Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
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
add comment

1 Answer

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.


share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.