Convex uniform honeycomb

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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 exist:

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[edit]

  • 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 complete 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 polychora in 4-space).

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

Names[edit]

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 polychoron#Geometric derivations for 46 nonprismatic Wythoffian uniform polychora)

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)[edit]

Fundamental domains in a cubic element of three groups.
Family correspondences

The fundamental infinite Coxeter groups for 3-space are:

  1. The {\tilde{C}}_3, [4,3,4], cubic, CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png (8 unique forms plus one alternation)
  2. The {\tilde{B}}_3, [4,31,1], alternated cubic, CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png (11 forms, 3 new)
  3. The {\tilde{A}}_3 cyclic group, [(3,3,3,3)] or [3[4]], CDel branch.pngCDel 3ab.pngCDel 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, CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png (2 new forms)
  2. The {\tilde{H}}_2×{\tilde{I}}_1, [6,3,2,∞] prismatic group, CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png (7 unique forms)
  3. The {\tilde{A}}_2×{\tilde{I}}_1, [(3,3,3),2,∞] prismatic group, CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png (No new forms)
  4. The {\tilde{I}}_1×{\tilde{I}}_1×{\tilde{I}}_1, [∞,2,∞,2,∞] prismatic group, CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel 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 C~3, [4,3,4] group (cubic)[edit]

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.

Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Pm3m
(221)
4:2 [4,3,4] CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node c4.png ×1 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 1, CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 2, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 3, CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 4,
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 5, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 6
Fm3m
(225)
2:2 [1+,4,3,4]
↔ [4,31,1]
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node c2.png
CDel nodes 10ru.pngCDel split2.pngCDel node c1.pngCDel 4.pngCDel node c2.png
Half CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 7, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png 11, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 12, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png 13
I43m
(217)
4o:2 [[(4,3,4,2+)]] CDel branch.pngCDel 4a4b.pngCDel nodes hh.png Half × 2 CDel branch.pngCDel 4a4b.pngCDel nodes hh.png (7),
Fd3m
(227)
2+:2 [[1+,4,3,4,1+]]
↔ [[3[4]]]
CDel branch.pngCDel 4a4b.pngCDel nodes h1h1.png
CDel branch 11.pngCDel 3ab.pngCDel branch.png
Quarter × 2 CDel branch.pngCDel 4a4b.pngCDel nodes h1h1.png 10,
Im3m
(229)
8o:2 [[4,3,4]] CDel branch c2.pngCDel 4a4b.pngCDel nodeab c1.png ×2

CDel branch.pngCDel 4a4b.pngCDel nodes 11.png (1), CDel branch 11.pngCDel 4a4b.pngCDel nodes.png 8, CDel branch 11.pngCDel 4a4b.pngCDel nodes 11.png 9


[4,3,4], space group Pm3m (221)
Reference
Indices
Honeycomb name
Coxeter diagram
and Schläfli symbol
Cell counts/vertex
and positions in cubic honeycomb
(0)
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
(1)
CDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
(2)
CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png
(3)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Alt Solids
(Partial)
Frames
(Perspective)
Vertex figure
J11,15
A1
W1
G22
δ4
cubic (chon)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0{4,3,4}
{4,3,4}
      (8)
Hexahedron.png
(4.4.4)
  Partial cubic honeycomb.png Cubic honeycomb.png Cubic honeycomb verf.png
octahedron
J12,32
A15
W14
G7
O1
rectified cubic (rich)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t1{4,3,4}
r{4,3,4}
(2)
Octahedron.png
(3.3.3.3)
    (4)
Cuboctahedron.png
(3.4.3.4)
  Rectified cubic honeycomb.png Rectified cubic tiling.png Rectified cubic honeycomb verf.png
cuboid
J13
A14
W15
G8
t1δ4
O15
truncated cubic (tich)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,1{4,3,4}
t{4,3,4}
(1)
Octahedron.png
(3.3.3.3)
    (4)
Truncated hexahedron.png
(3.8.8)
  Truncated cubic honeycomb.png Truncated cubic tiling.png Truncated cubic honeycomb verf.png
square pyramid
J14
A17
W12
G9
t0,2δ4
O14
cantellated cubic (srich)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,2{4,3,4}
rr{4,3,4}
(1)
Cuboctahedron.png
(3.4.3.4)
(2)
Hexahedron.png
(4.4.4)
  (2)
Small rhombicuboctahedron.png
(3.4.4.4)
  Cantellated cubic honeycomb.jpg Cantellated cubic tiling.png Cantellated cubic honeycomb verf.png
oblique triangular prism
J17
A18
W13
G25
t0,1,2δ4
O17
cantitruncated cubic (grich)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1,2{4,3,4}
tr{4,3,4}
(1)
Truncated octahedron.png
(4.6.6)
(1)
Hexahedron.png
(4.4.4)
  (2)
Great rhombicuboctahedron.png
(4.6.8)
  Cantitruncated Cubic Honeycomb.svg Cantitruncated cubic tiling.png Cantitruncated cubic honeycomb verf.png
irregular tetrahedron
J18
A19
W19
G20
t0,1,3δ4
O19
runcitruncated cubic (prich)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,1,3{4,3,4}
(1)
Small rhombicuboctahedron.png
(3.4.4.4)
(1)
Hexahedron.png
(4.4.4)
(2)
Octagonal prism.png
(4.4.8)
(1)
Truncated hexahedron.png
(3.8.8)
  Runcitruncated cubic honeycomb.jpg Runcitruncated cubic tiling.png Runcitruncated cubic honeycomb verf.png
oblique trapezoidal pyramid
J21,31,51
A2
W9
G1
4
O21
alternated cubic (octet)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
h{4,3,4}
      (8)
Tetrahedron.png
(3.3.3)
(6)
Octahedron.png
(3.3.3.3)
Tetrahedral-octahedral honeycomb.png Alternated cubic tiling.png Alternated cubic honeycomb verf.svg
cuboctahedron
J22,34
A21
W17
G10
h2δ4
O25
Cantic cubic (tatoh)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png
Cuboctahedron.png (1)
(3.4.3.4)
  Truncated octahedron.png (2)
(4.6.6)
Truncated tetrahedron.png (2)
(3.6.6)
Truncated Alternated Cubic Honeycomb.svg Truncated alternated cubic tiling.png Truncated alternated cubic honeycomb verf.png
rectangular pyramid
J23
A16
W11
G5
h3δ4
O26
Runcic cubic (ratoh)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.png
Hexahedron.png (1)
cube
  Small rhombicuboctahedron.png (3)
(3.4.4.4)
Tetrahedron.png (1)
(3.3.3)
Runcinated alternated cubic honeycomb.jpg Runcinated alternated cubic tiling.png Runcinated alternated cubic honeycomb verf.png
tapered triangular prism
J24
A20
W16
G21
h2,3δ4
O28
Runcicantic cubic (gratoh)
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Truncated hexahedron.png (1)
(3.8.8)
  Great rhombicuboctahedron.png(2)
(4.6.8)
Truncated tetrahedron.png (1)
(3.6.6)
Cantitruncated alternated cubic honeycomb.png Cantitruncated alternated cubic tiling.png Runcitruncated alternate cubic honeycomb verf.png
Irregular tetrahedron
Nonuniformb snub rectified cubic
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
sr{4,3,4}
Uniform polyhedron-43-h01.svg(1)
(3.3.3.3.3)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Tetrahedron.png(1)
(3.3.3)
CDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node.png
  Snub hexahedron.png(2)
(3.3.3.3.4)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
Tetrahedron.png(4)
(3.3.3)
Alternated cantitruncated cubic honeycomb.png Alternated cantitruncated cubic honeycomb verf.png
Irr. tridiminished icosahedron
Nonuniform Trirectified bisnub cubic
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
2s0{4,3,4}
Uniform polyhedron-43-h01.svg
(3.3.3.3.3)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Hexahedron.png
(4.4.4)
CDel node 1.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node h.png
Cube rotorotational symmetry.png
(4.4.4)
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node h.png
Rhombicuboctahedron uniform edge coloring.png
(3.4.4.4)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
Nonuniform Runcic cantitruncated cubic
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
sr3{4,3,4}
Rhombicuboctahedron uniform edge coloring.png
(3.4.4.4)
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
Cube rotorotational symmetry.png
(4.4.4)
CDel node h.pngCDel 2.pngCDel node h.pngCDel 4.pngCDel node 1.png
Hexahedron.png
(4.4.4)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node 1.png
Snub hexahedron.png
(3.3.3.3.4)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
[[4,3,4]] honeycombs, space group Im3m (229)
Reference
Indices
Honeycomb name
Coxeter diagram
CDel branch c1.pngCDel 4a4b.pngCDel nodeab c2.png
and Schläfli symbol
Cell counts/vertex
and positions in cubic honeycomb
(0,3)
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
(1,2)
CDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png
Alt Solids
(Partial)
Frames
(Perspective)
Vertex figure
J11,15
A1
W1
G22
δ4
O1
runcinated cubic
(same as regular cubic) (chon)
CDel branch.pngCDel 4a4b.pngCDel nodes 11.png
t0,3{4,3,4}
(2)
Hexahedron.png
(4.4.4)
(6)
Hexahedron.png
(4.4.4)
  Runcinated cubic honeycomb.png Cubic honeycomb.png Runcinated cubic honeycomb verf.png
octahedron
J16
A3
W2
G28
t1,2δ4
O16
bitruncated cubic (batch)
CDel branch 11.pngCDel 4a4b.pngCDel nodes.png
t1,2{4,3,4}
2t{4,3,4}
(4)
Truncated octahedron.png
(4.6.6)
    Bitruncated cubic honeycomb.png Bitruncated cubic tiling.png Bitruncated cubic honeycomb verf.png
(disphenoid)
J19
A22
W18
G27
t0,1,2,3δ4
O20
omnitruncated cubic (otch)
CDel branch 11.pngCDel 4a4b.pngCDel nodes 11.png
t0,1,2,3{4,3,4}
(2)
Great rhombicuboctahedron.png
(4.6.8)
(2)
Octagonal prism.png
(4.4.8)
  Omnitruncated cubic honeycomb.jpg Omnitruncated cubic tiling.png Omnitruncated cubic honeycomb verf.png
irregular tetrahedron
J21,31,51
A2
W9
G1
4
O27
Quarter cubic honeycomb
CDel branch.pngCDel 4a4b.pngCDel nodes h1h1.png
ht0ht3{4,3,4}
(2)
Uniform polyhedron-33-t0.png
(3.3.3)
(6)
Uniform polyhedron-33-t01.png
(3.6.6)
Quarter cubic honeycomb2.png Bitruncated alternated cubic tiling.png T01 quarter cubic honeycomb verf2.png
elongated triangular antiprism
J21,31,51
A2
W9
G1
4
O21
Alternated runcinated cubic
(same as alternated cubic)
CDel branch.pngCDel 4a4b.pngCDel nodes hh.png
ht0,3{4,3,4}
(4)
Uniform polyhedron-33-t0.png
(3.3.3)
(4)
Uniform polyhedron-33-t2.png
(3.3.3)
(6)
Uniform polyhedron-33-t1.png
(3.3.3.3)
Tetrahedral-octahedral honeycomb2.png Alternated cubic tiling.png Alternated cubic honeycomb verf.svg
cuboctahedron
Nonuniform CDel branch 11.pngCDel 4a4b.pngCDel nodes hh.png
2s0,3{(4,2,4,3)}
Nonuniforma Alternated bitruncated cubic
CDel branch hh.pngCDel 4a4b.pngCDel nodes.png
h2t{4,3,4}
Uniform polyhedron-43-h01.svg (4)
(3.3.3.3.3)
  Tetrahedron.png (4)
(3.3.3)
Alternated bitruncated cubic honeycomb2.png Alternated bitruncated cubic honeycomb verf.png
Nonuniform CDel branch hh.pngCDel 4a4b.pngCDel nodes 11.png
2s0,3{4,3,4}
Nonuniformc Alternated omnitruncated cubic
CDel branch hh.pngCDel 4a4b.pngCDel nodes hh.png
ht0,1,2,3{4,3,4}
Snub hexahedron.png (2)
(3.3.3.3.4)
Square antiprism.png (2)
(3.3.3.4)
Tetrahedron.png (4)
(3.3.3)
  Snub cubic honeycomb verf.png

B~4, [4,31,1] group[edit]

The {\tilde{B}}_4, [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.

Space
group
Fibrifold Extended
symmetry
Extended
diagram
Order Honeycombs
Fm3m
(225)
2:2 [4,31,1]
↔ [4,3,4,1+]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel nodes 10lu.png
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
×1 CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 1, CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png 2, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 3, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png 4
Fm3m
(225)
2:2 <[1+,4,31,1]>
↔ <[3[4]]>
CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodeab c1.png
CDel node 1.pngCDel split1.pngCDel nodeab c1.pngCDel split2.pngCDel node.png
×2 CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png (1), CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png (3)
Pm3m
(221)
4:2 <[4,31,1]> CDel node c3.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel nodeab c1.png ×2

CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png 5, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png 6, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png 7, CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png (6), CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png 9, CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 10, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png 11

[4,31,1] uniform honeycombs, space group Fm3m (225)
Referenced
indices
Honeycomb name
Coxeter diagrams
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0)
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
(1)
CDel nodea.pngCDel 2.pngCDel nodeb.pngCDel 2.pngCDel nodea.png
(0')
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
(3)
CDel nodea.pngCDel 3a.pngCDel branch.png
J21,31,51
A2
W9
G1
4
O21
Alternated cubic (octet)
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
    Octahedron.png (6)
(3.3.3.3)
Tetrahedron.png(8)
(3.3.3)
Tetrahedral-octahedral honeycomb.png Alternated cubic tiling.png Alternated cubic honeycomb verf.svg
cuboctahedron
J22,34
A21
W17
G10
h2δ4
O25
Cantic cubic (tatoh)
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cuboctahedron.png (1)
(3.4.3.4)
  Truncated octahedron.png (2)
(4.6.6)
Truncated tetrahedron.png (2)
(3.6.6)
Truncated Alternated Cubic Honeycomb.svg Truncated alternated cubic tiling.png Truncated alternated cubic honeycomb verf.png
rectangular pyramid
J23
A16
W11
G5
h3δ4
O26
Runcic cubic (ratoh)
CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Hexahedron.png (1)
cube
  Small rhombicuboctahedron.png (3)
(3.4.4.4)
Tetrahedron.png (1)
(3.3.3)
Runcinated alternated cubic honeycomb.jpg Runcinated alternated cubic tiling.png Runcinated alternated cubic honeycomb verf.png
tapered triangular prism
J24
A20
W16
G21
h2,3δ4
O28
Runcicantic cubic (gratoh)
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Truncated hexahedron.png (1)
(3.8.8)
  Great rhombicuboctahedron.png(2)
(4.6.8)
Truncated tetrahedron.png (1)
(3.6.6)
Cantitruncated alternated cubic honeycomb.png Cantitruncated alternated cubic tiling.png Runcitruncated alternate cubic honeycomb verf.png
Irregular tetrahedron
<[4,31,1]> uniform honeycombs, space group Pm3m (221)
Referenced
indices
Honeycomb name
Coxeter diagrams
CDel nodeab c1.pngCDel split2.pngCDel node c2.pngCDel 4.pngCDel node c3.pngCDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 3.pngCDel node c3.png
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,0')
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
(1)
CDel nodea.pngCDel 2.pngCDel nodeb.pngCDel 2.pngCDel nodea.png
(3)
CDel nodea.pngCDel 3a.pngCDel branch.png
Alt
J11,15
A1
W1
G22
δ4
O1
Cubic (chon)
CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Hexahedron.png (8)
(4.4.4)
      Bicolor cubic honeycomb.png Cubic tiling.png Cubic honeycomb verf.png
octahedron
J12,32
A15
W14
G7
t1δ4
O15
Rectified cubic (rich)
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Cuboctahedron.png (4)
(3.4.3.4)
  Uniform polyhedron-33-t1.png (2)
(3.3.3.3)
  Rectified cubic honeycomb4.png Rectified cubic tiling.png Rectified alternate cubic honeycomb verf.png
cuboid
Rectified cubic (rich)
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron.png (2)
(3.3.3.3)
  Uniform polyhedron-33-t02.png (4)
(3.4.3.4)
  Rectified cubic honeycomb3.png Cantellated alternate cubic honeycomb verf.png
cuboid
J13
A14
W15
G8
t0,1δ4
O14
Truncated cubic (tich)
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Truncated hexahedron.png (4)
(3.8.8)
  Uniform polyhedron-33-t1.png (1)
(3.3.3.3)
  Truncated cubic honeycomb2.png Truncated cubic tiling.png Bicantellated alternate cubic honeycomb verf.png
square pyramid
J14
A17
W12
G9
t0,2δ4
O17
Cantellated cubic (srich)
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Small rhombicuboctahedron.png (2)
(3.4.4.4)
Uniform polyhedron 222-t012.png (2)
(4.4.4)
Uniform polyhedron-33-t02.png (1)
(3.4.3.4)
  Cantellated cubic honeycomb.jpg Cantellated cubic tiling.png Runcicantellated alternate cubic honeycomb verf.png
obilique triangular prism
J16
A3
W2
G28
t0,2δ4
O16
Bitruncated cubic (batch)
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated octahedron.png (2)
(4.6.6)
  Uniform polyhedron-33-t012.png (2)
(4.6.6)
  Bitruncated cubic honeycomb3.png Bitruncated cubic tiling.png Cantitruncated alternate cubic honeycomb verf.png
isosceles tetrahedron
J17
A18
W13
G25
t0,1,2δ4
O18
Cantitruncated cubic (grich)
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Great rhombicuboctahedron.png (2)
(4.6.8)
Uniform polyhedron 222-t012.png (1)
(4.4.4)
Uniform polyhedron-33-t012.png(1)
(4.6.6)
  Cantitruncated Cubic Honeycomb.svg Cantitruncated cubic tiling.png Omnitruncated alternated cubic honeycomb verf.png
irregular tetrahedron
J21,31,51
A2
W9
G1
4
O21
Alternated cubic (octet)
CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png
Tetrahedron.png (8)
(3.3.3)
    Octahedron.png (6)
(3.3.3.3)
Tetrahedral-octahedral honeycomb2.png Alternated cubic tiling.png Alternated cubic honeycomb verf.svg
cuboctahedron
J22,34
A21
W17
G10
h2δ4
O25
Cantic cubic (tatoh)
CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.pngCDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.png
Truncated tetrahedron.png (2)
(3.6.6)
  Cuboctahedron.png (1)
(3.4.3.4)
Truncated octahedron.png (2)
(4.6.6)
Truncated Alternated Cubic Honeycomb.svg Truncated alternated cubic tiling.png Truncated alternated cubic honeycomb verf.png
rectangular pyramid
Nonuniforma Alternated bitruncated cubic
CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 4.pngCDel node.pngCDel node h0.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
Uniform polyhedron-43-h01.svg (2)
(3.3.3.3.3)
  Uniform polyhedron-33-s012.svg (2)
(3.3.3.3.3)
Tetrahedron.png (4)
(3.3.3)
Alternated bitruncated cubic honeycomb verf.png
Nonuniformb Alternated cantitruncated cubic
CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel node h0.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
Snub hexahedron.png (2)
(3.3.3.3.4)
Tetrahedron.png (1)
(3.3.3)
Uniform polyhedron-43-h01.svg (1)
(3.3.3.3.3)
Tetrahedron.png (4)
(3.3.3)
Alternated cantitruncated cubic honeycomb.png Alternated cantitruncated cubic honeycomb verf.png
Irr. tridiminished icosahedron

A~3, [3[4])] group[edit]

There are 5 forms[1] 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.


Space
group
Fibrifold Square
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
F43m
(216)
1o:2 a1 [3[4]] CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png ×1 (None)
Fd3m
(227)
2+:2 p2 [[3[4]]] CDel branch 11.pngCDel 3ab.pngCDel branch.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
×2 CDel branch 11.pngCDel 3ab.pngCDel branch.png 3
Fm3m
(225)
2:2 d2 <[3[4]]>
↔ [4,3,31,1]
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png
CDel node.pngCDel 4.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png
×2 CDel node.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node.png 1,CDel node 1.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node 1.png 2
Pm3m
(221)
4:2 d4 [2[3[4]]]
↔ [4,3,4]
CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png
×4 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.png 4
Im3m
(229)
8o:2 r8 [4[3[4]]]
↔ [[4,3,4]]
CDel branch c1.pngCDel 3ab.pngCDel branch c1.png
CDel branch c1.pngCDel 4a4b.pngCDel nodes.png
×8 CDel branch 11.pngCDel 3ab.pngCDel branch 11.png 5, CDel branch hh.pngCDel 3ab.pngCDel branch hh.png (*)
[[3[4]]] uniform honeycombs, space group Fd3m (227)
Referenced
indices
Honeycomb name
Coxeter diagrams
CDel branch c1-2.pngCDel 3ab.pngCDel branch c1-2.png
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,1)
CDel nodeb.pngCDel 3b.pngCDel branch.png
(2,3)
CDel branch.pngCDel 3a.pngCDel nodea.png
J25,33
A13
W10
G6
4
O27
quarter cubic (batatoh)
CDel branch 10r.pngCDel 3ab.pngCDel branch 10l.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
q{4,3,4}
Tetrahedron.png (2)
(3.3.3)
Truncated tetrahedron.png (6)
(3.6.6)
Quarter cubic honeycomb.png Bitruncated alternated cubic tiling.png T01 quarter cubic honeycomb verf.png
triangular antiprism
<[3[4]]> ↔ [4,31,1] uniform honeycombs, space group Fm3m (225)
Referenced
indices
Honeycomb name
Coxeter diagrams
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel node.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel 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
4
O21
alternated cubic (octet)
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
h{4,3,4}
Uniform polyhedron-33-t0.png (8)
(3.3.3)
Uniform polyhedron-33-t1.png (6)
(3.3.3.3)
Tetrahedral-octahedral honeycomb2.png Alternated cubic tiling.png Alternated cubic honeycomb verf.svg
cuboctahedron
J22,34
A21
W17
G10
h2δ4
O25
cantic cubic (tatoh)
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
h2{4,3,4}
Truncated tetrahedron.png (2)
(3.6.6)
Uniform polyhedron-33-t02.png (1)
(3.4.3.4)
Uniform polyhedron-33-t012.png (2)
(4.6.6)
Truncated Alternated Cubic Honeycomb2.png Truncated alternated cubic tiling.png 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
CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.pngCDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,2)
CDel nodeb.pngCDel 3b.pngCDel branch.png
(1,3)
CDel branch.pngCDel 3b.pngCDel nodeb.png
J12,32
A15
W14
G7
t1δ4
O1
rectified cubic (rich)
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node 1.pngCDel nodes.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
r{4,3,4}
Uniform polyhedron-33-t02.png (2)
(3.4.3.4)
Uniform polyhedron-33-t1.png (1)
(3.3.3.3)
Rectified cubic honeycomb2.png Rectified cubic tiling.png 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
CDel node c1.pngCDel split1.pngCDel nodeab c1.pngCDel split2.pngCDel node c1.pngCDel nodeab c1.pngCDel split2.pngCDel node c1.pngCDel 4.pngCDel node h0.pngCDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node h0.png
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0,1,2,3)
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Alt
J16
A3
W2
G28
t1,2δ4
O16
bitruncated cubic (batch)
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node h0.pngCDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
2t{4,3,4}
Uniform polyhedron-33-t012.png (4)
(4.6.6)
Bitruncated cubic honeycomb2.png Bitruncated cubic tiling.png T0123 quarter cubic honeycomb verf.png
isosceles tetrahedron
Nonuniforma Alternated cantitruncated cubic
CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.pngCDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 4.pngCDel node h0.pngCDel node h0.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png
h2t{4,3,4}
Uniform polyhedron-33-s012.png (4)
(3.3.3.3.3)
Uniform polyhedron-33-t0.png (4)
(3.3.3)
  Alternated bitruncated cubic honeycomb verf.png

Nonwythoffian forms (gyrated and elongated)[edit]

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)
Gyrated alternated cubic honeycomb.png Gyrated alternated cubic.png 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)
Gyroelongated alternated cubic honeycomb.png Gyroelongated alternated cubic tiling.png 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)
Elongated alternated cubic honeycomb.png Elongated alternated cubic tiling.png
J63
A?
G12
O12
{3,6}:g × {∞} gyrated triangular prismatic (gytoph) triangular prism (12) Gyrated triangular prismatic honeycomb.png Gyrated triangular prismatic tiling.png Gyrated triangular prismatic honeycomb verf.png
J64
A?
G15
O13
{3,6}:ge × {∞} gyroelongated triangular prismatic (gyetaph) triangular prism (6)
cube (4)
Gyroelongated triangular prismatic honeycomb.png Gyroelongated triangular prismatic tiling.png Gyroelongated alternated triangular prismatic honeycomb verf.png

Prismatic stacks[edit]

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[edit]

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

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

Indices Coxeter-Dynkin
and Schläfli
symbols
Honeycomb name Plane
tiling
Solids
(Partial)
Tiling
J41
A4
G11
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
{3,6} × {∞}
Triangular prismatic (tiph) (36) Triangular prismatic honeycomb.png Uniform tiling 63-t2.png
J42
A5
G26
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
{6,3} × {∞}
Hexagonal prismatic (hiph) (63) Hexagonal prismatic honeycomb.png Uniform tiling 63-t0.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t{3,6} × {∞}
Truncated triangular prismatic honeycomb.png Uniform tiling 63-t12.png
J43
A8
G18
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
r{6,3} × {∞}
Trihexagonal prismatic (thiph) (3.6.3.6) Triangular-hexagonal prismatic honeycomb.png Uniform tiling 63-t1.png
J46
A7
G19
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t{6,3} × {∞}
Truncated hexagonal prismatic (thaph) (3.12.12) Truncated hexagonal prismatic honeycomb.png Uniform tiling 63-t01.png
J47
A9
G16
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
rr{6,3} × {∞}
Rhombi-trihexagonal prismatic (rothaph) (3.4.6.4) Rhombitriangular-hexagonal prismatic honeycomb.png Uniform tiling 63-t02.png
J48
A12
G17
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
sr{6,3} × {∞}
Snub hexagonal prismatic (snathaph) (3.3.3.3.6) Snub triangular-hexagonal prismatic honeycomb.png Uniform tiling 63-snub.png
J49
A10
G23
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
tr{6,3} × {∞}
truncated trihexagonal prismatic (otathaph) (4.6.12) Omnitruncated triangular-hexagonal prismatic honeycomb.png Uniform tiling 63-t012.png
J65
A11'
G13
{3,6}:e × {∞} elongated triangular prismatic (etoph) (3.3.3.4.4) Elongated triangular prismatic honeycomb.png Tile 33344.svg
J52
A2'
G2
CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png
h3t{3,6,2,∞}
gyrated tetrahedral-octahedral (gytoh) (36) Gyrated alternated cubic honeycomb.png Uniform tiling 63-t2.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png
s2r{3,6,2,∞}
Nonuniform CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png
ht0,1,2,3{3,6,2,∞}

Enumeration of Wythoff forms[edit]

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]
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
[4,3,4]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node c4.png
6 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png22 | CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png7 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png8
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png9 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png25 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png20
[1+,4,3+,4,1+] (2) CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png1 | CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngb
[2+[4,3,4]]
CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node c1.png = CDel node c1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
(1) CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 22 [2+[(4,3+,4,2+)]] (1) CDel branch.pngCDel 4a4b.pngCDel branch hh.pngCDel label2.png1 | CDel branch.pngCDel 4a4b.pngCDel nodes hh.png6
[2+[4,3,4]]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c1.png
1 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png28 [2+[(4,3+,4,2+)]] (1) CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pnga
[2+[4,3,4]]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node c1.png
2 CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png27 [2+[4,3,4]]+ (1) CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngc
[4,31,1]
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
[4,31,1]
CDel node c3.pngCDel 4.pngCDel node c4.pngCDel split1.pngCDel nodeab c1-2.png
4 CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png1 | CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png7 | CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png10 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 10lu.png28
[1[4,31,1]]=[4,3,4]
CDel node c1.pngCDel 4.pngCDel node c2.pngCDel split1.pngCDel nodeab c3.png = CDel node c1.pngCDel 4.pngCDel node c2.pngCDel 3.pngCDel node c3.pngCDel 4.pngCDel node h0.png
(7) CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png22 | CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png7 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png22 | CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png7 | CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png9 | CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png28 | CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png25 [1[1+,4,31,1]]+ (2) CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png1 | CDel node h1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 10lu.png6 | CDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.pnga
[1[4,31,1]]+
=[4,3,4]+
(1) CDel node h.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.pngb
[3[4]]
CDel branch.pngCDel 3ab.pngCDel branch.png
[3[4]] (none)
[2+[3[4]]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
1 CDel branch 11.pngCDel 3ab.pngCDel branch.png6
[1[3[4]]]=[4,31,1]
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png = CDel node h0.pngCDel 3.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png
(2) CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png1 | CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.png10
[2[3[4]]]=[4,3,4]
CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png = CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node h0.png
(1) CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node 1.png7
[(2+,4)[3[4]]]=[2+[4,3,4]]
CDel branch c1.pngCDel 3ab.pngCDel branch c1.png = CDel node h0.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node h0.png
(1) CDel branch 11.pngCDel 3ab.pngCDel branch 11.png28 [(2+,4)[3[4]]]+
= [2+[4,3,4]]+
(1) CDel branch hh.pngCDel 3ab.pngCDel branch hh.pnga

Examples[edit]

All 28 of these tessellations are found in crystal arrangements.[citation needed]

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). [2] [3] [4] [5]. Octet trusses are now among the most common types of truss used in construction.

Frieze forms[edit]

Examples (partially drawn)
Cubic semicheck.png Tetroctahedric semicheck.png Trihexagonal prism slab honeycomb.png
Cubic slab honeycomb
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png
Alternated hexagonal slab honeycomb
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Trihexagonal slab honeycomb
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png

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

Families:

Partial snubs can generate other slabs that are vertex-transitive, but not uniform, for example, these have pyramid and cupola cells:

CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Runcic snub 263 honeycomb.png
Up and down triangular cupola and middle octahedra
A slice of a rectified cubic honeycomb.
Runcic snub 244 honeycomb.png
Up and down Square cupola and middle tetrahedra.
A slice of a Runcic cubic honeycomb.
Alternated cubic slab honeycomb.png
Up and down square pyramids and middle tetrahedrons.
A slice of an alternated cubic honeycomb.

Hyperbolic forms[edit]

The order-4 dodecahedral honeycomb, {5,3,4} in perspective
The paracompact hexagonal tiling honeycomb, {6,3,3}, in perspective

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] : CDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png - 9 forms
  • [5,3,4] : CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png - 15 forms
  • [5,3,5] : CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png - 9 forms
  • [5,31,1] : CDel nodes.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png - 11 forms (7 overlap with [5,3,4] family, 4 are unique)
  • [(4,3,3,3)] : CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.png - 9 forms
  • [(4,3,4,3)] : CDel label4.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png - 6 forms
  • [(5,3,3,3)] : CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.png - 9 forms
  • [(5,3,4,3)] : CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png - 9 forms
  • [(5,3,5,3)] : CDel label5.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png - 6 forms

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

Paracompact hyperbolic forms[edit]

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

References[edit]

  1. ^ [1], A000029 6-1 cases, skipping one with zero marks
  • 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)
  • George Olshevsky, (2006, Uniform Panoploid Tetracombs, Manuscript (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) [6]
  • 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

External links[edit]