Convex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills threedimensional Euclidean space with nonoverlapping convex uniform polyhedral cells.
Twentyeight such honeycombs are known:
 the familiar cubic honeycomb and 7 truncations thereof;
 the alternated cubic honeycomb and 4 truncations thereof;
 10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb);
 5 modifications of some of the above by elongation and/or gyration.
They can be considered the threedimensional 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 SemiRegular 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.[1]
 1994: Branko Grünbaum, in his paper Uniform tilings of 3space, 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 4polytopes in 4space).[2]
Only 14 of the convex uniform polyhedra appear in these patterns:
 three of the five Platonic solids,
 six of the thirteen Archimedean solids, and
 five of the infinite family of prisms.
Names
This set can be called the regular and semiregular honeycombs. It has been called the Archimedean honeycombs by analogy with the convex uniform (nonregular) 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 4polytope#Geometric derivations for 46 nonprismatic Wythoffian uniform 4polytopes)
For crossreferencing, they are given with list indices from Andreini (122), Williams(12,919), Johnson (1119, 2125, 3134, 4149, 5152, 6165), and Grünbaum(128). Coxeter uses δ_{4} for a cubic honeycomb, hδ_{4} for an alternated cubic honeycomb, qδ_{4} 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 3space are:
 The , [4,3,4], cubic,
(8 unique forms plus one alternation)  The , [4,3^{1,1}], alternated cubic,
(11 forms, 3 new)  The cyclic group, [(3,3,3,3)] or [3^{[4]}],
(5 forms, one new)
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 3space are:
 The ×, [4,4,2,∞] prismatic group,
(2 new forms)  The ×, [6,3,2,∞] prismatic group,
(7 unique forms)  The ×, [(3,3,3),2,∞] prismatic group,
(No new forms)  The ××, [∞,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.
C3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Pm3m (221) 
4^{−}:2  [4,3,4]  ×1  
Fm3m (225) 
2^{−}:2  [1^{+},4,3,4] ↔ [4,3^{1,1}] 
↔ 
Half  
I43m (217) 
4^{o}:2  [[(4,3,4,2^{+})]]  Half × 2  
Fd3m (227) 
2^{+}:2  [[1^{+},4,3,4,1^{+}]] ↔ [[3^{[4]}]] 
↔ 
Quarter × 2  
Im3m (229) 
8^{o}: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)  
J_{11,15} A_{1} W_{1} G_{22} δ_{4} 
cubic (chon) t_{0}{4,3,4} {4,3,4} 
(8) (4.4.4) 
octahedron 
Cube,  
J_{12,32} A_{15} W_{14} G_{7} O_{1} 
rectified cubic (rich) t_{1}{4,3,4} r{4,3,4} 
(2) (3.3.3.3) 
(4) (3.4.3.4) 
cuboid 
Square bipyramid  
J_{13} A_{14} W_{15} G_{8} t_{1}δ_{4} O_{15} 
truncated cubic (tich) t_{0,1}{4,3,4} t{4,3,4} 
(1) (3.3.3.3) 
(4) (3.8.8) 
square pyramid 
Isosceles square pyramid  
J_{14} A_{17} W_{12} G_{9} t_{0,2}δ_{4} O_{14} 
cantellated cubic (srich) t_{0,2}{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  
J_{17} A_{18} W_{13} G_{25} t_{0,1,2}δ_{4} O_{17} 
cantitruncated cubic (grich) t_{0,1,2}{4,3,4} tr{4,3,4} 
(1) (4.6.6) 
(1) (4.4.4) 
(2) (4.6.8) 
irregular tetrahedron 
Triangular pyramidille  
J_{18} A_{19} W_{19} G_{20} t_{0,1,3}δ_{4} O_{19} 
runcitruncated cubic (prich) t_{0,1,3}{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  
J_{21,31,51} A_{2} W_{9} G_{1} hδ_{4} O_{21} 
alternated cubic (octet) h{4,3,4} 
(8) (3.3.3) 
(6) (3.3.3.3) 
cuboctahedron 
Dodecahedrille  
J_{22,34} A_{21} W_{17} G_{10} h_{2}δ_{4} O_{25} 
Cantic cubic (tatoh) 
(3.4.3.4) 
(4.6.6) 
(3.6.6) 
rectangular pyramid 
Half oblate octahedrille  
J_{23} A_{16} W_{11} G_{5} h_{3}δ_{4} O_{26} 
Runcic cubic (ratoh) 
cube 
(3.4.4.4) 
(3.3.3) 
tapered triangular prism 
Quarter cubille  
J_{24} A_{20} W_{16} G_{21} h_{2,3}δ_{4} O_{28} 
Runcicantic cubic (gratoh) 
(3.8.8) 
(4.6.8) 
(3.6.6) 
Irregular tetrahedron 
Half pyramidille  
Nonuniform_{b}  snub rectified cubic sr{4,3,4} 
(3.3.3.3.3) 
(3.3.3) 
(3.3.3.3.4) 
(3.3.3) 
Irr. tridiminished icosahedron  
Nonuniform  Trirectified bisnub cubic 2s_{0}{4,3,4} 
(3.3.3.3.3) 
(4.4.4) 
(4.4.4) 
(3.4.4.4) 

Nonuniform  Runcic cantitruncated cubic sr_{3}{4,3,4} 
(3.4.4.4) 
(4.4.4) 
(4.4.4) 
(3.3.3.3.4) 
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  
J_{11,15} A_{1} W_{1} G_{22} δ_{4} O_{1} 
runcinated cubic (same as regular cubic) (chon) t_{0,3}{4,3,4} 
(2) (4.4.4) 
(6) (4.4.4) 
octahedron 
Cube  
J_{16} A_{3} W_{2} G_{28} t_{1,2}δ_{4} O_{16} 
bitruncated cubic (batch) t_{1,2}{4,3,4} 2t{4,3,4} 
(4) (4.6.6) 
(disphenoid) 
Oblate tetrahedrille  
J_{19} A_{22} W_{18} G_{27} t_{0,1,2,3}δ_{4} O_{20} 
omnitruncated cubic (otch) t_{0,1,2,3}{4,3,4} 
(2) (4.6.8) 
(2) (4.4.8) 
irregular tetrahedron 
Eighth pyramidille  
J_{21,31,51} A_{2} W_{9} G_{1} hδ_{4} O_{27} 
Quarter cubic honeycomb ht_{0}ht_{3}{4,3,4} 
(2) (3.3.3) 
(6) (3.6.6) 
elongated triangular antiprism 
Oblate cubille  
J_{21,31,51} A_{2} W_{9} G_{1} hδ_{4} O_{21} 
Alternated runcinated cubic (same as alternated cubic) ht_{0,3}{4,3,4} 
(4) (3.3.3) 
(4) (3.3.3) 
(6) (3.3.3.3) 
cuboctahedron  
Nonuniform  2s_{0,3}{(4,2,4,3)} 

Nonuniform_{a}  Alternated bitruncated cubic h2t{4,3,4} 
(3.3.3.3.3) 
(3.3.3) 

Nonuniform  2s_{0,3}{4,3,4} 

Nonuniform_{c}  Alternated omnitruncated cubic ht_{0,1,2,3}{4,3,4} 
(3.3.3.3.4) 
(3.3.3.4) 
(3.3.3) 
B^{~}_{3}, [4,3^{1,1}] group
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^{+},4,3^{1,1}], [4,(3^{1,1})^{+}], and [4,3^{1,1}]^{+}. 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  [4,3^{1,1}] ↔ [4,3,4,1^{+}] 
↔ 
×1  
Fm3m (225) 
2^{−}:2  <[1^{+},4,3^{1,1}]> ↔ <[3^{[4]}]> 
↔ 
×2  
Pm3m (221) 
4^{−}:2  <[4,3^{1,1}]>  ×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)  
J_{21,31,51} A_{2} W_{9} G_{1} hδ_{4} O_{21} 
Alternated cubic (octet) 
(3.3.3.3) 
(3.3.3) 
cuboctahedron  
J_{22,34} A_{21} W_{17} G_{10} h_{2}δ_{4} O_{25} 
Cantic cubic (tatoh) 
(3.4.3.4) 
(4.6.6) 
(3.6.6) 
rectangular pyramid  
J_{23} A_{16} W_{11} G_{5} h_{3}δ_{4} O_{26} 
Runcic cubic (ratoh) 
cube 
(3.4.4.4) 
(3.3.3) 
tapered triangular prism  
J_{24} A_{20} W_{16} G_{21} h_{2,3}δ_{4} O_{28} 
Runcicantic cubic (gratoh) 
(3.8.8) 
(4.6.8) 
(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  
J_{11,15} A_{1} W_{1} G_{22} δ_{4} O_{1} 
Cubic (chon) 
(4.4.4) 
octahedron  
J_{12,32} A_{15} W_{14} G_{7} t_{1}δ_{4} O_{15} 
Rectified cubic (rich) 
(3.4.3.4) 
(3.3.3.3) 
cuboid  
Rectified cubic (rich) 
(3.3.3.3) 
(3.4.3.4) 
cuboid  
J_{13} A_{14} W_{15} G_{8} t_{0,1}δ_{4} O_{14} 
Truncated cubic (tich) 
(3.8.8) 
(3.3.3.3) 
square pyramid  
J_{14} A_{17} W_{12} G_{9} t_{0,2}δ_{4} O_{17} 
Cantellated cubic (srich) 
(3.4.4.4) 
(4.4.4) 
(3.4.3.4) 
obilique triangular prism  
J_{16} A_{3} W_{2} G_{28} t_{0,2}δ_{4} O_{16} 
Bitruncated cubic (batch) 
(4.6.6) 
(4.6.6) 
isosceles tetrahedron  
J_{17} A_{18} W_{13} G_{25} t_{0,1,2}δ_{4} O_{18} 
Cantitruncated cubic (grich) 
(4.6.8) 
(4.4.4) 
(4.6.6) 
irregular tetrahedron  
J_{21,31,51} A_{2} W_{9} G_{1} hδ_{4} O_{21} 
Alternated cubic (octet) 
(3.3.3) 
(3.3.3.3) 
cuboctahedron  
J_{22,34} A_{21} W_{17} G_{10} h_{2}δ_{4} O_{25} 
Cantic cubic (tatoh) 
(3.6.6) 
(3.4.3.4) 
(4.6.6) 
rectangular pyramid  
Nonuniform_{a}  Alternated bitruncated cubic 
(3.3.3.3.3) 
(3.3.3.3.3) 
(3.3.3) 

Nonuniform_{b}  Alternated cantitruncated cubic 
(3.3.3.3.4) 
(3.3.3) 
(3.3.3.3.3) 
(3.3.3) 
Irr. tridiminished icosahedron 
A^{~}_{3}, [3^{[4]})] group
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) 
1^{o}:2  a1 
[3^{[4]}]  (None)  
Fm3m (225) 
2^{−}:2  d2 
<[3^{[4]}]> ↔ [4,3^{1,1}] 
↔ 
×2_{1} ↔ 

Fd3m (227) 
2^{+}:2  g2 
[[3^{[4]}]] or [2^{+}[3^{[4]}]] 
↔ 
×2_{2}  
Pm3m (221) 
4^{−}:2  d4 
<2[3^{[4]}]> ↔ [4,3,4] 
↔ 
×4_{1} ↔ 

I3 (204) 
8^{−o}  r8 
[4[3^{[4]}]]^{+} ↔ [[4,3<sup>+</sup>,4]] 
↔ 
½×8 ↔ ½×2 

Im3m (229) 
8^{o}:2  [4[3^{[4]}]] ↔ [[4,3,4]] 
×8 ↔ ×2 
Referenced indices 
Honeycomb name Coxeter diagrams 
Cells by location (and count around each vertex) 
Solids (Partial) 
Frames (Perspective) 
vertex figure  

(0,1) 
(2,3)  
J_{25,33} A_{13} W_{10} G_{6} qδ_{4} O_{27} 
quarter cubic (batatoh) q{4,3,4} 
(3.3.3) 
(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  
J_{21,31,51} A_{2} W_{9} G_{1} hδ_{4} O_{21} 
alternated cubic (octet) h{4,3,4} 
(3.3.3) 
(3.3.3.3) 
cuboctahedron  
J_{22,34} A_{21} W_{17} G_{10} h_{2}δ_{4} O_{25} 
cantic cubic (tatoh) h_{2}{4,3,4} 
(3.6.6) 
(3.4.3.4) 
(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)  
J_{12,32} A_{15} W_{14} G_{7} t_{1}δ_{4} O_{1} 
rectified cubic (rich) r{4,3,4} 
(3.4.3.4) 
(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  
J_{16} A_{3} W_{2} G_{28} t_{1,2}δ_{4} O_{16} 
bitruncated cubic (batch) 2t{4,3,4} 
(4.6.6) 
isosceles tetrahedron  
Nonuniform_{a}  Alternated cantitruncated cubic h2t{4,3,4} 
(3.3.3.3.3) 
(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 

J_{52} A_{2'} G_{2} O_{22} 
h{4,3,4}:g  gyrated alternated cubic (gytoh)  tetrahedron (8) octahedron (6) 
triangular orthobicupola  
J_{61} A_{?} G_{3} O_{24} 
h{4,3,4}:ge  gyroelongated alternated cubic (gyetoh)  triangular prism (6) tetrahedron (4) octahedron (3) 

J_{62} A_{?} G_{4} O_{23} 
h{4,3,4}:e  elongated alternated cubic (etoh)  triangular prism (6) tetrahedron (4) octahedron (3) 

J_{63} A_{?} G_{12} O_{12} 
{3,6}:g × {∞}  gyrated triangular prismatic (gytoph)  triangular prism (12)  
J_{64} A_{?} G_{15} O_{13} 
{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  CoxeterDynkin and Schläfli symbols 
Honeycomb name  Plane tiling 
Solids (Partial) 
Tiling 

J_{11,15} A_{1} G_{22} 
{4,4}×{∞} 
Cubic (Square prismatic) (chon) 
(4.4.4.4)  
r{4,4}×{∞} 

rr{4,4}×{∞} 

J_{45} A_{6} G_{24} 
t{4,4}×{∞} 
Truncated/Bitruncated square prismatic (tassiph)  (4.8.8)  
tr{4,4}×{∞} 

J_{44} A_{11} G_{14} 
sr{4,4}×{∞} 
Snub square prismatic (sassiph)  (3.3.4.3.4)  
Nonuniform  ht_{0,1,2,3}{4,4,2,∞} 
The G^{~}_{2}xI^{~}_{1}(∞), [6,3,2,∞] prismatic group
Indices  CoxeterDynkin and Schläfli symbols 
Honeycomb name  Plane tiling 
Solids (Partial) 
Tiling 

J_{41} A_{4} G_{11} 
{3,6} × {∞} 
Triangular prismatic (tiph)  (3^{6})  
J_{42} A_{5} G_{26} 
{6,3} × {∞} 
Hexagonal prismatic (hiph)  (6^{3})  
t{3,6} × {∞} 

J_{43} A_{8} G_{18} 
r{6,3} × {∞} 
Trihexagonal prismatic (thiph)  (3.6.3.6)  
J_{46} A_{7} G_{19} 
t{6,3} × {∞} 
Truncated hexagonal prismatic (thaph)  (3.12.12)  
J_{47} A_{9} G_{16} 
rr{6,3} × {∞} 
Rhombitrihexagonal prismatic (rothaph)  (3.4.6.4)  
J_{48} A_{12} G_{17} 
sr{6,3} × {∞} 
Snub hexagonal prismatic (snathaph)  (3.3.3.3.6)  
J_{49} A_{10} G_{23} 
tr{6,3} × {∞} 
truncated trihexagonal prismatic (otathaph)  (4.6.12)  
J_{65} A_{11'} G_{13} 
{3,6}:e × {∞} 
elongated triangular prismatic (etoph)  (3.3.3.4.4)  
J_{52} A_{2'} G_{2} 
h3t{3,6,2,∞} 
gyrated tetrahedraloctahedral (gytoh)  (3^{6})  
s2r{3,6,2,∞}  
Nonuniform  ht_{0,1,2,3}{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  [1^{+},4,3^{+},4,1^{+}]  (2)  
[2^{+}[4,3,4]]  (1)  [2^{+}[(4,3^{+},4,2^{+})]]  (1)  
[2^{+}[4,3,4]]  1  [2^{+}[(4,3^{+},4,2^{+})]]  (1)  
[2^{+}[4,3,4]]  2  [2^{+}[4,3,4]]^{+}  (1)  
[4,3^{1,1}]  [4,3^{1,1}]  4  
[1[4,3^{1,1}]]=[4,3,4]  (7)  [1[1^{+},4,3^{1,1}]]^{+}  (2)  
[1[4,3^{1,1}]]^{+} =[4,3,4]^{+}  (1)  
[3^{[4]}]  [3^{[4]}]  (none)  
[2^{+}[3^{[4]}]]  1  
[1[3^{[4]}]]=[4,3^{1,1}]  (2)  
[2[3^{[4]}]]=[4,3,4]  (1)  
[(2^{+},4)[3^{[4]}]]=[2^{+}[4,3,4]]  (1)  [(2^{+},4)[3^{[4]}]]^{+} = [2^{+}[4,3,4]]^{+} 
(1) 
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 closepacking of spheres. The spacefilling truss of packed octahedra and tetrahedra was apparently first discovered by Alexander Graham Bell and independently rediscovered 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:
 x: [4,4,2]
Cubic slab honeycombs (3 forms)  x: [6,3,2]
Trihexagonal slab honeycombs (8 forms)  x: [(3,3,3),2]
Triangular slab honeycombs (No new forms)  xx: [∞,2,2]
= Cubic column honeycombs (1 form)  x: [p,2,∞]
Polygonal column honeycombs  xx: [∞,2,∞,2] = [4,4,2] 
= (Same as cubic slab honeycomb family)
Cubic slab honeycomb 
Alternated hexagonal slab honeycomb 
Trihexagonal slab honeycomb 

(4) 4^{3}: cube (1) 4^{4}: square tiling 
(4) 3^{3}: tetrahedron (3) 3^{4}: octahedron (1) 3^{6}: 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 vertextransitive, 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.[4]
Frieze slabs  Prismatic stacks  

s_{3}{2,6,3}, 
s_{3}{2,4,4}, 
s{2,4,4}, 
3s_{4}{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 3space, generated as Wythoff constructions, and represented by ring permutations of the CoxeterDynkin 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  [5,3^{1,1}] :
 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 nonWythoffian 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:
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  
Loopntail graphs  4+4+4+2 = 14 
References
 "A242941  OEIS". oeis.org. Retrieved 20190203.
 George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
 , A000029 61 cases, skipping one with zero marks
 http://bendwavy.org/klitzing/explain/polytopetree.htm#scaliform
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292298, includes all the nonprismatic forms)
 Branko Grünbaum, (1994) Uniform tilings of 3space. Geombinatorics 4, 49  56.
 Norman Johnson (1991) Uniform Polytopes, Manuscript
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Chapter 5: Polyhedra packing and space filling)
 Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0500340331.
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (1.9 Uniform spacefillings)
 A. Andreini, (1905) Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 75–129. PDF
 D. M. Y. Sommerville, (1930) An Introduction to the Geometry of n Dimensions. New York, E. P. Dutton, . 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
 Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0520030567. Chapter 5. Joining polyhedra
 Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi (2009), p.5455. 12 packings of 2 or more uniform polyhedra with cubic symmetry
External links
Wikimedia Commons has media related to Uniform tilings of Euclidean 3space. 
 Weisstein, Eric W. "Honeycomb". MathWorld.
 Uniform Honeycombs in 3Space VRML models
 Elementary Honeycombs Vertex transitive space filling honeycombs with nonuniform cells.
 Uniform partitions of 3space, their relatives and embedding, 1999
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra
 octet truss animation
 Review: A. F. Wells, Threedimensional nets and polyhedra, H. S. M. Coxeter (Source: Bull. Amer. Math. Soc. Volume 84, Number 3 (1978), 466470.)
 Klitzing, Richard. "3D Euclidean tesselations".
 (sequence A242941 in the OEIS)
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 