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In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

Question, Web Fundamentals.svg Unsolved problem in mathematics:
Find the complete set of hyperbolic uniform honeycombs
(more unsolved problems in mathematics)
Four compact regular hyperbolic honeycombs
H3 534 CC center.png
{5,3,4}
H3 535 CC center.png
{5,3,5}
H3 435 CC center.png
{4,3,5}
H3 353 CC center.png
{3,5,3}
Poincaré ball model projections

Contents

Hyperbolic uniform honeycomb familiesEdit

Honeycombs are divided between compact and paracompact forms defined by Coxeter groups, the first category only including finite cells and vertex figures (finite subgroups), and the second includes affine subgroups.

Compact uniform honeycomb familiesEdit

The nine compact Coxeter groups are listed here with their Coxeter diagrams,[1] in order of the relative volumes of their fundamental simplex domains.[2]

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. Only two families are related as a mirror-removal halving: [5,31,1] ↔ [5,3,4,1+].

Indexed Fundamental
simplex
volume[3]
Witt
symbol
Coxeter
notation
Commutator
subgroup
Coxeter
diagram
Honeycombs
H1 0.0358850633   [5,3,4] [(5,3)+,4,1+]
= [5,31,1]+
        15 forms, 2 regular
H2 0.0390502856   [3,5,3] [3,5,3]+         9 forms, 1 regular
H3 0.0717701267   [5,31,1] [5,31,1]+       11 forms (7 overlap with [5,3,4] family, 4 are unique)
H4 0.0857701820   [(4,3,3,3)] [(4,3,3,3)]+      9 forms
H5 0.0933255395   [5,3,5] [5,3,5]+         9 forms, 1 regular
H6 0.2052887885   [(5,3,3,3)] [(5,3,3,3)]+      9 forms
H7 0.2222287320   [(4,3)[2]] [(4,3+,4,3+)]       6 forms
H8 0.3586534401   [(3,4,3,5)] [(3,4,3,5)]+       9 forms
H9 0.5021308905   [(5,3)[2]] [(5,3)[2]]+       6 forms

There are just two radical subgroups with nonsimplectic domains that can be generated by removing a set of two or more mirrors separated by all other mirrors by even-order branches. One is [(4,3,4,3*)], represented by Coxeter diagrams      an index 6 subgroup with a trigonal trapezohedron fundamental domain       , which can be extended by restoring one mirror as      . The other is [4,(3,5)*], index 120 with a dodecahedral fundamental domain.

Paracompact hyperbolic uniform honeycombsEdit

There are also 23 paracompact Coxeter groups of rank 4 that produce paracompact uniform honeycombs with infinite or unbounded facets or vertex figure, including ideal vertices at infinity.

Hyperbolic paracompact group summary
Type Coxeter groups
Linear graphs         |         |         |         |         |         |        
Tridental graphs       |       |      
Cyclic graphs       |       |      |       |       |       |       |       |    
Loop-n-tail graphs       |       |       |      

Other paracompact Coxeter groups exists as Vinberg polytope fundamental domains, including these triangular bipyramid fundamental domains (double tetrahedra) as rank 5 graphs including parallel mirrors. Uniform honeycombs exist as all permutations of rings in these graphs, with the constraint that at least one node must be ringed across infinite order branches.

Dimension Rank Graphs
H3 5
     ,      ,      ,      ,      
       ,        ,        ,        ,        ,        
       ,        ,        ,        ,        ,        ,        
     ,      ,      ,      ,      ,      ,      ,      ,      ,      ,      ,      ,      

[3,5,3] familyEdit

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, called a tetrahedrally diminished dodecahedron.[4]

The bitruncated and runcinated forms (5 and 6) contain the faces of two regular skew polyhedrons: {4,10|3} and {10,4|3}.

# Honeycomb name
Coxeter diagram
and Schläfli
symbols
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
     
1
      
2
     
3
     
1 icosahedral
       
t0{3,5,3}
      (12)
 
(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)
 
(5.6.6)
   
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)
 
(5.6.6)
   
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)
   
# Honeycomb name
Coxeter diagram
and Schläfli
symbols
Cell counts/vertex
and positions in honeycomb
Vertex figure Picture
0
     
1
      
2
     
3
     
Alt
[77] partially diminished icosahedral
pd{3,5,3}[5]
(12)
 
(3.3.3.5)
(4)
 
(5.5.5)
   
Nonuniform omnisnub icosahedral
       
ht0,1,2,3{3,5,3}
(1)
 
(3.3.3.3.5)
(1)
 
(3.3.3.3
(1)
 
(3.3.3.3)
(1)
 
(3.3.3.3.5)
(4)
 
+(3.3.3)
 

[5,3,4] familyEdit

There are 15 forms, generated by ring permutations of the Coxeter group: [5,3,4] or        .

This family is related to the group [5,31,1] by a half symmetry [5,3,4,1+], or             , when the last mirror after the order-4 branch is inactive, or as an alternation if the third mirror is inactive             .

# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
     
1
     
2
     
3
     
10 order-4 dodecahedral
            
- - - (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
       
(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)
   
# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
     
1
     
2
     
3
     
Alt
[34] alternated order-5 cubic
            
(20)
     
 
(3.3.3)
    (12)
 
(3.3.3.3.3)
   
[35] cantic order-5 cubic
            
(1)
 
(3.5.3.5)
- (2)
 
(5.6.6)
(2)
 
(3.6.6)
   
[36] runcic order-5 cubic
            
(1)
 
(5.5.5)
- (3)
 
(3.4.5.4)
(1)
 
(3.3.3)
   
[37] runcicantic order-5 cubic
            
(1)
 
(3.10.10)
- (2)
 
(4.6.10)
(1)
 
(3.6.6)
   
Nonuniform snub rectified order-4 dodecahedral
       
(1)
     
 
(3.3.3.3.3)
(1)
     
 
(3.3.3)
- (2)
     
 
(3.3.3.3.5)
(4)
 
+(3.3.3)
 
Irr. tridiminished icosahedron
Nonuniform runcic snub rectified order-4 dodecahedral
       
     
 
(3.4.4.4)
     
 
(4.4.4.4)
-      
 
(3.3.3.3.5)
 
+(3.3.3)
Nonuniform omnisnub order-5 cubic
       
(1)
     
 
(3.3.3.3.4)
(1)
     
 
(3.3.3.4)
(1)
     
 
(3.3.3.5)
(1)
     
 
(3.3.3.3.5)
(4)
 
+(3.3.3)
 

[5,3,5] familyEdit

There are 9 forms, generated by ring permutations of the Coxeter group: [5,3,5] or        

The bitruncated and runcinated forms (29 and 30) contain the faces of two regular skew polyhedrons: {4,6|5} and {6,4|5}.

# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
     
1
     
2
     
3
     
25 (Regular) 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)
 
(5.6.6)
    (2)
 
(5.6.6)
   
31 cantitruncated order-5 dodecahedral
       
t0,1,2{5,3,5}
(1)
 
(5.6.6)
(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)
   
# Name of honeycomb
Coxeter diagram
Cells by location and count per vertex Vertex figure Picture
0
     
1
     
2
     
3
     
Alt
Nonuniform omnisnub order-5 dodecahedral
       
ht0,1,2,3{5,3,5}
(1)
     
 
(3.3.3.3.5)
(1)
     
 
(3.3.3.5)
(1)
     
 
(3.3.3.5)
(1)
     
 
(3.3.3.3.5)
(4)
 
+(3.3.3)
 

[5,31,1] familyEdit

There are 11 forms (and only 4 not shared with [5,3,4] family), generated by ring permutations of the Coxeter group: [5,31,1] or      . If the branch ring states match, an extended symmetry can double into the [5,3,4] family,             .

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
     
1
   
0'
     
3
   
34 alternated order-5 cubic
            
- - (12)
 
(3.3.3.3.3)
(20)
 
(3.3.3)
   
35 cantic order-5 cubic
            
(1)
 
(3.5.3.5)
- (2)
 
(5.6.6)
(2)
 
(3.6.6)
   
36 runcic order-5 cubic
            
(1)
 
(5.5.5)
- (3)
 
(3.4.5.4)
(1)
 
(3.3.3)
   
37 runcicantic order-5 cubic
            
(1)
 
(3.10.10)
- (2)
 
(4.6.10)
(1)
 
(3.6.6)
   
# Honeycomb name
Coxeter diagram
            
Cells by location
(and count around each vertex)
vertex figure Picture
0
     
1
   
3
   
Alt
[10] Order-4 dodecahedral
            
(4)
 
(5.5.5)
- -    
[11] rectified order-4 dodecahedral
            
(2)
 
(3.5.3.5)
- (2)
 
(3.3.3.3)
   
[12] rectified order-5 cubic
            
(1)
 
(3.3.3.3.3)
- (5)
 
(3.4.3.4)
   
[15] bitruncated order-5 cubic
            
(1)
 
(5.6.6)
- (2)
 
(4.6.6)
   
[14] truncated order-4 dodecahedral
            
(2)
 
(3.10.10)
- (1)
 
(3.3.3.3)
   
[17] cantellated order-4 dodecahedral
            
(1)
 
(3.4.5.4)
(2)
 
(4.4.4)
(1)
 
(3.4.3.4)
   
[20] cantitruncated order-4 dodecahedral
            
(1)
 
(4.6.10)
(1)
 
(4.4.4)
(1)
 
(4.6.6)
   
Nonuniform snub rectified order-4 dodecahedral
            
(2)
 
(3.3.3.3.5)
(1)
 
(3.3.3)
(2)
 
(3.3.3.3.3)
(4)
 
+(3.3.3)
 
Irr. tridiminished icosahedron

[(4,3,3,3)] familyEdit

There are 9 forms, generated by ring permutations of the Coxeter group:     

The bitruncated and runcinated forms (41 and 42) contain the faces of two regular skew polyhedrons: {8,6|3} and {6,8|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
   
1
   
2
    
3
    
Alt
38 tetrahedral-cubic
    
{(3,3,3,4)}
(4)
 
(3.3.3)
- (4)
 
(4.4.4)
(6)
 
(3.4.3.4)
   
39 tetrahedral-octahedral
    
{(3,3,4,3)}
(12)
 
(3.3.3.3)
(8)
 
(3.3.3)
- (8)
 
(3.3.3.3)
   
40 cyclotruncated tetrahedral-cubic
    
ct{(3,3,3,4)}
(3)
 
(3.6.6)
(1)
 
(3.3.3)
(1)
 
(4.4.4)
(3)
 
(4.6.6)
   
41 cyclotruncated cube-tetrahedron
    
ct{(4,3,3,3)}
(1)
 
(3.3.3)
(1)
 
(3.3.3)
(3)
 
(3.8.8)
(3)
 
(3.8.8)
   
42 cyclotruncated tetrahedral-octahedral
    
ct{(3,3,4,3)}
(4)
 
(3.6.6)
(4)
 
(3.6.6)
(1)
 
(3.3.3.3)
(1)
 
(3.3.3.3)
   
43 rectified tetrahedral-cubic
    
r{(3,3,3,4)}
(1)
 
(3.3.3.3)
(2)
 
(3.4.3.4)
(1)
 
(3.4.3.4)
(2)
 
(3.4.4.4)
   
44 truncated tetrahedral-cubic
    
t{(3,3,3,4)}
(1)
 
(3.6.6)
(1)
 
(3.4.3.4)
(1)
 
(3.8.8)
(2)
 
(4.6.8)
   
45 truncated tetrahedral-octahedral
    
t{(3,3,4,3)}
(2)
 
(4.6.6)
(1)
 
(3.6.6)
(1)
 
(3.4.4.4)
(1)
 
(4.6.6)
   
46 omnitruncated tetrahedral-cubic
    
tr{(3,3,3,4)}
(1)
 
(4.6.6)
(1)
 
(4.6.6)
(1)
 
(4.6.8)
(1)
 
(4.6.8)
   
Nonuniform omnisnub tetrahedral-cubic
    
sr{(3,3,3,4)}
(1)
 
(3.3.3.3.3)
(1)
 
(3.3.3.3.3)
(1)
 
(3.3.3.3.4)
(1)
 
(3.3.3.3.4)
(4)
 
+(3.3.3)
 

[(5,3,3,3)] familyEdit

There are 9 forms, generated by ring permutations of the Coxeter group:     

The bitruncated and runcinated forms (50 and 51) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
   
1
   
2
    
3
    
47 tetrahedral-dodecahedral
    
(4)
 
(3.3.3)
- (4)
 
(5.5.5)
(6)
 
(3.5.3.5)
   
48 tetrahedral-icosahedral
    
(30)
 
(3.3.3.3)
(20)
 
(3.3.3)
- (12)
 
(3.3.3.3.3)
   
49 cyclotruncated tetrahedral-dodecahedral
    
(3)
 
(3.6.6)
(1)
 
(3.3.3)
(1)
 
(5.5.5)
(3)
 
(5.6.6)
   
52 rectified tetrahedral-dodecahedral
    
(1)
 
(3.3.3.3)
(2)
 
(3.4.3.4)
(1)
 
(3.5.3.5)
(2)
 
(3.4.5.4)
   
53 truncated tetrahedral-dodecahedral
    
(1)
 
(3.6.6)
(1)
 
(3.4.3.4)
(1)
 
(3.10.10)
(2)
 
(4.6.10)
   
54 truncated tetrahedral-icosahedral
    
(2)
 
(4.6.6)
(1)
 
(3.6.6)
(1)
 
(3.4.5.4)
(1)
 
(5.6.6)
   
# Honeycomb name
Coxeter diagram
    
Cells by location
(and count around each vertex)
vertex figure Picture
0,1
   
2,3
    
Alt
50 cyclotruncated dodecahedral-tetrahedral
    
(2)
 
(3.3.3)
(6)
 
(3.10.10)
   
51 cyclotruncated tetrahedral-icosahedral
    
(10)
 
(3.6.6)
(2)
 
(3.3.3.3.3)
   
55 omnitruncated tetrahedral-dodecahedral
    
(2)
 
(4.6.6)
(2)
 
(4.6.10)
   
Nonuniform omnisnub tetrahedral-dodecahedral
    
(2)
 
(3.3.3.3.3)
(2)
 
(3.3.3.3.5)
(4)
 
+(3.3.3)
 

[(4,3,4,3)] familyEdit

There are 6 forms, generated by ring permutations of the Coxeter group:      . There are 4 extended symmetries possible based on the symmetry of the rings:      ,      ,      , and      .

This symmetry family is also related to a radical subgroup, index 6,            , constructed by [(4,3,4,3*)], and represents a trigonal trapezohedron fundamental domain.

The truncated forms (57 and 58) contain the faces of two regular skew polyhedrons: {6,6|4} and {8,8|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Pictures
0
    
1
    
2
    
3
    
56 cubic-octahedral
     
(6)
 
(3.3.3.3)
- (8)
 
(4.4.4)
(12)
 
(3.4.3.4)
   
60 truncated cubic-octahedral
     
(1)
 
(4.6.6)
(1)
 
(3.4.4.4)
(1)
 
(3.8.8)
(2)
 
(4.6.8)
   
# Honeycomb name
Coxeter diagram
     
Cells by location
(and count around each vertex)
vertex figure Picture
0,3
    
1,2
    
Alt
57 cyclotruncated octahedral-cubic
     
(6)
 
(4.6.6)
(2)
 
(4.4.4)
   
Nonuniform cyclosnub octahedral-cubic
     
(4)
 
(3.3.3.3.3)
(2)
 
(3.3.3)
(4)
 
+(3.3.3.3)
 
# Honeycomb name
Coxeter diagram
     
Cells by location
(and count around each vertex)
vertex figure Picture
0,1
    
2,3
    
58 cyclotruncated cubic-octahedral
     
(2)
 
(3.3.3.3)
(6)
 
(3.8.8)
   
# Honeycomb name
Coxeter diagram
     
Cells by location
(and count around each vertex)
vertex figure Picture
0,2
    
1,3
    
59 rectified cubic-octahedral
     
(2)
 
(3.4.3.4)
(4)
 
(3.4.4.4)
   
# Honeycomb name
Coxeter diagram
     
Cells by location
(and count around each vertex)
vertex figure Picture
0,1,2,3
    
Alt
61 omnitruncated cubic-octahedral
     
(4)
 
(4.6.8)
   
Nonuniform omnisnub cubic-octahedral
     
(4)
 
(3.3.3.3.4)
(4)
 
+(3.3.3)
 

[(4,3,5,3)] familyEdit

There are 9 forms, generated by ring permutations of the Coxeter group:      

The truncated forms (65 and 66) contain the faces of two regular skew polyhedrons: {10,6|3} and {6,10|3}.

# Honeycomb name
Coxeter diagram
Cells by location
(and count around each vertex)
vertex figure Picture
0
    
1
    
2
    
3
    
62 octahedral-dodecahedral
     
(6)
 
(3.3.3.3)
- (8)
 
(5.5.5)
(1)
 
(3.5.3.5)
   
63 cubic-icosahedral
     
(30)
 
(3.4.3.4)
(20)
 
(4.4.4)
- (12)
 
(3.3.3.3.3)
   
64 cyclotruncated octahedral-dodecahedral
     
(3)
 
(4.6.6)
(1)
 
(4.4.4)
(1)
 
(5.5.5)
(3)
 
(5.6.6)
   
67 rectified octahedral-dodecahedral
     
(1)
 
(3.4.3.4)
(2)
 
(3.4.4.4)
(1)
 
(3.5.3.5)
(2)
 
(3.4.5.4)
   
68 truncated octahedral-dodecahedral
     
(1)