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.
Only 14 of the convex uniform polyhedra appear in these patterns:
The icosahedron, snub cube, and square antiprism appear in some alternations, but those honeycombs cannot be realised with all edges unit length.
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.
The fundamental infinite Coxeter groups for 3-space are:
There is a correspondence between all three families. Removing one mirror from produces , and removing one mirror from produces . 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:
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 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.
C3 honeycombs | |||||
---|---|---|---|---|---|
Space group |
Fibrifold | Extended symmetry |
Extended diagram |
Order | Honeycombs |
Pm3m (221) |
4−:2 | [4,3,4] | ×1 | 1, 2, 3, 4, 5, 6 | |
Fm3m (225) |
2−:2 | [1+,4,3,4] ↔ [4,31,1] |
↔ |
Half | 7, 11, 12, 13 |
I43m (217) |
4o:2 | [[(4,3,4,2+)]] | Half × 2 | (7), | |
Fd3m (227) |
2+:2 | [[1+,4,3,4,1+]] ↔ [[3[4]]] |
↔ |
Quarter × 2 | 10, |
Im3m (229) |
8o:2 | [[4,3,4]] | ×2 |
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) (3.6.6) |
(2) (4.6.6) |
rectangular pyramid |
Half oblate octahedrille | ||||
J23 A16 W11 G5 h3δ4 O26 |
Runcic cubic (sratoh) ↔ |
(1) (4.4.4) |
(1) (3.3.3) |
(3) (3.4.4.4) |
tapered triangular prism |
Quarter cubille | ||||
J24 A20 W16 G21 h2,3δ4 O28 |
Runcicantic cubic (gratoh) ↔ |
(1) (3.8.8) |
(1) (3.6.6) |
(2) (4.6.8) |
Irregular tetrahedron |
Half pyramidille | ||||
Nonuniformb | snub rectified cubic (serch) 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 | Cantic snub cubic (casch) 2s0{4,3,4} |
(1) (3.3.3.3.3) |
(2) (3.4.4.4) |
(3) (3.4.4) |
||||||
Nonuniform | Runcicantic snub cubic (rusch) |
(1) (3.4.3.4) |
(2) (4.4.4) |
(1) (3.3.3) |
(1) (3.6.6) |
(3) Tricup |
||||
Nonuniform | Runcic cantitruncated cubic (esch) sr3{4,3,4} |
(1) (3.3.3.3.4) |
(1) (4.4.4) |
(1) (4.4.4) |
(1) (3.4.4.4) |
(3) (3.4.4) |
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 (gippich) 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 (cytatoh) 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 (octet) (same as alternated cubic) ht0,3{4,3,4} |
(2) (3.3.3) |
(6) (3.3.3) |
(6) (3.3.3.3) |
cuboctahedron | |||
Nonuniform | Biorthosnub cubic honeycomb (gabreth) 2s0,3{(4,2,4,3)} |
(2) (4.6.6) |
(2) (4.4.4) |
(2) (4.4.6) |
||||
Nonuniforma | Alternated bitruncated cubic (bisch) h2t{4,3,4} |
(4) (3.3.3.3.3) |
(4) (3.3.3) |
|||||
Nonuniform | Cantic bisnub cubic (cabisch) 2s0,3{4,3,4} |
(2) (3.4.4.4) |
(2) (4.4.4) |
(2) (4.4.4) |
||||
Nonuniformc | Alternated omnitruncated cubic (snich) ht0,1,2,3{4,3,4} |
(2) (3.3.3.3.4) |
(2) (3.3.3.4) |
(4) (3.3.3) |
The , [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.
B3 honeycombs | |||||
---|---|---|---|---|---|
Space group |
Fibrifold | Extended symmetry |
Extended diagram |
Order | Honeycombs |
Fm3m (225) |
2−:2 | [4,31,1] ↔ [4,3,4,1+] |
↔ |
×1 | 1, 2, 3, 4 |
Fm3m (225) |
2−:2 | <[1+,4,31,1]> ↔ <[3[4]]> |
↔ |
×2 | (1), (3) |
Pm3m (221) |
4−:2 | <[4,31,1]> | ×2 |
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 (sratoh) ↔ |
(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 |
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 (bisch) ↔ |
(2) (3.3.3.3.3) |
(2) (3.3.3.3.3) |
(4) (3.3.3) |
||||
Nonuniformb | Alternated cantitruncated cubic (serch) ↔ |
(2) (3.3.3.3.4) |
(1) (3.3.3) |
(1) (3.3.3.3.3) |
(4) (3.3.3) |
Irr. tridiminished icosahedron |
There are 5 forms[3] constructed from the , [3[4]] Coxeter group, of which only the quarter cubic honeycomb is unique. There is one index 2 subgroup [3[4]]+ which generates the snub form, which is not uniform, but included for completeness.
A3 honeycombs | ||||||
---|---|---|---|---|---|---|
Space group |
Fibrifold | Square symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycomb diagrams |
F43m (216) |
1o:2 | a1 | [3[4]] | (None) | ||
Fm3m (225) |
2−:2 | d2 | <[3[4]]> ↔ [4,31,1] |
↔ |
×21 ↔ |
1, 2 |
Fd3m (227) |
2+:2 | g2 | [[3[4]]] or [2+[3[4]]] |
↔ |
×22 | 3 |
Pm3m (221) |
4−:2 | d4 | <2[3[4]]> ↔ [4,3,4] |
↔ |
×41 ↔ |
4 |
I3 (204) |
8−o | r8 | [4[3[4]]]+ ↔ [[4,3+,4]] |
↔ |
½×8 ↔ ½×2 |
(*) |
Im3m (229) |
8o:2 | [4[3[4]]] ↔ [[4,3,4]] |
×8 ↔ ×2 |
5 |
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 (cytatoh) ↔ q{4,3,4} |
(2) (3.3.3) |
(6) (3.6.6) |
triangular antiprism |
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 |
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 |
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 (bisch) ↔ ↔ h2t{4,3,4} |
(4) (3.3.3.3.3) |
(4) (3.3.3) |
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) |
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.
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,∞} |
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 (srothaph) | (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 (grothaph) | (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,∞} |
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 | 22 | 7 | 8 9 | 25 | 20 |
[1+,4,3+,4,1+] | (2) | 1 | b |
[2+[4,3,4]] = |
(1) | 22 | [2+[(4,3+,4,2+)]] | (1) | 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 | 1 | 7 | 10 | 28 | |||
[1[4,31,1]]=[4,3,4] = |
(7) | 22 | 7 | 22 | 7 | 9 | 28 | 25 | [1[1+,4,31,1]]+ | (2) | 1 | 6 | a | |
[1[4,31,1]]+ =[4,3,4]+ |
(1) | b | ||||
[3[4]] |
[3[4]] | (none) | ||||
[2+[3[4]]] |
1 | 6 | ||||
[1[3[4]]]=[4,31,1] = |
(2) | 1 | 10 | ||||
[2[3[4]]]=[4,3,4] = |
(1) | 7 | ||||
[(2+,4)[3[4]]]=[2+[4,3,4]] = |
(1) | 28 | [(2+,4)[3[4]]]+ = [2+[4,3,4]]+ |
(1) | a |
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). [3] [4] [5] [6]. Octet trusses are now among the most common types of truss used in construction.
If cells are allowed to be uniform tilings, more uniform honeycombs can be defined:
Families:
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: triangular tiling |
(2) 3.4.4: triangular prism (2) 4.4.6: hexagonal prism (1) (3.6)2: trihexagonal tiling |
The first two forms shown above are semiregular (uniform with only regular facets), and were listed by Thorold Gosset in 1900 respectively as the 3-ic semi-check and tetroctahedric semi-check.[4]
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.[5]
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 |
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:
Several non-Wythoffian forms outside the list of 76 are known; it is not known how many there are.
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:
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 |
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |