by **Marek14** » Fri Jan 16, 2015 8:56 pm

The pentagonal polytopes aren't impossible because there would be no possible tiling with them, but rather because this tiling can't be derived from the basic cubic tiling. There is a dearth of regular tilings; in 2D there are triangular and hexagonal tilings which can only have integral coordinates when the plane is put into 3D space and in 4D you have 16-cell and 24-cell tiling which can be both derived from tesseractic tiling. (16-cell tiling is alternated tesseractic and 24-cell tiling is the dual of 16-cell.) Basically, my argument was why triangle/hexagon can be integer, not why the other can't.

In 6 to 8 dimensions you have tilings based on Gosset polytopes.

I realized that the set of cubic/orthoplex polytopes with integer coordinates is smaller than I thought, btw. For example, you can't have integer rhombicuboctahedron because it has octagons hidden inside.

So in 3D, you have:

Cube - (1,1,1) with sign changes; (1,1,1) joined to (-1,1,1), (1,-1,1) and (1,1,-1)

Octahedron - (1,0,0) with sign changes and permutations; (1,0,0) joined to (0,1,0), (0,-1,0), (0,0,1) and (0,0,-1)

Cuboctahedron - (1,1,0) with sign changes and permutations; (1,1,0) joined to (1,0,1), (1,0,-1), (0,1,1) and (0,1,-1)

Truncated octahedron - (2,1,0) with sign changes and permutations; joined to (2,0,1), (2,0,-1) and (1,2,0)

In 4D, you have:

1000 - tesseract - (1,1,1,1) with sign changes; (1,1,1,1) joined to (-1,1,1,1), (1,-1,1,1), (1,1,-1,1) and (1,1,1,-1)

0100 - rectified tesseract - (1,1,1,0) with permutations and sign changes; (1,1,1,0) joined to (1,1,0,1), (1,1,0,-1), (1,0,1,1), (1,0,1,-1), (0,1,1,1) and (0,1,1,-1)

0010 - 24-cell - (1,1,0,0) with permutations and sign changes; (1,1,0,0) joined to (1,0,1,0), (1,0,-1,0), (1,0,0,1), (1,0,0,-1), (0,1,1,0), (0,1,-1,0), (0,1,0,1) and (0,1,0,-1)

0001 - 16-cell - (1,0,0,0) with permutations and sign changes; (1,0,0,0) joined to (0,1,0,0), (0,-1,0,0), (0,0,1,0), (0,0,-1,0), (0,0,0,1) and (0,0,0,-1)

0110 - bitruncated tesseract - (2,2,1,0) with permutations and sign changes; (2,2,1,0) joined to (2,2,0,1), (2,2,0,-1), (2,1,2,0) and (1,2,2,0); note that this contains truncated tetrahedra

0101 - rectified 24-cell - (2,1,1,0) with permutations and sign changes; (2,1,1,0) joined to (2,1,0,1), (2,1,0,-1), (2,0,1,1), (2,0,1,-1), (1,2,1,0) and (1,1,2,0)

0011 - truncated 16-cell - (2,1,0,0) with permutations and sign changes; (2,1,0,0) joined to (2,0,0,1), (2,0,0,-1), (2,0,1,0), (2,0,-1,0) and (1,2,0,0)

0111 - truncated 24-cell - (3,2,1,0) with permutations and sign changes; (3,2,1,0) joined to (3,2,0,1), (3,2,0,-1), (3,1,2,0) and (2,3,1,0)