A121273 Number of different n-dimensional convex regular polytopes that can tile n-dimensional space.

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

Author

Sergio Pimentel, Aug 23 2006

Keywords

• nonn

Comments

The only n-dimensional convex regular polytope that can tile n-dimensional space for all n>4 is the n-hypercube

Examples

			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.
		

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.

Crossrefs

Cf. A053016, A060296.

Formula

a(n)=3 for n = 2 & 4. a(n)=1 for all other n.