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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 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

  • 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 4-polytopes in 4-space).

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

Names

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

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

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

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

File:Coxeter-Dynkin 3-space groups.png
Fundamental domains in a cubic element of three groups.

The fundamental infinite Coxeter groups for 3-space are:

  1. The <math>{\tilde{C}}_3</math>, [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 <math>{\tilde{B}}_3</math>, [4,31,1], alternated cubic, File:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png (11 forms, 3 new)
  3. The <math>{\tilde{A}}_3</math> 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 <math>{\tilde{C}}_3</math> produces <math>{\tilde{B}}_3</math>, and removing one mirror from <math>{\tilde{B}}_3</math> produces <math>{\tilde{A}}_3</math>. 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 <math>{\tilde{C}}_2</math>×<math>{\tilde{I}}_1</math>, [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 <math>{\tilde{H}}_2</math>×<math>{\tilde{I}}_1</math>, [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 <math>{\tilde{A}}_2</math>×<math>{\tilde{I}}_1</math>, [(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 <math>{\tilde{I}}_1</math>×<math>{\tilde{I}}_1</math>×<math>{\tilde{I}}_1</math>, [∞,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 C~3, [4,3,4] group (cubic)

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

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

File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel nodes 11.png (1), File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel nodes.png 8, File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel 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)
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)
Frames
(Perspective)
Vertex figure
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)
30px
(4.4.4)
  75px 75px 75px
octahedron
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)
30px
(3.3.3.3)
    (4)
30px
(3.4.3.4)
  75px 75px 75px
cuboid
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)
30px
(3.3.3.3)
    (4)
30px
(3.8.8)
  75px 75px 75px
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)
30px
(3.4.3.4)
(2)
30px
(4.4.4)
  (2)
30px
(3.4.4.4)
  75px 75px 75px
oblique triangular prism
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)
30px
(4.6.6)
(1)
30px
(4.4.4)
  (2)
30px
(4.6.8)
  75px 75px 75px
irregular tetrahedron
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)
30px
(3.4.4.4)
(1)
30px
(4.4.4)
(2)
30px
(4.4.8)
(1)
30px
(3.8.8)
  75px 75px 75px
oblique trapezoidal pyramid
J21,31,51
A2
W9
G1
4
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)
30px
(3.3.3)
(6)
30px
(3.3.3.3)
76px 75px 75px
cuboctahedron
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 3.pngFile:CDel node.pngFile:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png
30px (1)
(3.4.3.4)
  30px (2)
(4.6.6)
30px (2)
(3.6.6)
75px 75px 60px
rectangular pyramid
J23
A16
W11
G5
h3δ4
O26
Runcic cubic (ratoh)
File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png
30px (1)
cube
  30px (3)
(3.4.4.4)
30px (1)
(3.3.3)
75px 75px 60px
tapered triangular prism
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 3.pngFile:CDel node 1.pngFile:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
30px (1)
(3.8.8)
  30px(2)
(4.6.8)
30px (1)
(3.6.6)
75px 75px 60px
Irregular tetrahedron
Nonuniformb snub rectified cubic
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}
30px(1)
(3.3.3.3.3)
File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png
30px(1)
(3.3.3)
File:CDel node h.pngFile:CDel 2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png
  30px(2)
(3.3.3.3.4)
File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
30px(4)
(3.3.3)
75px 75px
Irr. tridiminished icosahedron
Nonuniform Trirectified bisnub cubic
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}
30px
(3.3.3.3.3)
File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png
30px
(4.4.4)
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h.png
30px
(4.4.4)
File:CDel node 1.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 2.pngFile:CDel node h.png
30px
(3.4.4.4)
File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.png
Nonuniform Runcic cantitruncated cubic
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}
30px
(3.4.4.4)
File:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.png
30px
(4.4.4)
File:CDel node h.pngFile:CDel 2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node 1.png
30px
(4.4.4)
File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 2.pngFile:CDel node 1.png
30px
(3.3.3.3.4)
File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png
[<span/>[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
(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 Solids
(Partial)
Frames
(Perspective)
Vertex figure
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)
30px
(4.4.4)
(6)
30px
(4.4.4)
  75px 75px 75px
octahedron
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)
30px
(4.6.6)
    75px 75px 75px
(disphenoid)
J19
A22
W18
G27
t0,1,2,3δ4
O20
omnitruncated cubic (otch)
File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel nodes 11.png
t0,1,2,3{4,3,4}
(2)
30px
(4.6.8)
(2)
30px
(4.4.8)
  75px 75px 75px
irregular tetrahedron
J21,31,51
A2
W9
G1
4
O27
Quarter cubic honeycomb
File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel nodes h1h1.png
ht0ht3{4,3,4}
(2)
30px
(3.3.3)
(6)
30px
(3.6.6)
76px 75px 75px
elongated triangular antiprism
J21,31,51
A2
W9
G1
4
O21
Alternated runcinated cubic
(same as alternated cubic)
File:CDel branch.pngFile:CDel 4a4b.pngFile:CDel nodes hh.png
ht0,3{4,3,4}
(4)
30px
(3.3.3)
(4)
30px
(3.3.3)
(6)
30px
(3.3.3.3)
76px 75px 75px
cuboctahedron
Nonuniform File:CDel branch 11.pngFile:CDel 4a4b.pngFile:CDel nodes hh.png
2s0,3{(4,2,4,3)}
Nonuniforma Alternated bitruncated cubic
File:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes.png
h2t{4,3,4}
30px (4)
(3.3.3.3.3)
  30px (4)
(3.3.3)
75px 75px
Nonuniform File:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes 11.png
2s0,3{4,3,4}
Nonuniformc Alternated omnitruncated cubic
File:CDel branch hh.pngFile:CDel 4a4b.pngFile:CDel nodes hh.png
ht0,1,2,3{4,3,4}
30px (2)
(3.3.3.3.4)
30px (2)
(3.3.3.4)
30px (4)
(3.3.3)
  75px

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

The <math>{\tilde{B}}_4</math>, [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+]
File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel split1.pngFile:CDel nodes 10lu.png
File:CDel node c1.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h1.png
×1 File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png 1, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png 2, File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10lu.png 3, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10lu.png 4
Fm3m
(225)
2:2 <[1+,4,31,1]>
↔ <[3[4]]>
File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodeab c1.png
File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node.png
×2 File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png (1), File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png (3)
Pm3m
(221)
4:2 <[4,31,1]> File:CDel node c3.pngFile:CDel 4.pngFile:CDel node c2.pngFile:CDel split1.pngFile:CDel nodeab c1.png ×2

File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png 5, File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png 6, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png 7, File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png (6), File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.png 9, File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.png 10, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel 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)
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
4
O21
Alternated cubic (octet)
File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
    30px (6)
(3.3.3.3)
30px(8)
(3.3.3)
76px 75px 60px
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.pngFile:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
30px (1)
(3.4.3.4)
  30px (2)
(4.6.6)
30px (2)
(3.6.6)
75px 75px 60px
rectangular pyramid
J23
A16
W11
G5
h3δ4
O26
Runcic cubic (ratoh)
File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png
30px (1)
cube
  30px (3)
(3.4.4.4)
30px (1)
(3.3.3)
75px 75px 60px
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.pngFile:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
30px (1)
(3.8.8)
  30px(2)
(4.6.8)
30px (1)
(3.6.6)
75px 75px 60px
Irregular tetrahedron
<[4,31,1]> 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.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 3.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.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
30px (8)
(4.4.4)
      75px 75px 60px
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.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png
30px (4)
(3.4.3.4)
  30px (2)
(3.3.3.3)
  75px 75px 60px
cuboid
Rectified cubic (rich)
File:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png
30px (2)
(3.3.3.3)
  30px (4)
(3.4.3.4)
  75px 60px
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.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png
30px (4)
(3.8.8)
  30px (1)
(3.3.3.3)
  75px 75px 60px
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.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png
30px (2)
(3.4.4.4)
30px (2)
(4.4.4)
30px (1)
(3.4.3.4)
  75px 75px 60px
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.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png
30px (2)
(4.6.6)
  30px (2)
(4.6.6)
  75px 75px 60px
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.pngFile: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
30px (2)
(4.6.8)
30px (1)
(4.4.4)
30px(1)
(4.6.6)
  75px 75px 60px
irregular tetrahedron
J21,31,51
A2
W9
G1
4
O21
Alternated cubic (octet)
File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png
30px (8)
(3.3.3)
    30px (6)
(3.3.3.3)
75px 75px 60px
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.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.png
30px (2)
(3.6.6)
  30px (1)
(3.4.3.4)
30px (2)
(4.6.6)
75px 75px 60px
rectangular pyramid
Nonuniforma Alternated bitruncated cubic
File:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel node h0.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node.png
30px (2)
(3.3.3.3.3)
  30px (2)
(3.3.3.3.3)
30px (4)
(3.3.3)
60px
Nonuniformb Alternated cantitruncated cubic
File:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile: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
30px (2)
(3.3.3.3.4)
30px (1)
(3.3.3)
30px (1)
(3.3.3.3.3)
30px (4)
(3.3.3)
75px 60px
Irr. tridiminished icosahedron

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

There are 5 forms[1] constructed from the <math>{\tilde{A}}_3</math>, [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]] File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.png ×1 (None)
Fd3m
(227)
2+:2 p2 [[3[4]]] File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch.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
×2 File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch.png 3
Fm3m
(225)
2:2 d2 <[3[4]]>
↔ [4,3,31,1]
File:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c1-2.pngFile:CDel split2.pngFile:CDel node c3.png
File:CDel node.pngFile:CDel 4.pngFile:CDel node c3.pngFile:CDel split1.pngFile:CDel nodeab c1-2.png
×2 File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10luru.pngFile:CDel split2.pngFile:CDel node.png 1,File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 10luru.pngFile:CDel split2.pngFile:CDel node 1.png 2
Pm3m
(221)
4:2 d4 [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.pngFile:CDel 4.pngFile:CDel node c1.pngFile:CDel 3.pngFile:CDel node c2.pngFile:CDel 4.pngFile:CDel node.png
×4 File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node.png 4
Im3m
(229)
8o:2 r8 [4[3[4]]]
↔ [<span/>[4,3,4]]
File:CDel branch c1.pngFile:CDel 3ab.pngFile:CDel branch c1.png
File:CDel branch c1.pngFile:CDel 4a4b.pngFile:CDel nodes.png
×8 File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel branch 11.png 5, File:CDel branch hh.pngFile:CDel 3ab.pngFile:CDel branch hh.png (*)
[<span/>[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
4
O27
quarter cubic (batatoh)
File:CDel branch 10r.pngFile:CDel 3ab.pngFile:CDel branch 10l.pngFile: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}
30px (2)
(3.3.3)
30px (6)
(3.6.6)
75px 75px 75px
triangular antiprism
<[3[4]]> ↔ [4,31,1] 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.pngFile: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
4
O21
alternated cubic (octet)
File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile: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}
30px (8)
(3.3.3)
30px (6)
(3.3.3.3)
75px 75px 75px
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.pngFile:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile: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}
30px (2)
(3.6.6)
30px (1)
(3.4.3.4)
30px (2)
(4.6.6)
75px 75px 75px
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.pngFile: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.pngFile:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile: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}
30px (2)
(3.4.3.4)
30px (1)
(3.3.3.3)
75px 75px 75px
cuboid
[4[3[4]]] ↔ [<span/>[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.pngFile:CDel nodeab c1.pngFile:CDel split2.pngFile:CDel node c1.pngFile:CDel 4.pngFile:CDel node h0.pngFile: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.pngFile:CDel nodes 11.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node h0.pngFile: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}
30px (4)
(4.6.6)
75px 75px 75px
isosceles tetrahedron
Nonuniforma Alternated cantitruncated cubic
File:CDel node h.pngFile:CDel split1.pngFile:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel nodes hh.pngFile:CDel split2.pngFile:CDel node h.pngFile:CDel 4.pngFile:CDel node h0.pngFile: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}
30px (4)
(3.3.3.3.3)
30px (4)
(3.3.3)
  75px

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)
70px 100px 80px
triangular orthobicupola
J61
A?
G3
O24
h{4,3,4}:ge gyroelongated alternated cubic (gyetoh) triangular prism (6)
tetrahedron (4)
octahedron (3)
70px 100px 80px
J62
A?
G4
O23
h{4,3,4}:e elongated alternated cubic (etoh) triangular prism (6)
tetrahedron (4)
octahedron (3)
70px 80px
J63
A?
G12
O12
{3,6}:g × {∞} gyrated triangular prismatic (gytoph) triangular prism (12) 70px 100px 80px
J64
A?
G15
O13
{3,6}:ge × {∞} gyroelongated triangular prismatic (gyetaph) triangular prism (6)
cube (4)
70px 100px 80px

Prismatic stacks

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

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

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

Indices Coxeter-Dynkin
and Schläfli
symbols
Honeycomb name Plane
tiling
Solids
(Partial)
Tiling
J11,15
A1
G22
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) 80px 50px
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}×{∞}
50px
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}×{∞}
50px
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) 80px 50px
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}×{∞}
50px
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) 80px 50px
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 G~2xI~1(∞), [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) 60px 60px
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) 60px 60px
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} × {∞}
60px 60px
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) 60px 60px
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) 60px 60px
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 (rothaph) (3.4.6.4) 60px 60px
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) 60px 60px
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 (otathaph) (4.6.12) 60px 60px
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) 60px 60px
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) 60px 60px
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

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

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

Families:

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
180px 180px 180px
180px
(4) 43: cube
(1) 44: square tiling
180px
(4) 33: tetrahedron
(3) 34: octahedron
(1) 36: hexagonal tiling
180px
(2) 3.4.4: triangular prism
(2) 4.4.6: hexagonal prism
(1) (3.6)2: trihexagonal tiling

Scaliform honeycomb

A scaliform honeycomb is vertex-transitive, like a uniform honeycomb, with regular polygon faces while cells and higher elements are only required to be orbiforms, equilateral, with their vertices lying on hyperspheres. For 3D honeycombs, this allows a subset of Johnson solids along with the uniform polyhedra. Some scaliforms can be generated by an alternation process, leaving, for example, pyramid and cupola gaps.[2]

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
200px 200px 200px 200px
40px 40px 40px 40px 40px 40px 40px 40px 40px 40px 40px 40px
200px
(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
200px
(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
200px
(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
200px
(1) 3.3.3.3: square pyramid
(4) 3.3.4: square pyramid
(4) 3.3.3: tetrahedron
(4) 4.4.4: cube

Hyperbolic forms

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

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

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

Paracompact hyperbolic forms

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

Simplectic hyperbolic paracompact group summary
Type Coxeter groups Unique honeycomb count
Linear graphs 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

  1. ^ [1], A000029 6-1 cases, skipping one with zero marks
  2. ^ http://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform
  • 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
  • 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