Uniform 5-polytope - Regular and Uniform Honeycombs

Regular and Uniform Honeycombs

There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.

Fundamental groups
# Coxeter group Coxeter-Dynkin diagram
1 ]
2
3 h
4 q
5
Prismatic groups
# Coxeter group Coxeter-Dynkin diagram
1 x x
2 x x
3 x ]x
4 xx xx
5 xx xx
6 xx ]xx
7 xxx xxx
8 x ]x]
9 x ]x
10 x ]x
11 x x
12 x x
13 x x

There are three regular honeycombs of Euclidean 4-space:

  • tesseractic honeycomb, with symbols {4,3,3,4}, = . There are 19 uniform honeycombs in this family.
  • 24-cell honeycomb, with symbols {3,4,3,3}, . There are 31 reflective uniform honeycombs in this family, and one alternated form.
    • Snub 24-cell honeycomb, with symbols h0,1{3,4,3,3}, constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex.
  • 4-demicube honeycomb, with symbols {3,3,4,3},

Other families that generate uniform honeycombs:

  • There are 23 uniform honeycombs, 4 unique in the demitesseractic honeycomb family. With symbols h{4,32,4} it is geometrically identical to the hexadecachoric honeycomb, =
  • There are 7 uniform honeycombs from the, family, all unique, including:
    • 4-simplex honeycomb
    • Truncated 4-simplex honeycomb
    • Omnitruncated 4-simplex honeycomb
  • There are 9 uniform honeycombs in the : family, all repeated in other families, including the demitesseractic honeycomb.

Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.

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