jul 7 '11 at 13:55

1 Answer 1

No. A full count only applies to simple cases with given constraints.

Regarding the space-filling tetrahedra please note that there is an infinite number of those. Among those who fill space by isometries, it is known that there are 9 topological families.

http://link.springer.com/article/10.1007%2FPL00009423#page-1

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For tetrahedra, have you seen: Senechal, Marjorie. "Which tetrahedra fill space?." Mathematics Magazine (1981): 227-243. –  Joseph Malkevitch Jul 19 at 20:21

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