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A121273
Number of different n-dimensional convex regular polytopes that can tile n-dimensional space.
0
1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,2
COMMENTS
The only n-dimensional convex regular polytope that can tile n-dimensional space for all n>4 is the n-hypercube
LINKS
Eric Weisstein's World of Mathematics, Space-Filling Polyhedron.
Wikipedia, Regular Polytopes.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
FORMULA
a(n)=3 for n = 2 & 4. a(n)=1 for all other n.
EXAMPLE
a(2)=3 because the plane can be tiled by equilateral triangles, squares or regular hexagons. a(3)=1 since the only platonic solid that can tile 3-dimensional space is the cube. a(4)=3 because the 4-dimensional space can be tiled by hypercubes (tesseracts), hyperoctahedra or 24-cell polytopes.
CROSSREFS
Sequence in context: A073272 A268442 A175623 * A063065 A324724 A051718
KEYWORD
nonn
AUTHOR
Sergio Pimentel, Aug 23 2006
STATUS
approved