# Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

Twenty-eight such honeycombs are known:

They can be considered the three-dimensional analogue to the uniform tilings of the plane.

The Voronoi diagram of any lattice forms a convex uniform honeycomb in which the cells are zonohedra.

## History

• 1900: Thorold Gosset enumerated the list of semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions, including one regular cubic honeycomb, and two semiregular forms with tetrahedra and octahedra.
• 1905: Alfredo Andreini enumerated 25 of these tessellations.
• 1991: Norman Johnson's manuscript Uniform Polytopes identified the list of 28.
• 1994: Branko Grünbaum, in his paper Uniform tilings of 3-space, also independently enumerated all 28, after discovering errors in Andreini's publication. He found the 1905 paper, which listed 25, had 1 wrong, and 4 being missing. Grünbaum states in this paper that Norman Johnson deserves priority for achieving the same enumeration in 1991. He also mentions that I. Alexeyev of Russia had contacted him regarding a putative enumeration of these forms, but that Grünbaum was unable to verify this at the time.
• 2006: George Olshevsky, in his manuscript Uniform Panoploid Tetracombs, along with repeating the derived list of 11 convex uniform tilings, and 28 convex uniform honeycombs, expands a further derived list of 143 convex uniform tetracombs (Honeycombs of uniform 4-polytopes in 4-space).

Only 14 of the convex uniform polyhedra appear in these patterns:

### Names

This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (non-regular) polyhedra, commonly called Archimedean solids. Recently Conway has suggested naming the set as the Architectonic tessellations and the dual honeycombs as the Catoptric tessellations.

The individual honeycombs are listed with names given to them by Norman Johnson. (Some of the terms used below are defined in Uniform 4-polytope#Geometric derivations for 46 nonprismatic Wythoffian uniform 4-polytopes)

For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28). Coxeter uses δ4 for a cubic honeycomb, hδ4 for an alternated cubic honeycomb, qδ4 for a quarter cubic honeycomb, with subscripts for other forms based on the ring patterns of the Coxeter diagram.

## Compact Euclidean uniform tessellations (by their infinite Coxeter group families)

The fundamental infinite Coxeter groups for 3-space are:

1. The ${\tilde {C}}_{3}$ , [4,3,4], cubic, (8 unique forms plus one alternation)
2. The ${\tilde {B}}_{3}$ , [4,31,1], alternated cubic, (11 forms, 3 new)
3. The ${\tilde {A}}_{3}$ cyclic group, [(3,3,3,3)] or [3], (5 forms, one new)

There is a correspondence between all three families. Removing one mirror from ${\tilde {C}}_{3}$ produces ${\tilde {B}}_{3}$ , and removing one mirror from ${\tilde {B}}_{3}$ produces ${\tilde {A}}_{3}$ . This allows multiple constructions of the same honeycombs. If cells are colored based on unique positions within each Wythoff construction, these different symmetries can be shown.

In addition there are 5 special honeycombs which don't have pure reflectional symmetry and are constructed from reflectional forms with elongation and gyration operations.

The total unique honeycombs above are 18.

The prismatic stacks from infinite Coxeter groups for 3-space are:

1. The ${\tilde {C}}_{2}$ ×${\tilde {I}}_{1}$ , [4,4,2,∞] prismatic group, (2 new forms)
2. The ${\tilde {H}}_{2}$ ×${\tilde {I}}_{1}$ , [6,3,2,∞] prismatic group, (7 unique forms)
3. The ${\tilde {A}}_{2}$ ×${\tilde {I}}_{1}$ , [(3,3,3),2,∞] prismatic group, (No new forms)
4. The ${\tilde {I}}_{1}$ ×${\tilde {I}}_{1}$ ×${\tilde {I}}_{1}$ , [∞,2,∞,2,∞] prismatic group, (These all become a cubic honeycomb)

In addition there is one special elongated form of the triangular prismatic honeycomb.

The total unique prismatic honeycombs above (excluding the cubic counted previously) are 10.

Combining these counts, 18 and 10 gives us the total 28 uniform honeycombs.

### The C~3, [4,3,4] group (cubic)

The regular cubic honeycomb, represented by Schläfli symbol {4,3,4}, offers seven unique derived uniform honeycombs via truncation operations. (One redundant form, the runcinated cubic honeycomb, is included for completeness though identical to the cubic honeycomb.) The reflectional symmetry is the affine Coxeter group [4,3,4]. There are four index 2 subgroups that generate alternations: [1+,4,3,4], [(4,3,4,2+)], [4,3+,4], and [4,3,4]+, with the first two generated repeated forms, and the last two are nonuniform.

[4,3,4], space group Pm3m (221)
Reference
Indices
Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in cubic honeycomb
Frames
(Perspective)
Vertex figure Dual cell
(0)
(1)
(2)
(3)
Alt Solids
(Partial)
J11,15
A1
W1
G22
δ4
cubic (chon)

t0{4,3,4}
{4,3,4}
(8)

(4.4.4)

octahedron

Cube,
J12,32
A15
W14
G7
O1
rectified cubic (rich)

t1{4,3,4}
r{4,3,4}
(2)

(3.3.3.3)
(4)

(3.4.3.4)

cuboid

Square bipyramid
J13
A14
W15
G8
t1δ4
O15
truncated cubic (tich)

t0,1{4,3,4}
t{4,3,4}
(1)

(3.3.3.3)
(4)

(3.8.8)

square pyramid

Isosceles square pyramid
J14
A17
W12
G9
t0,2δ4
O14
cantellated cubic (srich)

t0,2{4,3,4}
rr{4,3,4}
(1)

(3.4.3.4)
(2)

(4.4.4)
(2)

(3.4.4.4)

oblique triangular prism

Triangular bipyramid
J17
A18
W13
G25
t0,1,2δ4
O17
cantitruncated cubic (grich)

t0,1,2{4,3,4}
tr{4,3,4}
(1)

(4.6.6)
(1)

(4.4.4)
(2)

(4.6.8)

irregular tetrahedron

Triangular pyramidille
J18
A19
W19
G20
t0,1,3δ4
O19
runcitruncated cubic (prich)

t0,1,3{4,3,4}
(1)

(3.4.4.4)
(1)

(4.4.4)
(2)

(4.4.8)
(1)

(3.8.8)

oblique trapezoidal pyramid

Square quarter pyramidille
J21,31,51
A2
W9
G1
hδ4
O21
alternated cubic (octet)

h{4,3,4}
(8)

(3.3.3)
(6)

(3.3.3.3)

cuboctahedron

Dodecahedrille
J22,34
A21
W17
G10
h2δ4
O25
Cantic cubic (tatoh)
(1)
(3.4.3.4)
(2)
(4.6.6)
(2)
(3.6.6)

rectangular pyramid

Half oblate octahedrille
J23
A16
W11
G5
h3δ4
O26
Runcic cubic (ratoh)
(1)
cube
(3)
(3.4.4.4)
(1)
(3.3.3)

tapered triangular prism

Quarter cubille
J24
A20
W16
G21
h2,3δ4
O28
Runcicantic cubic (gratoh)
(1)
(3.8.8)
(2)
(4.6.8)
(1)
(3.6.6)

Irregular tetrahedron

Half pyramidille
Nonuniformb snub rectified cubic

sr{4,3,4}
(1)
(3.3.3.3.3)
(1)
(3.3.3)
(2)
(3.3.3.3.4)
(4)
(3.3.3)

Irr. tridiminished icosahedron
Nonuniform Trirectified bisnub cubic

2s0{4,3,4}

(3.3.3.3.3)

(4.4.4)

(4.4.4)

(3.4.4.4)
Nonuniform Runcic cantitruncated cubic

sr3{4,3,4}

(3.4.4.4)

(4.4.4)

(4.4.4)

(3.3.3.3.4)
[[4,3,4]] honeycombs, space group Im3m (229)
Reference
Indices
Honeycomb name
Coxeter diagram

and Schläfli symbol
Cell counts/vertex
and positions in cubic honeycomb
Solids
(Partial)
Frames
(Perspective)
Vertex figure Dual cell
(0,3)

(1,2)

Alt
J11,15
A1
W1
G22
δ4
O1
runcinated cubic
(same as regular cubic) (chon)

t0,3{4,3,4}
(2)

(4.4.4)
(6)

(4.4.4)

octahedron

Cube
J16
A3
W2
G28
t1,2δ4
O16
bitruncated cubic (batch)

t1,2{4,3,4}
2t{4,3,4}
(4)

(4.6.6)

(disphenoid)

Oblate tetrahedrille
J19
A22
W18
G27
t0,1,2,3δ4
O20
omnitruncated cubic (otch)

t0,1,2,3{4,3,4}
(2)

(4.6.8)
(2)

(4.4.8)

irregular tetrahedron

Eighth pyramidille
J21,31,51
A2
W9
G1
hδ4
O27
Quarter cubic honeycomb

ht0ht3{4,3,4}
(2)

(3.3.3)
(6)

(3.6.6)

elongated triangular antiprism

Oblate cubille
J21,31,51
A2
W9
G1
hδ4
O21
Alternated runcinated cubic
(same as alternated cubic)

ht0,3{4,3,4}
(4)

(3.3.3)
(4)

(3.3.3)
(6)

(3.3.3.3)

cuboctahedron
Nonuniform
2s0,3{(4,2,4,3)}
Nonuniforma Alternated bitruncated cubic

h2t{4,3,4}
(4)
(3.3.3.3.3)
(4)
(3.3.3)
Nonuniform
2s0,3{4,3,4}
Nonuniformc Alternated omnitruncated cubic

ht0,1,2,3{4,3,4}
(2)
(3.3.3.3.4)
(2)
(3.3.3.4)
(4)
(3.3.3)

### B~3, [4,31,1] group

The ${\tilde {B}}_{3}$ , [4,3] group offers 11 derived forms via truncation operations, four being unique uniform honeycombs. There are 3 index 2 subgroups that generate alternations: [1+,4,31,1], [4,(31,1)+], and [4,31,1]+. The first generates repeated honeycomb, and the last two are nonuniform but included for completeness.

The honeycombs from this group are called alternated cubic because the first form can be seen as a cubic honeycomb with alternate vertices removed, reducing cubic cells to tetrahedra and creating octahedron cells in the gaps.

Nodes are indexed left to right as 0,1,0',3 with 0' being below and interchangeable with 0. The alternate cubic names given are based on this ordering.

[4,31,1] uniform honeycombs, space group Fm3m (225)
Referenced
indices
Honeycomb name
Coxeter diagrams
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0)
(1)
(0')
(3)
J21,31,51
A2
W9
G1
hδ4
O21
Alternated cubic (octet)
(6)
(3.3.3.3)
(8)
(3.3.3)

cuboctahedron
J22,34
A21
W17
G10
h2δ4
O25
Cantic cubic (tatoh)
(1)
(3.4.3.4)
(2)
(4.6.6)
(2)
(3.6.6)

rectangular pyramid
J23
A16
W11
G5
h3δ4
O26
Runcic cubic (ratoh)
(1)
cube
(3)
(3.4.4.4)
(1)
(3.3.3)

tapered triangular prism
J24
A20
W16
G21
h2,3δ4
O28
Runcicantic cubic (gratoh)
(1)
(3.8.8)
(2)
(4.6.8)
(1)
(3.6.6)

Irregular tetrahedron
<[4,31,1]> uniform honeycombs, space group Pm3m (221)
Referenced
indices
Honeycomb name
Coxeter diagrams
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,0')
(1)
(3)
Alt
J11,15
A1
W1
G22
δ4
O1
Cubic (chon)
(8)
(4.4.4)

octahedron
J12,32
A15
W14
G7
t1δ4
O15
Rectified cubic (rich)
(4)
(3.4.3.4)
(2)
(3.3.3.3)

cuboid
Rectified cubic (rich)
(2)
(3.3.3.3)
(4)
(3.4.3.4)

cuboid
J13
A14
W15
G8
t0,1δ4
O14
Truncated cubic (tich)
(4)
(3.8.8)
(1)
(3.3.3.3)

square pyramid
J14
A17
W12
G9
t0,2δ4
O17
Cantellated cubic (srich)
(2)
(3.4.4.4)
(2)
(4.4.4)
(1)
(3.4.3.4)

obilique triangular prism
J16
A3
W2
G28
t0,2δ4
O16
Bitruncated cubic (batch)
(2)
(4.6.6)
(2)
(4.6.6)

isosceles tetrahedron
J17
A18
W13
G25
t0,1,2δ4
O18
Cantitruncated cubic (grich)
(2)
(4.6.8)
(1)
(4.4.4)
(1)
(4.6.6)

irregular tetrahedron
J21,31,51
A2
W9
G1
hδ4
O21
Alternated cubic (octet)
(8)
(3.3.3)
(6)
(3.3.3.3)

cuboctahedron
J22,34
A21
W17
G10
h2δ4
O25
Cantic cubic (tatoh)
(2)
(3.6.6)
(1)
(3.4.3.4)
(2)
(4.6.6)

rectangular pyramid
Nonuniforma Alternated bitruncated cubic
(2)
(3.3.3.3.3)
(2)
(3.3.3.3.3)
(4)
(3.3.3)
Nonuniformb Alternated cantitruncated cubic
(2)
(3.3.3.3.4)
(1)
(3.3.3)
(1)
(3.3.3.3.3)
(4)
(3.3.3)

Irr. tridiminished icosahedron

### A~3, [3)] group

There are 5 forms constructed from the ${\tilde {A}}_{3}$ , [3] Coxeter group, of which only the quarter cubic honeycomb is unique. There is one index 2 subgroup [3]+ which generates the snub form, which is not uniform, but included for completeness.

[[3]] uniform honeycombs, space group Fd3m (227)
Referenced
indices
Honeycomb name
Coxeter diagrams
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,1)
(2,3)
J25,33
A13
W10
G6
qδ4
O27
quarter cubic (batatoh)

q{4,3,4}
(2)
(3.3.3)
(6)
(3.6.6)

triangular antiprism
<[3]> ↔ [4,31,1] uniform honeycombs, space group Fm3m (225)
Referenced
indices
Honeycomb name
Coxeter diagrams
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
0 (1,3) 2
J21,31,51
A2
W9
G1
hδ4
O21
alternated cubic (octet)

h{4,3,4}
(8)
(3.3.3)
(6)
(3.3.3.3)

cuboctahedron
J22,34
A21
W17
G10
h2δ4
O25
cantic cubic (tatoh)

h2{4,3,4}
(2)
(3.6.6)
(1)
(3.4.3.4)
(2)
(4.6.6)

Rectangular pyramid
[2[3]] ↔ [4,3,4] uniform honeycombs, space group Pm3m (221)
Referenced
indices
Honeycomb name
Coxeter diagrams
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,2)
(1,3)
J12,32
A15
W14
G7
t1δ4
O1
rectified cubic (rich)

r{4,3,4}
(2)
(3.4.3.4)
(1)
(3.3.3.3)

cuboid
[4[3]] ↔ [[4,3,4]] uniform honeycombs, space group Im3m (229)
Referenced
indices
Honeycomb name
Coxeter diagrams
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,1,2,3)
Alt
J16
A3
W2
G28
t1,2δ4
O16
bitruncated cubic (batch)

2t{4,3,4}
(4)
(4.6.6)

isosceles tetrahedron
Nonuniforma Alternated cantitruncated cubic

h2t{4,3,4}
(4)
(3.3.3.3.3)
(4)
(3.3.3)

### Nonwythoffian forms (gyrated and elongated)

Three more uniform honeycombs are generated by breaking one or another of the above honeycombs where its faces form a continuous plane, then rotating alternate layers by 60 or 90 degrees (gyration) and/or inserting a layer of prisms (elongation).

The elongated and gyroelongated alternated cubic tilings have the same vertex figure, but are not alike. In the elongated form, each prism meets a tetrahedron at one triangular end and an octahedron at the other. In the gyroelongated form, prisms that meet tetrahedra at both ends alternate with prisms that meet octahedra at both ends.

The gyroelongated triangular prismatic tiling has the same vertex figure as one of the plain prismatic tilings; the two may be derived from the gyrated and plain triangular prismatic tilings, respectively, by inserting layers of cubes.

Referenced
indices
symbol Honeycomb name cell types (# at each vertex) Solids
(Partial)
Frames
(Perspective)
vertex figure
J52
A2'
G2
O22
h{4,3,4}:g gyrated alternated cubic (gytoh) tetrahedron (8)
octahedron (6)

triangular orthobicupola
J61
A?
G3
O24
h{4,3,4}:ge gyroelongated alternated cubic (gyetoh) triangular prism (6)
tetrahedron (4)
octahedron (3)
J62
A?
G4
O23
h{4,3,4}:e elongated alternated cubic (etoh) triangular prism (6)
tetrahedron (4)
octahedron (3)
J63
A?
G12
O12
{3,6}:g × {∞} gyrated triangular prismatic (gytoph) triangular prism (12)
J64
A?
G15
O13
{3,6}:ge × {∞} gyroelongated triangular prismatic (gyetaph) triangular prism (6)
cube (4)

### Prismatic stacks

Eleven prismatic tilings are obtained by stacking the eleven uniform plane tilings, shown below, in parallel layers. (One of these honeycombs is the cubic, shown above.) The vertex figure of each is an irregular bipyramid whose faces are isosceles triangles.

#### The C~2×I~1(∞), [4,4,2,∞], prismatic group

There are only 3 unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding to each form.

Indices Coxeter-Dynkin
and Schläfli
symbols
Honeycomb name Plane
tiling
Solids
(Partial)
Tiling
J11,15
A1
G22

{4,4}×{∞}
Cubic
(Square prismatic) (chon)
(4.4.4.4)

r{4,4}×{∞}

rr{4,4}×{∞}
J45
A6
G24

t{4,4}×{∞}
Truncated/Bitruncated square prismatic (tassiph) (4.8.8)

tr{4,4}×{∞}
J44
A11
G14

sr{4,4}×{∞}
Snub square prismatic (sassiph) (3.3.4.3.4)
Nonuniform
ht0,1,2,3{4,4,2,∞}

#### The G~2xI~1(∞), [6,3,2,∞] prismatic group

Indices Coxeter-Dynkin
and Schläfli
symbols
Honeycomb name Plane
tiling
Solids
(Partial)
Tiling
J41
A4
G11

{3,6} × {∞}
Triangular prismatic (tiph) (36)
J42
A5
G26

{6,3} × {∞}
Hexagonal prismatic (hiph) (63)

t{3,6} × {∞}
J43
A8
G18

r{6,3} × {∞}
Trihexagonal prismatic (thiph) (3.6.3.6)
J46
A7
G19

t{6,3} × {∞}
Truncated hexagonal prismatic (thaph) (3.12.12)
J47
A9
G16

rr{6,3} × {∞}
Rhombi-trihexagonal prismatic (rothaph) (3.4.6.4)
J48
A12
G17

sr{6,3} × {∞}
Snub hexagonal prismatic (snathaph) (3.3.3.3.6)
J49
A10
G23

tr{6,3} × {∞}
truncated trihexagonal prismatic (otathaph) (4.6.12)
J65
A11'
G13

{3,6}:e × {∞}
elongated triangular prismatic (etoph) (3.3.3.4.4)
J52
A2'
G2

h3t{3,6,2,∞}
gyrated tetrahedral-octahedral (gytoh) (36)

s2r{3,6,2,∞}
Nonuniform
ht0,1,2,3{3,6,2,∞}

### Enumeration of Wythoff forms

All nonprismatic Wythoff constructions by Coxeter groups are given below, along with their alternations. Uniform solutions are indexed with Branko Grünbaum's listing. Green backgrounds are shown on repeated honeycombs, with the relations are expressed in the extended symmetry diagrams.

Coxeter group Extended
symmetry
Honeycombs Chiral
extended
symmetry
Alternation honeycombs
[4,3,4]
[4,3,4]
6 7 | 8
9 | 25 | 20
[1+,4,3+,4,1+](2) b
[2+[4,3,4]]
=
(1) 22 [2+[(4,3+,4,2+)]](1) 6
[2+[4,3,4]]
1 28 [2+[(4,3+,4,2+)]](1) a
[2+[4,3,4]]
2 27 [2+[4,3,4]]+(1) c
[4,31,1]
[4,31,1]
4 7 | 10 | 28
[1[4,31,1]]=[4,3,4]
=
(7) 22 | 7 | 22 | 7 | 9 | 28 | 25 [1[1+,4,31,1]]+(2) 6 | a
[1[4,31,1]]+
=[4,3,4]+
(1) b
[3]
[3] (none)
[2+[3]]
1 6
[1[3]]=[4,31,1]
=
(2) 10
[2[3]]=[4,3,4]
=
(1) 7
[(2+,4)[3]]=[2+[4,3,4]]
=
(1) 28 [(2+,4)[3]]+
= [2+[4,3,4]]+
(1)a

### Examples

All 28 of these tessellations are found in crystal arrangements.

The alternated cubic honeycomb is of special importance since its vertices form a cubic close-packing of spheres. The space-filling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently re-discovered by Buckminster Fuller (who called it the octet truss and patented it in the 1940s). . Octet trusses are now among the most common types of truss used in construction.

## Frieze forms

If cells are allowed to be uniform tilings, more uniform honeycombs can be defined:

Families:

• ${\tilde {C}}_{2}$ x$A_{1}$ : [4,4,2] Cubic slab honeycombs (3 forms)
• ${\tilde {G}}_{2}$ x$A_{1}$ : [6,3,2] Tri-hexagonal slab honeycombs (8 forms)
• ${\tilde {A}}_{2}$ x$A_{1}$ : [(3,3,3),2] Triangular slab honeycombs (No new forms)
• ${\tilde {I}}_{1}$ x$A_{1}$ x$A_{1}$ : [∞,2,2] = Cubic column honeycombs (1 form)
• $I_{2}(p)$ x${\tilde {I}}_{1}$ : [p,2,∞] Polygonal column honeycombs
• ${\tilde {C}}_{2}$ x${\tilde {C}}_{2}$ x$A_{1}$ : [∞,2,∞,2] = [4,4,2] - = (Same as cubic slab honeycomb family)
Examples (partially drawn)
Cubic slab honeycomb
Alternated hexagonal slab honeycomb
Trihexagonal slab honeycomb

(4) 43: cube
(1) 44: square tiling

(4) 33: tetrahedron
(3) 34: octahedron
(1) 36: hexagonal tiling

(2) 3.4.4: triangular prism
(2) 4.4.6: hexagonal prism
(1) (3.6)2: trihexagonal tiling

## Scaliform honeycomb

A scaliform honeycomb is vertex-transitive, like a uniform honeycomb, with regular polygon faces while cells and higher elements are only required to be orbiforms, equilateral, with their vertices lying on hyperspheres. For 3D honeycombs, this allows a subset of Johnson solids along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, pyramid and cupola gaps.

Euclidean honeycomb scaliforms
Frieze slabs Prismatic stacks
s3{2,6,3}, s3{2,4,4}, s{2,4,4}, 3s4{4,4,2,},

(1) 3.4.3.4: triangular cupola
(2) 3.4.6: triangular cupola
(1) 3.3.3.3: octahedron
(1) 3.6.3.6: trihexagonal tiling

(1) 3.4.4.4: square cupola
(2) 3.4.8: square cupola
(1) 3.3.3: tetrahedron
(1) 4.8.8: truncated square tiling

(1) 3.3.3.3: square pyramid
(4) 3.3.4: square pyramid
(4) 3.3.3: tetrahedron
(1) 4.4.4.4: square tiling

(1) 3.3.3.3: square pyramid
(4) 3.3.4: square pyramid
(4) 3.3.3: tetrahedron
(4) 4.4.4: cube

## Hyperbolic forms

There are 9 Coxeter group families of compact uniform honeycombs in hyperbolic 3-space, generated as Wythoff constructions, and represented by ring permutations of the Coxeter-Dynkin diagrams for each family.

From these 9 families, there are a total of 76 unique honeycombs generated:

• [3,5,3] : - 9 forms
• [5,3,4] : - 15 forms
• [5,3,5] : - 9 forms
• [5,31,1] : - 11 forms (7 overlap with [5,3,4] family, 4 are unique)
• [(4,3,3,3)] : - 9 forms
• [(4,3,4,3)] : - 6 forms
• [(5,3,3,3)] : - 9 forms
• [(5,3,4,3)] : - 9 forms
• [(5,3,5,3)] : - 6 forms

The full list of hyperbolic uniform honeycombs has not been proven and an unknown number of non-Wythoffian forms exist. One known example is in the {3,5,3} family.

### Paracompact hyperbolic forms

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

Simplectic hyperbolic paracompact group summary
Type Coxeter groups Unique honeycomb count
Linear graphs | | | | | 4×15+6+8+8 = 82
Tridental graphs | 4+4+0 = 8
Cyclic graphs | | | | | | | 4×9+5+1+4+1+0 = 47
Loop-n-tail graphs | | 4+4+4+2 = 14