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.
# | Coxeter group | Coxeter-Dynkin diagram | ||
---|---|---|---|---|
1 | ] | |||
2 | ||||
3 | h | |||
4 | q | |||
5 |
# | 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.
Read more about this topic: Uniform 5-polytope
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