Convex uniform honeycomb

File:Tetrahedral-octahedral honeycomb.png

The alternated cubic honeycomb is one of 28 space-filling uniform tessellations in Euclidean 3-space, composed of alternating yellow tetrahedra and red octahedra.

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:

• 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 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:

• three of the five Platonic solids (the tetrahedron, cube, and octahedron),
• six of the thirteen Archimedean solids (the ones with reflective tetrahedral or octahedral symmetry), and
• five of the infinite family of prisms (the 3-, 4-, 6-, 8-, and 12-gonal ones; the 4-gonal prism duplicates the cube).

The icosahedron, snub cube, and square antiprism appear in some alternations, but those honeycombs cannot be realised with all edges unit length.

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)

File:Coxeter-Dynkin 3-space groups.png

Fundamental domains in a cubic element of three groups.

File:Coxeter diagram affine rank4 correspondence.png

Family correspondences

The fundamental infinite Coxeter groups for 3-space are:

1. The ${\tilde{C}}_3$, [4,3,4], cubic, File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png (8 unique forms plus one alternation)
2. The ${\tilde{B}}_3$, [4,3^1,1], alternated cubic, File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png (11 forms, 3 new)
3. The ${\tilde{A}}_3$ cyclic group, [(3,3,3,3)] or [3^[4]], File:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.png (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, File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png (2 new forms)
2. The ${\tilde{G}}_2$×${\tilde{I}}_1$, [6,3,2,∞] prismatic group, File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png (7 unique forms)
3. The ${\tilde{A}}_2$×${\tilde{I}}_1$, [(3,3,3),2,∞] prismatic group, File:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png (No new forms)
4. The ${\tilde{I}}_1$×${\tilde{I}}_1$×${\tilde{I}}_1$, [∞,2,∞,2,∞] prismatic group, File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png (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 C̃₃, [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) File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	(1) File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	(2) File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png	(3) File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	Alt	Solids (Partial)
J11,15 A1 W1 G22 δ4	cubic (chon) File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png t0{4,3,4} {4,3,4}				(8) File:Hexahedron.png (4.4.4)		File:Partial cubic honeycomb.png	File:Cubic honeycomb.png	File:Cubic honeycomb verf.svg octahedron	File:Cubic full domain.png Cube, File:CDel node f1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
J12,32 A15 W14 G7 O1	rectified cubic (rich) File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png t1{4,3,4} r{4,3,4}	(2) File:Octahedron.png (3.3.3.3)			(4) File:Cuboctahedron.png (3.4.3.4)		File:Rectified cubic honeycomb.png	File:Rectified cubic tiling.png	File:Rectified cubic honeycomb verf.png cuboid	File:Cubic square bipyramid.png Square bipyramid File:CDel node f1.pngFile:CDel 2.pngFile:CDel node f1.pngFile:CDel 4.pngFile:CDel node.png
J13 A14 W15 G8 t1δ4 O15	truncated cubic (tich) File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png t0,1{4,3,4} t{4,3,4}	(1) File:Octahedron.png (3.3.3.3)			(4) File:Truncated hexahedron.png (3.8.8)		File:Truncated cubic honeycomb.png	File:Truncated cubic tiling.png	File:Truncated cubic honeycomb verf.png square pyramid	File:Cubic square pyramid.png Isosceles square pyramid
J14 A17 W12 G9 t0,2δ4 O14	cantellated cubic (srich) File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png t0,2{4,3,4} rr{4,3,4}	(1) File:Cuboctahedron.png (3.4.3.4)	(2) File:Hexahedron.png (4.4.4)		(2) File:Small rhombicuboctahedron.png (3.4.4.4)		File:Cantellated cubic honeycomb.jpg	File:Cantellated cubic tiling.png	File:Cantellated cubic honeycomb verf.png oblique triangular prism	File:Quarter oblate octahedrille cell.png Triangular bipyramid
J17 A18 W13 G25 t0,1,2δ4 O17	cantitruncated cubic (grich) File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png t0,1,2{4,3,4} tr{4,3,4}	(1) File:Truncated octahedron.png (4.6.6)	(1) File:Hexahedron.png (4.4.4)		(2) File:Great rhombicuboctahedron.png (4.6.8)		File:Cantitruncated Cubic Honeycomb.svg	File:Cantitruncated cubic tiling.png	File:Cantitruncated cubic honeycomb verf.png irregular tetrahedron	File:Triangular pyramidille cell1.png Triangular pyramidille
J18 A19 W19 G20 t0,1,3δ4 O19	runcitruncated cubic (prich) File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png t0,1,3{4,3,4}	(1) File:Small rhombicuboctahedron.png (3.4.4.4)	(1) File:Hexahedron.png (4.4.4)	(2) File:Octagonal prism.png (4.4.8)	(1) File:Truncated hexahedron.png (3.8.8)		File:Runcitruncated cubic honeycomb.jpg	File:Runcitruncated cubic tiling.png	File:Runcitruncated cubic honeycomb verf.png oblique trapezoidal pyramid	File:Square quarter pyramidille cell.png Square quarter pyramidille
J21,31,51 A2 W9 G1 hδ4 O21	alternated cubic (octet) File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png h{4,3,4}				(8) File:Tetrahedron.png (3.3.3)	(6) File:Octahedron.png (3.3.3.3)	File:Tetrahedral-octahedral honeycomb.png	File:Alternated cubic tiling.png	File:Alternated cubic honeycomb verf.svg cuboctahedron	File:Dodecahedrille cell.png Dodecahedrille
J22,34 A21 W17 G10 h2δ4 O25	Cantic cubic (tatoh) File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	(1) File:Cuboctahedron.png(3.4.3.4)			(2) File:Truncated tetrahedron.png(3.6.6)	(2) File:Truncated octahedron.png(4.6.6)	File:Truncated Alternated Cubic Honeycomb.svg	File:Truncated alternated cubic tiling.png	File:Truncated alternated cubic honeycomb verf.png rectangular pyramid	File:Half oblate octahedrille cell.png Half oblate octahedrille
J23 A16 W11 G5 h3δ4 O26	Runcic cubic (sratoh) File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	(1) File:Hexahedron.png (4.4.4)			(1) File:Tetrahedron.png (3.3.3)	(3) File:Small rhombicuboctahedron.png (3.4.4.4)	File:Runcinated alternated cubic honeycomb.jpg	File:Runcinated alternated cubic tiling.png	File:Runcinated alternated cubic honeycomb verf.png tapered triangular prism	File:Quarter cubille cell.png Quarter cubille
J24 A20 W16 G21 h2,3δ4 O28	Runcicantic cubic (gratoh) File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png	(1) File:Truncated hexahedron.png (3.8.8)			(1) File:Truncated tetrahedron.png (3.6.6)	(2) File:Great rhombicuboctahedron.png (4.6.8)	File:Cantitruncated alternated cubic honeycomb.png	File:Cantitruncated alternated cubic tiling.png	File:Runcitruncated alternate cubic honeycomb verf.png Irregular tetrahedron	File:Half pyramidille cell.png Half pyramidille
Nonuniformb	snub rectified cubic (serch) File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png sr{4,3,4}	(1) File:Uniform polyhedron-43-h01.svg (3.3.3.3.3) File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png	(1) File:Tetrahedron.png (3.3.3) File:CDel node h.pngFile:CDel 2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png		(2) File:Snub hexahedron.png (3.3.3.3.4) File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png	(4) File:Tetrahedron.png (3.3.3)	File:Alternated cantitruncated cubic honeycomb.png		File:Alternated cantitruncated cubic honeycomb verf.png Irr. tridiminished icosahedron
Nonuniform	Cantic snub cubic (casch) File:CDel node 1.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png 2s0{4,3,4}	(1) File:Uniform polyhedron-43-h01.svg (3.3.3.3.3) File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png			(2) File:Rhombicuboctahedron uniform edge coloring.png (3.4.4.4) File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.png	(3) File:Triangular prism.png (3.4.4)
Nonuniform	Runcicantic snub cubic (rusch) File:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h.png	(1) File:Cuboctahedron.png (3.4.3.4)	(2) File:Cube rotorotational symmetry.png (4.4.4)	(1) File:Tetrahedron.png (3.3.3)	(1) File:Truncated tetrahedron.png (3.6.6)	(3) File:Triangular cupola.png Tricup
Nonuniform	Runcic cantitruncated cubic (esch) File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.png sr3{4,3,4}	(1) File:Snub hexahedron.png (3.3.3.3.4) File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png	(1) File:Tetragonal prism.png (4.4.4) File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 2.pngFile:CDel node 1.png	(1) File:Cube rotorotational symmetry.png (4.4.4) File:CDel node h.pngFile:CDel 2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.png	(1) File:Rhombicuboctahedron uniform edge coloring.png (3.4.4.4) File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.png	(3) File:Triangular prism.png (3.4.4)

[[4,3,4]] honeycombs, space group Im3m (229)

Reference Indices	Honeycomb name Coxeter diagram File:CDel branch c1.pngFile:CDel 4a4b.pngFile:CDel nodeab c2.png and Schläfli symbol	Cell counts/vertex and positions in cubic honeycomb			Solids (Partial)	Frames (Perspective)	Vertex figure	Dual cell
		(0,3) File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	(1,2) File:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png	Alt
J11,15 A1 W1 G22 δ4 O1	runcinated cubic (same as regular cubic) (chon) File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel nodes 11.png t0,3{4,3,4}	(2) File:Hexahedron.png (4.4.4)	(6) File:Hexahedron.png (4.4.4)		File:Runcinated cubic honeycomb.png	File:Cubic honeycomb.png	File:Runcinated cubic honeycomb verf.png octahedron	File:Cubic full domain.png Cube
J16 A3 W2 G28 t1,2δ4 O16	bitruncated cubic (batch) File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel nodes.png t1,2{4,3,4} 2t{4,3,4}	(4) File:Truncated octahedron.png (4.6.6)			File:Bitruncated cubic honeycomb.png	File:Bitruncated cubic tiling.png	File:Bitruncated cubic honeycomb verf.png (disphenoid)	File:Oblate tetrahedrille cell.png Oblate tetrahedrille
J19 A22 W18 G27 t0,1,2,3δ4 O20	omnitruncated cubic (gippich) File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel nodes 11.png t0,1,2,3{4,3,4}	(2) File:Great rhombicuboctahedron.png (4.6.8)	(2) File:Octagonal prism.png (4.4.8)		File:Omnitruncated cubic honeycomb.jpg	File:Omnitruncated cubic tiling.png	File:Omnitruncated cubic honeycomb verf.png irregular tetrahedron	File:Fundamental tetrahedron1.png Eighth pyramidille
J21,31,51 A2 W9 G1 hδ4 O27	Quarter cubic honeycomb (cytatoh) File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel nodes h1h1.png ht0ht3{4,3,4}	(2) File:Uniform polyhedron-33-t0.png (3.3.3)	(6) File:Uniform polyhedron-33-t01.png (3.6.6)		File:Quarter cubic honeycomb2.png	File:Bitruncated alternated cubic tiling.png	File:T01 quarter cubic honeycomb verf2.png elongated triangular antiprism	File:Oblate cubille cell.png Oblate cubille
J21,31,51 A2 W9 G1 hδ4 O21	Alternated runcinated cubic (octet) (same as alternated cubic) File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel nodes hh.png ht0,3{4,3,4}	(2) File:Uniform polyhedron-33-t0.png (3.3.3)	(6) File:Uniform polyhedron-33-t2.png (3.3.3)	(6) File:Uniform polyhedron-33-t1.svg (3.3.3.3)	File:Tetrahedral-octahedral honeycomb2.png	File:Alternated cubic tiling.png	File:Alternated cubic honeycomb verf.svg cuboctahedron
Nonuniform	Biorthosnub cubic honeycomb (gabreth) File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel nodes hh.png 2s0,3{(4,2,4,3)}	(2) File:Truncated octahedron.png (4.6.6)	(2) File:Cube rotorotational symmetry.png (4.4.4)	(2) File:Cantic snub hexagonal hosohedron2.png (4.4.6)
Nonuniforma	Alternated bitruncated cubic (bisch) File:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes.png h2t{4,3,4}	File:Uniform polyhedron-43-h01.svg (4) (3.3.3.3.3)		File:Tetrahedron.png (4) (3.3.3)	File:Alternated bitruncated cubic honeycomb2.png		File:Alternated bitruncated cubic honeycomb verf.png	File:Ten-of-diamonds decahedron in cube.png
Nonuniform	Cantic bisnub cubic (cabisch) File:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes 11.png 2s0,3{4,3,4}	(2) File:Rhombicuboctahedron uniform edge coloring.png (3.4.4.4)	(2) File:Tetragonal prism.png (4.4.4)	(2) File:Cube rotorotational symmetry.png (4.4.4)
Nonuniformc	Alternated omnitruncated cubic (snich) File:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes hh.png ht0,1,2,3{4,3,4}	(2) File:Snub hexahedron.png (3.3.3.3.4)	(2) File:Square antiprism.png (3.3.3.4)	(4) File:Tetrahedron.png (3.3.3)			File:Snub cubic honeycomb verf.png

B̃₃, [4,3^1,1] 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⁺,4,3^1,1], [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.

[4,3^1,1] 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) File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 4a.pngFile:CDel nodea.png	(1) File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodeb.pngFile:CDel 2.pngFile:CDel nodea.png	(0') File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 4a.pngFile:CDel nodea.png	(3) File:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.png
J21,31,51 A2 W9 G1 hδ4 O21	Alternated cubic (octet) File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png			File:Octahedron.png (6) (3.3.3.3)	File:Tetrahedron.png(8) (3.3.3)	File:Tetrahedral-octahedral honeycomb.png	File:Alternated cubic tiling.png	File:Alternated cubic honeycomb verf.svg cuboctahedron
J22,34 A21 W17 G10 h2δ4 O25	Cantic cubic (tatoh) File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png	File:Cuboctahedron.png (1) (3.4.3.4)		File:Truncated octahedron.png (2) (4.6.6)	File:Truncated tetrahedron.png (2) (3.6.6)	File:Truncated Alternated Cubic Honeycomb.svg	File:Truncated alternated cubic tiling.png	File:Truncated alternated cubic honeycomb verf.png rectangular pyramid
J23 A16 W11 G5 h3δ4 O26	Runcic cubic (sratoh) File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	File:Hexahedron.png (1) cube		File:Small rhombicuboctahedron.png (3) (3.4.4.4)	File:Tetrahedron.png (1) (3.3.3)	File:Runcinated alternated cubic honeycomb.jpg	File:Runcinated alternated cubic tiling.png	File:Runcinated alternated cubic honeycomb verf.png tapered triangular prism
J24 A20 W16 G21 h2,3δ4 O28	Runcicantic cubic (gratoh) File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:Truncated hexahedron.png (1) (3.8.8)		File:Great rhombicuboctahedron.png(2) (4.6.8)	File:Truncated tetrahedron.png (1) (3.6.6)	File:Cantitruncated alternated cubic honeycomb.png	File:Cantitruncated alternated cubic tiling.png	File:Runcitruncated alternate cubic honeycomb verf.png Irregular tetrahedron

<[4,3^1,1]> uniform honeycombs, space group Pm3m (221)

Referenced indices	Honeycomb name Coxeter diagrams File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node c3.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node c3.png	Cells by location (and count around each vertex)				Solids (Partial)	Frames (Perspective)	vertex figure
		(0,0') File:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 4a.pngFile:CDel nodea.png	(1) File:CDel nodea.pngFile:CDel 2.pngFile:CDel nodeb.pngFile:CDel 2.pngFile:CDel nodea.png	(3) File:CDel nodea.pngFile:CDel 3a.pngFile:CDel branch.png	Alt
J11,15 A1 W1 G22 δ4 O1	Cubic (chon) File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:Hexahedron.png (8) (4.4.4)				File:Bicolor cubic honeycomb.png	File:Cubic tiling.png	File:Cubic honeycomb verf.svg octahedron
J12,32 A15 W14 G7 t1δ4 O15	Rectified cubic (rich) File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	File:Cuboctahedron.png (4) (3.4.3.4)		File:Uniform polyhedron-33-t1.svg (2) (3.3.3.3)		File:Rectified cubic honeycomb4.png	File:Rectified cubic tiling.png	File:Rectified alternate cubic honeycomb verf.png cuboid
	Rectified cubic (rich) File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	File:Octahedron.png (2) (3.3.3.3)		File:Uniform polyhedron-33-t02.png (4) (3.4.3.4)		File:Rectified cubic honeycomb3.png		File:Cantellated alternate cubic honeycomb verf.png cuboid
J13 A14 W15 G8 t0,1δ4 O14	Truncated cubic (tich) File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png	File:Truncated hexahedron.png (4) (3.8.8)		File:Uniform polyhedron-33-t1.svg (1) (3.3.3.3)		File:Truncated cubic honeycomb2.png	File:Truncated cubic tiling.png	File:Bicantellated alternate cubic honeycomb verf.png square pyramid
J14 A17 W12 G9 t0,2δ4 O17	Cantellated cubic (srich) File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	File:Small rhombicuboctahedron.png (2) (3.4.4.4)	File:Uniform polyhedron 222-t012.png (2) (4.4.4)	File:Uniform polyhedron-33-t02.png (1) (3.4.3.4)		File:Cantellated cubic honeycomb.jpg	File:Cantellated cubic tiling.png	File:Runcicantellated alternate cubic honeycomb verf.png obilique triangular prism
J16 A3 W2 G28 t0,2δ4 O16	Bitruncated cubic (batch) File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	File:Truncated octahedron.png (2) (4.6.6)		File:Uniform polyhedron-33-t012.png (2) (4.6.6)		File:Bitruncated cubic honeycomb3.png	File:Bitruncated cubic tiling.png	File:Cantitruncated alternate cubic honeycomb verf.png isosceles tetrahedron
J17 A18 W13 G25 t0,1,2δ4 O18	Cantitruncated cubic (grich) File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png	File:Great rhombicuboctahedron.png (2) (4.6.8)	File:Uniform polyhedron 222-t012.png (1) (4.4.4)	File:Uniform polyhedron-33-t012.png(1) (4.6.6)		File:Cantitruncated Cubic Honeycomb.svg	File:Cantitruncated cubic tiling.png	File:Omnitruncated alternated cubic honeycomb verf.png irregular tetrahedron
J21,31,51 A2 W9 G1 hδ4 O21	Alternated cubic (octet) File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png ↔ File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png	File:Tetrahedron.png (8) (3.3.3)			File:Octahedron.png (6) (3.3.3.3)	File:Tetrahedral-octahedral honeycomb2.png	File:Alternated cubic tiling.png	File:Alternated cubic honeycomb verf.svg cuboctahedron
J22,34 A21 W17 G10 h2δ4 O25	Cantic cubic (tatoh) File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png ↔ File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.png	File:Truncated tetrahedron.png (2) (3.6.6)		File:Cuboctahedron.png (1) (3.4.3.4)	File:Truncated octahedron.png (2) (4.6.6)	File:Truncated Alternated Cubic Honeycomb.svg	File:Truncated alternated cubic tiling.png	File:Truncated alternated cubic honeycomb verf.png rectangular pyramid
Nonuniforma	Alternated bitruncated cubic (bisch) File:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png	File:Uniform polyhedron-43-h01.svg (2) (3.3.3.3.3)		File:Uniform polyhedron-33-s012.svg (2) (3.3.3.3.3)	File:Tetrahedron.png (4) (3.3.3)			File:Alternated bitruncated cubic honeycomb verf.png
Nonuniformb	Alternated cantitruncated cubic (serch) File:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h.png	File:Snub hexahedron.png (2) (3.3.3.3.4)	File:Tetrahedron.png (1) (3.3.3)	File:Uniform polyhedron-43-h01.svg (1) (3.3.3.3.3)	File:Tetrahedron.png (4) (3.3.3)	File:Alternated cantitruncated cubic honeycomb.png		File:Alternated cantitruncated cubic honeycomb verf.png Irr. tridiminished icosahedron

Ã₃, [3^[4]] group

There are 5 forms constructed from the ${\tilde{A}}_3$, [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.

[[3^[4]]] uniform honeycombs, space group Fd3m (227)

Referenced indices	Honeycomb name Coxeter diagrams File:CDel branch c1-2.pngFile:CDel 3ab.pngFile:CDel branch c1-2.png	Cells by location (and count around each vertex)		Solids (Partial)	Frames (Perspective)	vertex figure
		(0,1) File:CDel nodeb.pngFile:CDel 3b.pngFile:CDel branch.png	(2,3) File:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.png
J25,33 A13 W10 G6 qδ4 O27	quarter cubic (cytatoh) File:CDel branch 10r.pngFile:CDel 3ab.pngFile:CDel branch 10l.png ↔ File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png q{4,3,4}	File:Tetrahedron.png (2) (3.3.3)	File:Truncated tetrahedron.png (6) (3.6.6)	File:Quarter cubic honeycomb.png	File:Bitruncated alternated cubic tiling.png	File:T01 quarter cubic honeycomb verf.png triangular antiprism

<[3^[4]]> ↔ [4,3^1,1] uniform honeycombs, space group Fm3m (225)

Referenced indices	Honeycomb name Coxeter diagrams File:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.png ↔ File:CDel node.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c1-2.png	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) File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png h{4,3,4}		File:Uniform polyhedron-33-t0.png (8) (3.3.3)	File:Uniform polyhedron-33-t1.svg (6) (3.3.3.3)	File:Tetrahedral-octahedral honeycomb2.png	File:Alternated cubic tiling.png	File:Alternated cubic honeycomb verf.svg cuboctahedron
J22,34 A21 W17 G10 h2δ4 O25	cantic cubic (tatoh) File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.png ↔ File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png h2{4,3,4}	File:Truncated tetrahedron.png (2) (3.6.6)	File:Uniform polyhedron-33-t02.png (1) (3.4.3.4)	File:Uniform polyhedron-33-t012.png (2) (4.6.6)	File:Truncated Alternated Cubic Honeycomb2.png	File:Truncated alternated cubic tiling.png	File:T012 quarter cubic honeycomb verf.png Rectangular pyramid

[2[3^[4]]] ↔ [4,3,4] uniform honeycombs, space group Pm3m (221)

Referenced indices	Honeycomb name Coxeter diagrams File:CDel node c1.pngFile:CDel split1.pngFile:CDel nodeab c2.pngFile:CDel split2.pngFile:CDel node c1.png ↔ File:CDel node.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node.png	Cells by location (and count around each vertex)		Solids (Partial)	Frames (Perspective)	vertex figure
		(0,2) File:CDel nodeb.pngFile:CDel 3b.pngFile:CDel branch.png	(1,3) File:CDel branch.pngFile:CDel 3b.pngFile:CDel nodeb.png
J12,32 A15 W14 G7 t1δ4 O1	rectified cubic (rich) File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.png ↔ File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png ↔ File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png r{4,3,4}	File:Uniform polyhedron-33-t02.png (2) (3.4.3.4)	File:Uniform polyhedron-33-t1.svg (1) (3.3.3.3)	File:Rectified cubic honeycomb2.png	File:Rectified cubic tiling.png	File:T02 quarter cubic honeycomb verf.png cuboid

[4[3^[4]]] ↔ [[4,3,4]] uniform honeycombs, space group Im3m (229)

Referenced indices	Honeycomb name Coxeter diagrams File:CDel node c1.pngFile:CDel split1.pngFile:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c1.png ↔ File:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node h0.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node h0.png	Cells by location (and count around each vertex)		Solids (Partial)	Frames (Perspective)	vertex figure
		(0,1,2,3) File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	Alt
J16 A3 W2 G28 t1,2δ4 O16	bitruncated cubic (batch) File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.png ↔ File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node h0.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node h0.png 2t{4,3,4}	File:Uniform polyhedron-33-t012.png (4) (4.6.6)		File:Bitruncated cubic honeycomb2.png	File:Bitruncated cubic tiling.png	File:T0123 quarter cubic honeycomb verf.png isosceles tetrahedron
Nonuniforma	Alternated cantitruncated cubic (bisch) File:CDel node h.pngFile:CDel split1.pngFile:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.png ↔ File:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h0.png ↔ File:CDel node h0.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h0.png h2t{4,3,4}	File:Uniform polyhedron-33-s012.png (4) (3.3.3.3.3)	File:Uniform polyhedron-33-t0.png (4) (3.3.3)			File:Alternated bitruncated cubic honeycomb verf.png

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

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 C̃₂×Ĩ₁(∞), [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	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png {4,4}×{∞}	Cubic (Square prismatic) (chon)	(4.4.4.4)	File:Partial cubic honeycomb.png	File:Uniform tiling 44-t0.svg
	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png r{4,4}×{∞}				File:Uniform tiling 44-t1.png
	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png rr{4,4}×{∞}				File:Uniform tiling 44-t02.svg
J45 A6 G24	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png t{4,4}×{∞}	Truncated/Bitruncated square prismatic (tassiph)	(4.8.8)	File:Truncated square prismatic honeycomb.png	File:Uniform tiling 44-t01.png
	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png tr{4,4}×{∞}				File:Uniform tiling 44-t012.png
J44 A11 G14	File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png sr{4,4}×{∞}	Snub square prismatic (sassiph)	(3.3.4.3.4)	File:Snub square prismatic honeycomb.png	File:Uniform tiling 44-snub.png
Nonuniform	File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel infin.pngFile:CDel node.png ht0,1,2,3{4,4,2,∞}

The G̃₂xĨ₁(∞), [6,3,2,∞] prismatic group

Indices	Coxeter-Dynkin and Schläfli symbols	Honeycomb name	Plane tiling	Solids (Partial)	Tiling
J41 A4 G11	File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png {3,6} × {∞}	Triangular prismatic (tiph)	(36)	File:Triangular prismatic honeycomb.png	File:Uniform tiling 63-t2.png
J42 A5 G26	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png {6,3} × {∞}	Hexagonal prismatic (hiph)	(63)	File:Hexagonal prismatic honeycomb.png	File:Uniform tiling 63-t0.svg
	File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png t{3,6} × {∞}			File:Truncated triangular prismatic honeycomb.png	File:Uniform tiling 63-t12.svg
J43 A8 G18	File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png r{6,3} × {∞}	Trihexagonal prismatic (thiph)	(3.6.3.6)	File:Triangular-hexagonal prismatic honeycomb.png	File:Uniform tiling 63-t1.png
J46 A7 G19	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png t{6,3} × {∞}	Truncated hexagonal prismatic (thaph)	(3.12.12)	File:Truncated hexagonal prismatic honeycomb.png	File:Uniform tiling 63-t01.png
J47 A9 G16	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png rr{6,3} × {∞}	Rhombi-trihexagonal prismatic (srothaph)	(3.4.6.4)	File:Rhombitriangular-hexagonal prismatic honeycomb.png	File:Uniform tiling 63-t02.png
J48 A12 G17	File:CDel node h.pngFile:CDel 6.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png sr{6,3} × {∞}	Snub hexagonal prismatic (snathaph)	(3.3.3.3.6)	File:Snub triangular-hexagonal prismatic honeycomb.png	File:Uniform tiling 63-snub.png
J49 A10 G23	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png tr{6,3} × {∞}	truncated trihexagonal prismatic (grothaph)	(4.6.12)	File:Omnitruncated triangular-hexagonal prismatic honeycomb.png	File:Uniform tiling 63-t012.svg
J65 A11' G13	File:CDel node.pngFile:CDel infin.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png {3,6}:e × {∞}	elongated triangular prismatic (etoph)	(3.3.3.4.4)	File:Elongated triangular prismatic honeycomb.png	File:Tile 33344.svg
J52 A2' G2	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel infin.pngFile:CDel node.png h3t{3,6,2,∞}	gyrated tetrahedral-octahedral (gytoh)	(36)	File:Gyrated alternated cubic honeycomb.png	File:Uniform tiling 63-t2.png
	File:CDel node.pngFile:CDel 6.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel infin.pngFile:CDel node.png s2r{3,6,2,∞}
Nonuniform	File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel infin.pngFile:CDel node.png 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] File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	[4,3,4] File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel 4.pngFile:CDel node c4.png	6	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png22 | File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png7 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png8 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png9 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png25 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png20	[1+,4,3+,4,1+]	(2)	File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png1 | File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngb
	[2+[4,3,4]] File:CDel node c1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node c1.png = File:CDel node c1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	(1)	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png 22	[2+[(4,3+,4,2+)]]	(1)	File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel branch hh.pngFile:CDel label2.png1 | File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel nodes hh.png6
	[2+[4,3,4]] File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node c1.png	1	File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png28	[2+[(4,3+,4,2+)]]	(1)	File:CDel node.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pnga
	[2+[4,3,4]] File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node c1.png	2	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png27	[2+[4,3,4]]+	(1)	File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngc
[4,31,1] File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	[4,31,1] File:CDel node c3.pngFile:CDel 4.pngFile:CDel node c4.pngFile:CDel split1.pngFile:CDel nodeab c1-2.png	4	File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png1 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png7 | File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10lu.png10 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10lu.png28
	[1[4,31,1]]=[4,3,4] File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel split1.pngFile:CDel nodeab c3.png = File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel 4.pngFile:CDel node h0.png	(7)	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png22 | File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png7 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png22 | File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png7 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png9 | File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.png28 | File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.png25	[1[1+,4,31,1]]+	(2)	File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png1 | File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png6 | File:CDel node.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel split1.pngFile:CDel nodes hh.pnga
				[1[4,31,1]]+ =[4,3,4]+	(1)	File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel split1.pngFile:CDel nodes hh.pngb
[3[4]] File:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.png	[3[4]]	(none)
	[2+[3[4]]] File:CDel branch c1.pngFile:CDel 3ab.pngFile:CDel branch c2.png	1	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch.png6
	[1[3[4]]]=[4,31,1] File:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.png = File:CDel node h0.pngFile:CDel 3.pngFile:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c1-2.png	(2)	File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png1 | File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.png10
	[2[3[4]]]=[4,3,4] File:CDel node c1.pngFile:CDel split1.pngFile:CDel nodeab c2.pngFile:CDel split2.pngFile:CDel node c1.png = File:CDel node h0.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node h0.png	(1)	File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.png7
	[(2+,4)[3[4]]]=[2+[4,3,4]] File:CDel branch c1.pngFile:CDel 3ab.pngFile:CDel branch c1.png = File:CDel node h0.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node h0.png	(1)	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 11.png28	[(2+,4)[3[4]]]+ = [2+[4,3,4]]+	(1)	File:CDel branch hh.pngFile:CDel 3ab.pngFile:CDel branch hh.pnga

Examples

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.

Frieze forms

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

• ${\tilde{C}}_2$×$A_1$: [4,4,2] File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png Cubic slab honeycombs (3 forms)
• ${\tilde{G}}_2$×$A_1$: [6,3,2] File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png Tri-hexagonal slab honeycombs (8 forms)
• ${\tilde{A}}_2$×$A_1$: [(3,3,3),2] File:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel 2.pngFile:CDel node.png Triangular slab honeycombs (No new forms)
• ${\tilde{I}}_1$×$A_1$×$A_1$: [∞,2,2] File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png = File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png Cubic column honeycombs (1 form)
• $I_2(p)$×${\tilde{I}}_1$: [p,2,∞] File:CDel node.pngFile:CDel p.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.png Polygonal column honeycombs (analogous to duoprisms: these look like a single infinite tower of p-gonal prisms, with the remaining space filled with apeirogonal prisms)
• ${\tilde{I}}_1$×${\tilde{I}}_1$×$A_1$: [∞,2,∞,2] = [4,4,2] - File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png = File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png (Same as cubic slab honeycomb family)

Examples (partially drawn)

Cubic slab honeycomb File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.png	Alternated hexagonal slab honeycomb File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	Trihexagonal slab honeycomb File:CDel node.pngFile:CDel 6.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.png
File:Cubic semicheck.png	File:Tetroctahedric semicheck.png	File:Trihexagonal prism slab honeycomb.png
File:X4o4o2ox vertex figure.png (4) 43: cube (1) 44: square tiling	File:O6x3o2x vertex figure.png (4) 33: tetrahedron (3) 34: octahedron (1) 36: triangular tiling	File:O3o6s2s vertex figure.png (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.

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}, File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	s3{2,4,4}, File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png	s{2,4,4}, File:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	3s4{4,4,2,∞}, File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 2x.pngFile:CDel node h.pngFile:CDel infin.pngFile:CDel node 1.png
File:Runcic snub 263 honeycomb.png	File:Runcic snub 244 honeycomb.png	File:Alternated cubic slab honeycomb.png	File:Elongated square antiprismatic celluation.png
File:Triangular cupola.png File:Octahedron.png File:Uniform polyhedron-63-t1-1.svg	File:Square cupola.png File:Tetrahedron.png File:Uniform tiling 44-t01.png	File:Square pyramid.png File:Tetrahedron.png File:Uniform tiling 44-t0.svg	File:Square pyramid.png File:Tetrahedron.png File:Hexahedron.png
File:S2s6o3x vertex figure.png (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	File:S2s4o4x vertex figure.png (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	File:O4o4s2s vertex figure.png (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	File:O4o4s2six vertex figure.png (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

File:Hyperbolic orthogonal dodecahedral honeycomb.png

The order-4 dodecahedral honeycomb, {5,3,4} in perspective

File:Hyperbolic 3d hexagonal tiling.png

The paracompact hexagonal tiling honeycomb, {6,3,3}, in perspective

Main article: Uniform honeycombs in hyperbolic space

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] : File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png - 9 forms
• [5,3,4] : File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png - 15 forms
• [5,3,5] : File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png - 9 forms
• [5,3^1,1] : File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png - 11 forms (7 overlap with [5,3,4] family, 4 are unique)
• [(4,3,3,3)] : File:CDel label4.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.png - 9 forms
• [(4,3,4,3)] : File:CDel label4.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png - 6 forms
• [(5,3,3,3)] : File:CDel label5.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.png - 9 forms
• [(5,3,4,3)] : File:CDel label5.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png - 9 forms
• [(5,3,5,3)] : File:CDel label5.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label5.png - 6 forms

Several non-Wythoffian forms outside the list of 76 are known; it is not known how many there are.

Paracompact hyperbolic forms

Main article: Paracompact uniform honeycombs

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	File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png | File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png | File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png | File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png | File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png | File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png | File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png	4×15+6+8+8 = 82
Tridental graphs	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1-44.pngFile:CDel nodes.png | File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png | File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1-44.pngFile:CDel nodes.png	4+4+0 = 8
Cyclic graphs	File:CDel label6.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel 2.png | File:CDel label6.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label4.png | File:CDel label4.pngFile:CDel branch.pngFile:CDel 4-4.pngFile:CDel branch.png | File:CDel label6.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label5.png | File:CDel label6.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel branch.pngFile:CDel label6.png | File:CDel label4.pngFile:CDel branch.pngFile:CDel 4-4.pngFile:CDel branch.pngFile:CDel label4.png | File:CDel node.pngFile:CDel split1-44.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png | File:CDel node.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel split2.pngFile:CDel node.png | File:CDel branch.pngFile:CDel splitcross.pngFile:CDel branch.png	4×9+5+1+4+1+0 = 47
Loop-n-tail graphs	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.png | File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.png | File:CDel node.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.png | File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel branch.png	4+4+4+2 = 14

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292–298, includes all the nonprismatic forms)
• Branko Grünbaum, (1994) Uniform tilings of 3-space. 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 0-486-23729-X. (Chapter 5: Polyhedra packing and space filling)
• Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0-500-34033-1.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [7]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
• 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 [8]
• 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 0-520-03056-7. Chapter 5. Joining polyhedra
• Crystallography of Quasicrystals: Concepts, Methods and Structures by Walter Steurer, Sofia Deloudi (2009), p. 54-55. 12 packings of 2 or more uniform polyhedra with cubic symmetry

External links

• Weisstein, Eric W. "Honeycomb". MathWorld.
• Uniform Honeycombs in 3-Space VRML models
• Elementary Honeycombs Vertex transitive space filling honeycombs with non-uniform cells.
• Uniform partitions of 3-space, their relatives and embedding, 1999
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra
• octet truss animation
• Review: A. F. Wells, Three-dimensional nets and polyhedra, H. S. M. Coxeter (Source: Bull. Amer. Math. Soc. Volume 84, Number 3 (1978), 466-470.)
• Klitzing, Richard. "3D Euclidean tesselations".
• (sequence A242941 in the OEIS)

Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde{A}}_{n-1}$	${\tilde{C}}_{n-1}$	${\tilde{B}}_{n-1}$	${\tilde{D}}_{n-1}$	${\tilde{G}}_2$ / ${\tilde{F}}_4$ / ${\tilde{E}}_{n-1}$
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