In geometry, a convex uniform honeycomb is a tessellation of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter–Dynkin diagrams for each family.
Contents |
By Coxeter group and Coxeter–Dynkin diagrams, the nine are:[1]
# | Witt symbol |
Coxeter symbol |
Coxeter graph |
Honeycombs |
---|---|---|---|---|
1 | [4,3,5] | 15 forms | ||
2 | [5,3,5] | 9 forms | ||
2 | [3,5,3] | 9 forms | ||
4 | [5,31,1] | 11 forms (7 overlap with [5,3,4] family, 4 are unique) | ||
5 | [(3,3,3,4)] | 9 forms | ||
6 | [(3,3,3,5)] | 9 forms | ||
7 | [(3,4,3,4)] | 6 forms | ||
8 | [(3,4,3,5)] | 9 forms | ||
9 | [(3,5,3,5)] | 6 forms |
These 9 families generate a total of 76 unique uniform honeycombs.
The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is cited with the {3,5,3} family below.
There are also 23 noncompact Coxeter groups of rank 4. These families can produce uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity. These forms are not listed in this article.
Type | Coxeter groups | Group count | Honeycomb count |
---|---|---|---|
linear graphs | , , , , , , | 7 | 15+9+15+15+9+15+9=87 |
bifurcating graphs | , , | 3 | 11+11+7=29 |
cyclic graphs | , , , , , , | 7 | |
mixed graphs | , , , , , | 6 |
There are 9 forms, generated by ring permutations of the Coxeter group: [3,5,3] or
One related non-wythoffian form is constructed from the {3,5,3} vertex figure with 4 (tetrahedrally arranged) vertices removed, creating pentagonal antiprisms and dodecahedra filling in the gaps.[2]
# | Honeycomb name Coxeter–Dynkin and Schläfli symbols |
Cell counts/vertex and positions in honeycomb |
|||||
---|---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
Vertex figure | picture | ||
1 | icosahedral (Regular) t0{3,5,3} |
(20) (3.3.3.3.3) |
|||||
2 | rectified icosahedral t1{3,5,3} |
(2) (5.5.5) |
(3) (3.5.3.5) |
||||
3 | truncated icosahedral t0,1{3,5,3} |
(1) (5.5.5) |
(3) (4.6.6) |
||||
4 | cantellated icosahedral t0,2{3,5,3} |
(1) (3.5.3.5) |
(2) (4.4.3) |
(2) (3.5.4.5) |
|||
5 | Runcinated icosahedral t0,3{3,5,3} |
(1) (3.3.3.3.3) |
(5) (4.4.3) |
(5) (4.4.3) |
(1) (3.3.3.3.3) |
||
6 | bitruncated icosahedral t1,2{3,5,3} |
(2) (3.10.10) |
(2) (3.10.10) |
||||
7 | cantitruncated icosahedral t0,1,2{3,5,3} |
(1) (3.10.10) |
(1) (4.4.3) |
(2) (4.6.10) |
|||
8 | runcitruncated icosahedral t0,1,3{3,5,3} |
(1) (3.5.4.5) |
(1) (4.4.3) |
(2) (4.4.6) |
(1) (4.6.6) |
||
9 | omnitruncated icosahedral t0,1,2,3{3,5,3} |
(1) (4.6.10) |
(1) (4.4.6) |
(1) (4.4.6) |
(1) (4.6.10) |
||
[77] | partially truncated icosahedral pt{3,5,3} |
(4) (5.5.5) |
(12) (3.3.3.5) |
There are 15 forms, generated by ring permutations of the Coxeter group: [5,3,4] or
# | Name of honeycomb Coxeter–Dynkin diagram |
Cells by location and count per vertex | Vertex figure | Picture | |||
---|---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
||||
10 | order-4 dodecahedral (Regular) |
- | - | - | (8) (5.5.5) |
||
11 | Rectified order-4 dodecahedral |
(2) (3.3.3.3) |
- | - | (4) (3.5.3.5) |
||
12 | Rectified order-5 cubic |
(5) (3.4.3.4) |
- | - | (2) (3.3.3.3.3) |
||
13 | order-5 cubic (Regular) |
(20) (4.4.4) |
- | - | - | ||
14 | Truncated order-4 dodecahedral |
(1) (3.3.3.3) |
- | - | (4) (3.10.10) |
||
15 | Bitruncated order-5 cubic |
(2) (4.6.6) |
- | - | (2) (5.6.6) |
||
16 | Truncated order-5 cubic |
(5) (3.8.8) |
- | - | (1) (3.3.3.3.3) |
||
17 | Cantellated order-4 dodecahedral |
(1) (3.4.3.4) |
(2) (4.4.4) |
- | (2) (3.4.5.4) |
||
18 | Cantellated order-5 cubic |
(2) (3.4.4.4) |
- | (2) (4.4.5) |
(1) (3.5.3.5) |
||
19 | Runcinated order-5 cubic |
(1) (4.4.4) |
(3) (4.4.4) |
(3) (4.4.5) |
(1) (5.5.5) |
||
20 | Cantitruncated order-4 dodecahedral |
(1) (4.6.6) |
(1) (4.4.4) |
- | (2) (4.6.10) |
||
21 | Cantitruncated order-5 cubic |
(2) (4.6.8) |
- | (1) (4.4.5) |
(1) (5.6.6) |
||
22 | Runcitruncated order-4 dodecahedral |
(1) (3.4.4.4) |
(1) (4.4.4) |
(2) (4.4.10) |
(1) (3.10.10) |
||
23 | Runcitruncated order-5 cubic |
(1) (3.8.8) |
(2) (4.4.8) |
(1) (4.4.5) |
(1) (3.4.5.4) |
||
24 | Omnitruncated order-5 cubic |
(1) (4.6.8) |
(1) (4.4.8) |
(1) (4.4.10) |
(1) (4.6.10) |
||
[34] | alternated order-5 cubic |
(20) (3.3.3) |
(12) (3.3.3.3) |
There are 9 forms, generated by ring permutations of the Coxeter group: [5,3,5] or
# | Name of honeycomb Coxeter–Dynkin diagram |
Cells by location and count per vertex | Vertex figure | |||
---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
|||
25 | Order-5 dodecahedral t0{5,3,5} |
(20) (5.5.5) |
||||
26 | rectified order-5 dodecahedral t1{5,3,5} |
(2) (3.3.3.3.3) |
(5) (3.5.3.5) |
|||
27 | truncated order-5 dodecahedral t0,1{5,3,5} |
(1) (3.3.3.3.3) |
(5) (3.10.10) |
|||
28 | cantellated order-5 dodecahedral t0,2{5,3,5} |
(1) (3.5.3.5) |
(2) (4.4.5) |
(2) (3.5.4.5) |
||
29 | Runcinated order-5 dodecahedral t0,3{5,3,5} |
(1) (5.5.5) |
(3) (4.4.5) |
(3) (4.4.5) |
(1) (5.5.5) |
|
30 | bitruncated order-5 dodecahedral t1,2{5,3,5} |
(2) (4.6.6) |
(2) (4.6.6) |
|||
31 | cantitruncated order-5 dodecahedral t0,1,2{5,3,5} |
(1) (4.6.6) |
(1) (4.4.5) |
(2) (4.6.10) |
||
32 | runcitruncated order-5 dodecahedral t0,1,3{5,3,5} |
(1) (3.5.4.5) |
(1) (4.4.5) |
(2) (4.4.10) |
(1) (3.10.10) |
|
33 | omnitruncated order-5 dodecahedral t0,1,2,3{5,3,5} |
(1) (4.6.10) |
(1) (4.4.10) |
(1) (4.4.10) |
(1) (4.6.10) |
There are 11 forms (4 of which are not seen above), generated by ring permutations of the Coxeter group: [5,31,1] or
# | Honeycomb name Coxeter–Dynkin diagram |
Cells by location (and count around each vertex) |
vertex figure | Picture | |||
---|---|---|---|---|---|---|---|
0 |
1 |
0' |
3 |
||||
34 | alternated order-5 cubic |
(12) (3.3.3.3) |
(20) (3.3.3) |
||||
35 | truncated alternated order-5 cubic |
(1) (3.5.3.5) |
(2) (5.6.6) |
(2) (3.6.6) |
|||
[11] | rectified order-4 dodecahedral (rectified alternated order-5 cubic) |
(2) (3.5.3.5) |
(2) (3.5.3.5) |
(2) (3.3.3.3) |
|||
[12] | rectified order-5 cubic (cantellated alternated order-5 cubic) |
(1) (3.3.3.3.3) |
(1) (3.3.3.3.3) |
(5) (3.4.3.4) |
|||
[15] | bitruncated order-5 cubic (cantitruncated alternated order-5 cubic) |
(1) (5.6.6) |
(1) (5.6.6) |
(2) (4.6.6) |
|||
[14] | truncated order-4 dodecahedral (bicantellated alternated order-5 cubic) |
(2) (3.10.10) |
(2) (3.10.10) |
(1) (3.3.3.3) |
|||
[10] | Order-4 dodecahedral (trirectified alternated order-5 cubic) |
(4) (5.5.5) |
(4) (5.5.5) |
||||
36 | runcinated alternated order-5 cubic |
(1) (3.3.3) |
(3) (3.4.4.4) |
(1) (3.3.3) |
|||
[17] | cantellated order-4 dodecahedral (runcicantellated alternated order-5 cubic) |
(1) (3.4.5.4) |
(2) (4.4.4) |
(1) (3.4.5.4) |
(1) (3.4.3.4) |
||
37 | runcitruncated alternated order-5 cubic |
(1) (3.10.10) |
(2) (4.6.10) |
(1) (3.6.6) |
|||
[20] | cantitruncated order-4 dodecahedral (omnitruncated alternated order-5 cubic) |
(1) (4.6.10) |
(1) (4.4.4) |
(1) (4.6.10) |
(1) (4.6.6) |
There are 9 forms, generated by ring permutations of the Coxeter group:
# | Honeycomb name Coxeter–Dynkin diagram |
Cells by location (and count around each vertex) |
vertex figure | |||
---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
|||
38 | (4) (3.3.3) |
- | (4) (4.4.4) |
(6) (3.4.3.4) |
||
39 | (12) (3.3.3.3) |
(8) (3.3.3) |
- | (8) (3.3.3.3) |
||
40 | (3) (3.6.6) |
(1) (3.3.3) |
(1) (4.4.4) |
(3) (4.6.6) |
||
41 | (1) (3.3.3) |
(1) (3.3.3) |
(3) (3.8.8) |
(3) (3.8.8) |
||
42 | (4) (3.6.6) |
(4) (3.6.6) |
(1) (3.3.3.3) |
(1) (3.3.3.3) |
||
43 | (1) (3.3.3.3) |
(2) (3.4.3.4) |
(1) (3.4.3.4) |
(2) (3.4.4.4) |
||
44 | (1) (3.6.6) |
(1) (3.4.3.4) |
(1) (3.8.8) |
(2) (4.6.8) |
||
45 | (2) (4.6.6) |
(1) (3.6.6) |
(1) (3.4.4.4) |
(1) (4.6.6) |
||
46 | (1) (4.6.6) |
(1) (4.6.6) |
(1) (4.6.8) |
(1) (4.6.8) |
There are 9 forms, generated by ring permutations of the Coxeter group:
# | Honeycomb name Coxeter–Dynkin diagram |
Cells by location (and count around each vertex) |
vertex figure | |||
---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
|||
47 | (4) (3.3.3) |
- | (4) (5.5.5) |
(6) (3.5.3.5) |
||
48 | (30) (3.3.3.3) |
(20) (3.3.3) |
- | (12) (3.3.3.3.3) |
||
49 | (3) (3.6.6) |
(1) (3.3.3) |
(1) (5.5.5) |
(3) (5.6.6) |
||
50 | (1) (3.3.3) |
(1) (3.3.3) |
(3) (3.10.10) |
(3) (3.10.10) |
||
51 | (5) (3.6.6) |
(5) (3.6.6) |
(1) (3.3.3.3.3) |
(1) (3.3.3.3.3) |
||
52 | (1) (3.3.3.3) |
(2) (3.4.3.4) |
(1) (3.5.3.5) |
(2) (3.4.5.4) |
||
53 | (1) (3.6.6) |
(1) (3.4.3.4) |
(1) (3.10.10) |
(2) (4.6.10) |
||
54 | (2) (4.6.6) |
(1) (3.6.6) |
(1) (3.4.5.4) |
(1) (5.6.6) |
||
55 | (1) (4.6.6) |
(1) (4.6.6) |
(1) (4.6.10) |
(1) (4.6.10) |
There are 6 forms, generated by ring permutations of the Coxeter group:
# | Honeycomb name Coxeter–Dynkin diagram |
Cells by location (and count around each vertex) |
vertex figure | |||
---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
|||
56 | (6) (3.3.3.3) |
- | (8) (4.4.4) |
(12) (3.4.3.4) |
||
57 | (3) (4.6.6) |
(1) (4.4.4) |
(1) (4.4.4) |
(3) (4.6.6) |
||
58 | (1) (3.3.3.3) |
(1) (3.3.3.3) |
(3) (3.8.8) |
(3) (3.8.8) |
||
59 | (1) (3.4.3.4) |
(2) (3.4.4.4) |
(1) (3.4.3.4) |
(2) (3.4.4.4) |
||
60 | (1) (4.6.6) |
(1) (3.4.4.4) |
(1) (3.8.8) |
(2) (4.6.8) |
||
61 | (1) (4.6.8) |
(1) (4.6.8) |
(1) (4.6.8) |
(1) (4.6.8) |
There are 9 forms, generated by ring permutations of the Coxeter group:
# | Honeycomb name Coxeter–Dynkin diagram |
Cells by location (and count around each vertex) |
vertex figure | |||
---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
|||
62 | (6) (3.3.3.3) |
- | (8) (5.5.5) |
(1) (3.5.3.5) |
||
63 | (30) (3.4.3.4) |
(20) (4.4.4) |
- | (12) (3.3.3.3.3) |
||
64 | (3) (4.6.6) |
(1) (4.4.4) |
(1) (5.5.5) |
(3) (5.6.6) |
||
65 | (1) (3.3.3.3) |
(1) (3.3.3.3) |
(4) (3.10.10) |
(4) (3.10.10) |
||
66 | (5) (3.8.8) |
(5) (3.8.8) |
(1) (3.3.3.3.3) |
(1) (3.3.3.3.3) |
||
67 | (1) (3.4.3.4) |
(2) (3.4.4.4) |
(1) (3.5.3.5) |
(2) (3.4.5.4) |
||
68 | (1) (4.6.6) |
(1) (3.4.4.4) |
(1) (3.10.10) |
(2) (4.6.10) |
||
69 | (2) (4.6.8) |
(1) (3.8.8) |
(1) (3.4.5.4) |
(1) (5.6.6) |
||
70 | (1) (4.6.8) |
(1) (4.6.8) |
(1) (4.6.10) |
(1) (4.6.10) |
There are 6 forms, generated by ring permutations of the Coxeter group:
# | Honeycomb name Coxeter–Dynkin diagram |
Cells by location (and count around each vertex) |
vertex figure | |||
---|---|---|---|---|---|---|
0 |
1 |
2 |
3 |
|||
71 | (12) (3.3.3.3.3) |
- | (20) (5.5.5) |
(30) (3.5.3.5) |
||
72 | (3) (5.6.6) |
(1) (5.5.5) |
(1) (5.5.5) |
(3) (5.6.6) |
||
73 | (1) (3.3.3.3.3) |
(1) (3.3.3.3.3) |
(3) (3.10.10) |
(3) (3.10.10) |
||
74 | (1) (3.5.3.5) |
(2) (3.4.5.4) |
(1) (3.5.3.5) |
(2) (3.4.5.4) |
||
75 | (1) (5.6.6) |
(1) (3.4.5.4) |
(1) (3.10.10) |
(2) (4.6.10) |
||
76 | (1) (4.6.10) |
(1) (4.6.10) |
(1) (4.6.10) |
(1) (4.6.10) |