Convex uniform honeycomb
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:

  • the familiar cubic honeycomb and 7 truncations thereof;
  • the alternated cubic honeycomb and 4 truncations thereof;
  • 10 prismatic forms based on the uniform plane tilings (11 if including the cubic honeycomb);
  • 5 modifications of some of the above by elongation and/or gyration.

They can be considered the three-dimensional analogue to the uniform tilings of the plane.

Contents

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 polychorons 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 polychoron#Geometric derivations.)

For cross-referencing, they are given with list indices from [A]ndreini (1-22), [W]illiams(1-2,9-19), [J]ohnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and [G]runbaum(1-28).

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

Fundamental domains in a cubic element of three groups.

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 nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel 4a.pngCDel nodea.png (11 forms, 3 new)
  3. The {\tilde{A}}_3 cyclic group, [(3,3,3,3)], CDel branch.pngCDel 3ab.pngCDel branch.png (5 forms, one new)

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}}_2x{\tilde{I}}_1, [4,4]x[∞] 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}}_2x{\tilde{I}}_1, [6,3]x[∞] 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}}_2x{\tilde{I}}_1, (3 3 3)x[∞] prismatic group, CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png (No new forms)
  4. The {\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1, [∞]x[∞]x[∞] 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)

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.)


Reference
Indices
Honeycomb name
Coxeter-Dynkin
and Schläfli
symbols
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
Solids
(Partial)
Frames
(Perspective)
Vertex figure
J11,15
A1
W1
G22
cubic
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0{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
rectified cubic
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t1{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
truncated cubic
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
t0,1{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
cantellated cubic
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,2{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
obilique triangular prism
J11,15 runcinated cubic
(same as regular cubic)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,3{4,3,4}
(1)
Hexahedron.png
(4.4.4)
(3)
Hexahedron.png
(4.4.4)
(3)
Hexahedron.png
(4.4.4)
(1)
Hexahedron.png
(4.4.4)
Runcinated cubic honeycomb.png Cubic tiling.png Runcinated cubic honeycomb verf.png
octahedron
J16
A3
W2
G28
bitruncated cubic
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t1,2{4,3,4}
(2)
Truncated octahedron.png
(4.6.6)
    (2)
Truncated octahedron.png
(4.6.6)
Bitruncated cubic honeycomb.png Bitruncated cubic tiling.png Bitruncated cubic honeycomb verf.png
(disphenoid tetrahedron)
J17
A18
W13
G25
cantitruncated cubic
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t0,1,2{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
runcitruncated cubic
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
J19
A22
W18
G27
omnitruncated cubic
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
t0,1,2,3{4,3,4}
(1)
Great rhombicuboctahedron.png
(4.6.8)
(1)
Octagonal prism.png
(4.4.8)
(1)
Octagonal prism.png
(4.4.8)
(1)
Great rhombicuboctahedron.png
(4.6.8)
Omnitruncated cubic honeycomb.jpg Omnitruncated cubic tiling.png Omnitruncated cubic honeycomb verf.png
irregular tetrahedron
J21,31,51
A2
W9
G1
alternated cubic
CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
h0{4,3,4}
(6)
Octahedron.png
(3.3.3.3)
    (8)
Tetrahedron.png
(3.3.3)
Tetrahedral-octahedral honeycomb.png Alternated cubic tiling.png Alternated cubic honeycomb verf.svg
cuboctahedron

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

The {\tilde{B}}_4 group offers 11 derived forms via truncation operations, four being unique uniform honeycombs.

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.

Referenced
indices
Honeycomb name
Coxeter-Dynkin
diagram
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0)
CDel branch.pngCDel 3a.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
alternated cubic
CDel nodes 10ru.pngCDel split2.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
truncated alternated cubic
CDel 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
J12,32
A15
W14
G7
rectified cubic
(rectified alternate cubic)
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png
Cuboctahedron.png (2)
(3.4.3.4)
  Cuboctahedron.png (2)
(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
J12,32
A15
W14
G7
rectified cubic
(cantellated alternate cubic)
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png
Octahedron.png (1)
(3.3.3.3)
  Octahedron.png (1)
(3.3.3.3)
Uniform polyhedron-33-t02.png (4)
(3.4.3.4)
Rectified cubic honeycomb3.png Rectified cubic tiling.png Cantellated alternate cubic honeycomb verf.png
cuboid
J16
A3
W2
G28
bitruncated cubic
(cantitruncated alternate cubic)
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png
Truncated octahedron.png (1)
(4.6.6)
  Truncated octahedron.png (1)
(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
J13
A14
W15
G8
truncated cubic
(bicantellated alternate cubic)
CDel nodes.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Truncated hexahedron.png (2)
(3.8.8)
  Truncated hexahedron.png (2)
(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
J11,15
A1
W1
G22
cubic
(trirectified alternate cubic)
CDel nodes.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.png
Hexahedron.png (4)
(4.4.4)
  Hexahedron.png (4)
(4.4.4)
  Bicolor cubic honeycomb.png Cubic tiling.png Cubic honeycomb verf.png
octahedron
J23
A16
W11
G5
runcinated alternated cubic
CDel 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
J14
A17
W12
G9
cantellated cubic
(runcicantellated alternate cubic)
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node 1.png
Small rhombicuboctahedron.png (1)
(3.4.4.4)
Uniform polyhedron 222-t012.png (2)
(4.4.4)
Small rhombicuboctahedron.png (1)
(3.4.4.4)
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
J24
A20
W16
G21
cantitruncated alternated cubic
(or runcitruncated alternate cubic)
CDel 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.jpg Cantitruncated alternated cubic tiling.png Runcitruncated alternate cubic honeycomb verf.png
Irregular tetrahedron
J17
A18
W13
G25
cantitruncated cubic
(omnitruncated alternated cubic)
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Great rhombicuboctahedron.png (1)
(4.6.8)
Uniform polyhedron 222-t012.png (1)
(4.4.4)
Great rhombicuboctahedron.png (1)
(4.6.8)
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

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

There are 5 forms[1] constructed from the {\tilde{A}}_3 group, of which only the quarter cubic honeycomb is unique.

Referenced
indices
Honeycomb name
Coxeter-Dynkin
diagram
Cells by location
(and count around each vertex)
Solids
(Partial)
Frames
(Perspective)
vertex figure
(0)
CDel nodeb.pngCDel 3b.pngCDel branch.png
(1)
CDel branch.pngCDel 3b.pngCDel nodeb.png
(2)
CDel branch.pngCDel 3a.pngCDel nodea.png
(3)
CDel nodea.pngCDel 3a.pngCDel branch.png
J21,31,51
A2
W9
G1
alternated cubic
CDel branch 10r.pngCDel 3ab.pngCDel branch.png
  Uniform polyhedron-33-t0.png (4)
(3.3.3)
Uniform polyhedron-33-t1.png (6)
(3.3.3.3)
Uniform polyhedron-33-t2.png (4)
(3.3.3)
Tetrahedral-octahedral honeycomb2.png Alternated cubic tiling.png Alternated cubic honeycomb verf.svg
cuboctahedron
J12,32
A15
W14
G7
rectified cubic
CDel branch 10r.pngCDel 3ab.pngCDel branch 01l.png
Uniform polyhedron-33-t02.png (2)
(3.4.3.4)
Uniform polyhedron-33-t1.png (1)
(3.3.3.3)
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
J25,33
A13
W10
G6
quarter cubic
CDel branch 10r.pngCDel 3ab.pngCDel branch 10l.png
Tetrahedron.png (1)
(3.3.3)
Tetrahedron.png (1)
(3.3.3)
Truncated tetrahedron.png (3)
(3.6.6)
Truncated tetrahedron.png (3)
(3.6.6)
Quarter cubic honeycomb.png Bitruncated alternated cubic tiling.png T01 quarter cubic honeycomb verf.png
triangular antiprism
J22,34
A21
W17
G10
truncated alternated cubic
CDel branch 11.pngCDel 3ab.pngCDel branch 10l.png
Truncated tetrahedron.png (1)
(3.6.6)
Uniform polyhedron-33-t02.png (1)
(3.4.3.4)
Truncated tetrahedron.png (1)
(3.6.6)
Uniform polyhedron-33-t012.png (2)
(4.6.6)
Truncated Alternated Cubic Honeycomb.svg Truncated alternated cubic tiling.png T012 quarter cubic honeycomb verf.png
Rectangular pyramid
J16
A3
W2
G28
bitruncated cubic
CDel branch 11.pngCDel 3ab.pngCDel branch 11.png
Uniform polyhedron-33-t012.png (1)
(4.6.6)
Uniform polyhedron-33-t012.png (1)
(4.6.6)
Uniform polyhedron-33-t012.png (1)
(4.6.6)
Uniform polyhedron-33-t012.png (1)
(4.6.6)
Bitruncated cubic honeycomb2.png Bitruncated cubic tiling.png T0123 quarter cubic honeycomb verf.png
isosceles tetrahedron

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
h{4,3,4}:g gyrated alternated cubic tetrahedron (8)
octahedron (6)
Gyrated alternated cubic.jpg Gyrated alternated cubic.png VF-gyrated alternated cubic.png
triangular orthobicupola
J61
A?
G3
h{4,3,4}:ge gyroelongated alternated cubic triangular prism (6)
tetrahedron (4)
octahedron (3)
Gyroelongated alternated cubic honeycomb.png Gyroelongated alternated cubic tiling.png -
J62
A?
G4
h{4,3,4}:e elongated alternated cubic triangular prism (6)
tetrahedron (4)
octahedron (3)
Elongated alternated cubic honeycomb.png Elongated alternated cubic tiling.png VF-extended alternated cubic.png
J63
A?
G12
{3,6}:g x {∞} gyrated triangular prismatic triangular prism (12) Gyrated triangular prismatic honeycomb.png Gyrated triangular prismatic tiling.png VF-gyrated prismatic triangular.png
J64
A?
G15
{3,6}:ge x {∞} gyroelongated triangular prismatic triangular prism (6)
cube (4)
Gyroelongated triangular prismatic honeycomb.png Gyroelongated triangular prismatic tiling.png VF-prismatic extended triangular.png

Prismatic stacks

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

The C~2xI~1(∞), [4,4] x [∞], prismatic group

There's 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} x {∞}
Cubic
(Square prismatic)
(4.4.4.4) Partial cubic honeycomb.png Uniform tiling 44-t0.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
t0,1{4,4} x {∞}
Truncated/Bitruncated square prismatic (4.8.8) Truncated square prismatic honeycomb.png Uniform tiling 44-t01.png
J11,15
A1
G22
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t1{4,4} x {∞}
Cubic
(Rectified square prismatic)
(4.4.4.4) Square prismatic 2-color honeycomb.png Uniform tiling 44-t1.png
J11,15
A1
G22
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t0,2{4,4} x {∞}
Cubic
(Cantellated square prismatic)
(4.4.4.4) Partial cubic honeycomb.png Uniform tiling 44-t02.png
J45
A6
G24
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t0,1,2{4,4} x {∞}
Truncated square prismatic
(Omnitruncated square prismatic)
(4.8.8) Truncated square prismatic honeycomb.png 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
s{4,4} x {∞}
Snub square prismatic (3.3.4.3.4) Snub square prismatic honeycomb.png Uniform tiling 44-snub.png

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

Indices Coxeter-Dynkin
and Schläfli
symbols
Honeycomb name Plane
tiling
Solids
(Partial)
Tiling
J42
A5
G26
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t0{6,3} x {∞}
Hexagonal prismatic (63) Hexagonal prismatic honeycomb.png Uniform tiling 63-t0.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
t0,1{6,3} x {∞}
Truncated hexagonal prismatic (3.12.12) Truncated hexagonal prismatic honeycomb.png Uniform tiling 63-t01.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
t1{6,3} x {∞}
Trihexagonal prismatic (3.6.3.6) Triangular-hexagonal prismatic honeycomb.png Uniform tiling 63-t1.png
J42
A5
G26
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t1,2{6,3} x {∞}
Truncated triangular prismatic
Hexagonal prismatic
(6.6.6) Truncated triangular prismatic honeycomb.png Uniform tiling 63-t12.png
J41
A4
G11
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t2{6,3} x {∞}
Triangular prismatic (36) Triangular prismatic honeycomb.png Uniform tiling 63-t2.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
t0,2{6,3} x {∞}
Rhombi-trihexagonal prismatic (3.4.6.4) Rhombitriangular-hexagonal prismatic honeycomb.png Uniform tiling 63-t02.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
t0,1,2{6,3} x {∞}
Omnitruncated trihexagonal prismatic (4.6.12) Omnitruncated triangular-hexagonal prismatic honeycomb.png Uniform tiling 63-t012.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
s{6,3} x {∞}
Snub trihexagonal prismatic (3.3.3.3.6) Snub triangular-hexagonal prismatic honeycomb.png Uniform tiling 63-snub.png
J65
A11'
G13
{3,6}:e x {∞} elongated triangular prismatic (3.3.3.4.4) Elongated triangular prismatic honeycomb.png Tile 33344.svg

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.

Noncompact forms

Examples (partially drawn)
Cubic semicheck.png Tetroctahedric semicheck.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 6.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.png

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

Families:

  • {\tilde{C}}_2xA1: [4,4]x[ ] CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png Cubic prismatic slab honeycomb (3 forms)
  • {\tilde{G}}_2xA1: [6,3]x[ ] CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node.png Tri-hexagonal prismatic slab honeycomb (8 forms)
  • {\tilde{A}}_2xA1: (3 3 3)x[ ] CDel node.pngCDel split1.pngCDel branch.pngCDel 2.pngCDel node.png Triangular prismatic slab (No new forms)
  • {\tilde{I}}_1xA1xA1: [∞]x[ ]x[ ] CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node.png = CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png Cubic column honeycomb (1 form)
  • I2(p)x{\tilde{I}}_1: [p]x[∞] CDel node.pngCDel p.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.png Prismatic column honeycomb
  • {\tilde{C}}_2x{\tilde{C}}_2xA1: [∞]x[∞]x[ ] = [4,4]x[ ] - CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.png = CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node.png (Same as cubic slab honeycomb family)

Hyperbolic forms

The {5,3,4} honeycomb in 3D hyperbolic space, viewed 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 exist. One known example is in the {3,5,3} family.

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

Hyperbolic noncompact groups
7 CDel label4.pngCDel branch.pngCDel 4-3.pngCDel branch.pngCDel 2.png, CDel label4.pngCDel branch.pngCDel 4-3.pngCDel branch.pngCDel label4.png, CDel label4.pngCDel branch.pngCDel 4-4.pngCDel branch.pngCDel label4.png, CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel 2.png, CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label4.png, CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label5.png, CDel label6.pngCDel branch.pngCDel 3ab.pngCDel branch.pngCDel label6.png
7 CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.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 4.pngCDel node.png,CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png, CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
6 CDel branch.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png, CDel branch.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png, CDel branch.pngCDel split2.pngCDel node.pngCDel 5.pngCDel node.png, CDel branch.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png, CDel node.pngCDel split1.pngCDel branch.pngCDel split2.pngCDel node.png, CDel tet.png
3 CDel nodes.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png, CDel nodes.pngCDel split2-44.pngCDel node.pngCDel 3.pngCDel node.png, CDel nodes.pngCDel split2-44.pngCDel node.pngCDel 4.pngCDel node.png

References

  1. ^ [1], A000029 6-1 cases, skipping one with zero marks
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, (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)
  • 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 [6]
    • (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.
  • 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

External links


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