Open main menu
Cubic honeycomb
Cubic honeycomb.pngPartial cubic honeycomb.png
Type Regular honeycomb
Family Hypercube honeycomb
Indexing[1] J11,15, A1
W1, G22
Schläfli symbol {4,3,4}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cell type {4,3}
Face type {4}
Vertex figure Cubic honeycomb verf.png
(octahedron)
Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group , [4,3,4]
Dual self-dual
Cell: Cubic full domain.png
Properties vertex-transitive, regular

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

Contents

Cartesian coordinatesEdit

The Cartesian coordinates of the vertices are:

(i, j, k)
for all integral values: i,j,k, with edges parallel to the axes and with an edge length of 1.

Related honeycombsEdit

It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane.

It is one of 28 uniform honeycombs using convex uniform polyhedral cells.

Isometries of simple cubic latticesEdit

Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:

Crystal system Monoclinic
Triclinic
Orthorhombic Tetragonal Rhombohedral Cubic
Unit cell Parallelepiped Rectangular cuboid Square cuboid Trigonal
trapezohedron
Cube
Point group
Order
Rotation subgroup
[ ], (*)
Order 2
[ ]+, (1)
[2,2], (*222)
Order 8
[2,2]+, (222)
[4,2], (*422)
Order 16
[4,2]+, (422)
[3], (*33)
Order 6
[3]+, (33)
[4,3], (*432)
Order 48
[4,3]+, (432)
Diagram          
Space group
Rotation subgroup
Pm (6)
P1 (1)
Pmmm (47)
P222 (16)
P4/mmm (123)
P422 (89)
R3m (160)
R3 (146)
Pm3m (221)
P432 (207)
Coxeter notation - [∞]a×[∞]b×[∞]c [4,4]a×[∞]c - [4,3,4]a
Coxeter diagram -                       -        

Uniform coloringsEdit

There is a large number of uniform colorings, derived from different symmetries. These include:

Coxeter notation
Space group
Coxeter diagram Schläfli symbol Partial
honeycomb
Colors by letters
[4,3,4]
Pm3m (221)
       
         =        
{4,3,4}   1: aaaa/aaaa
[4,31,1] = [4,3,4,1+]
Fm3m (225)
      =         {4,31,1}   2: abba/baab
[4,3,4]
Pm3m (221)
        t0,3{4,3,4}   4: abbc/bccd
[[4,3,4]]
Pm3m (229)
    t0,3{4,3,4} 4: abbb/bbba
[4,3,4,2,∞]          
or         
{4,4}×t{∞}   2: aaaa/bbbb
[4,3,4,2,∞]           t1{4,4}×{∞}   2: abba/abba
[∞,2,∞,2,∞]          t{∞}×t{∞}×{∞}   4: abcd/abcd
[∞,2,∞,2,∞] = [4,(3,4)*]          =         t{∞}×t{∞}×t{∞}   8: abcd/efgh

ProjectionsEdit

The cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a triangular tiling. A square symmetry projection forms a square tiling.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid      
Frame      

Related polytopes and honeycombsEdit

It is related to the regular 4-polytope tesseract, Schläfli symbol {4,3,3}, which exists in 4-space, and only has 3 cubes around each edge. It's also related to the order-5 cubic honeycomb, Schläfli symbol {4,3,5}, of hyperbolic space with 5 cubes around each edge.

It is in a sequence of polychora and honeycomb with octahedral vertex figures.

It in a sequence of regular polytopes and honeycombs with cubic cells.

Related Euclidean tessellationsEdit

The [4,3,4],        , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.

The [4,31,1],      , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

This honeycomb is one of five distinct uniform honeycombs[2] constructed by the   Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:


Rectified cubic honeycombEdit

Rectified cubic honeycomb
Type Uniform honeycomb
Cells Octahedron  
Cuboctahedron  
Schläfli symbol r{4,3,4} or t1{4,3,4}
r{4,31,1}
2r{4,31,1}
r{3[4]}
Coxeter diagrams        
      =        
      =        
      =       =       =        
Vertex figure  
Cuboid
Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group  , [4,3,4]
Dual oblate octahedrille
Cell:  
Properties vertex-transitive, edge-transitive

The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1.

John Horton Conway calls this honeycomb a cuboctahedrille, and its dual an oblate octahedrille.

  

ProjectionsEdit

The rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid      
Frame      

SymmetryEdit

There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below.

Symmetry [4,3,4]
 
[1+,4,3,4]
[4,31,1],  
[4,3,4,1+]
[4,31,1],  
[1+,4,3,4,1+]
[3[4]],  
Space group Pm3m
(221)
Fm3m
(225)
Fm3m
(225)
F43m
(216)
Coloring        
Coxeter
diagram
                               
                 
Vertex figure        
Vertex
figure
symmetry
D4h
[4,2]
(*224)
order 16
D2h
[2,2]
(*222)
order 8
C4v
[4]
(*44)
order 8
C2v
[2]
(*22)
order 4

This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra, creating two triangular cupolae. This scaliform honeycomb is represented by Coxeter diagram        , and symbol s3{2,6,3}, with coxeter notation symmetry [2+,6,3].

 .

Truncated cubic honeycombEdit

Truncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol t{4,3,4} or t0,1{4,3,4}
t{4,31,1}
Coxeter diagrams        
      =        
Cell type 3.8.8, {3,4}
Face type {3}, {4}, {8}
Vertex figure  
Isosceles square pyramid
Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group  , [4,3,4]
Dual Pyramidille
Cell:  
Properties vertex-transitive

The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1.

John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.

  

ProjectionsEdit

The truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid      
Frame      

SymmetryEdit

There is a second uniform coloring by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells.

Construction Bicantellated alternate cubic Truncated cubic honeycomb
Coxeter group [4,31,1],   [4,3,4],  
=<[4,31,1]>
Space group Fm3m Pm3m
Coloring    
Coxeter diagram       =                
Vertex figure    

Alternated bitruncated cubic honeycombEdit

Alternated bitruncated cubic honeycomb
Type Nonuniform honeycomb
Schläfli symbol 2s{4,3,4}
2s{4,31,1}
sr{3[4]}
Coxeter diagrams        
      =        
      =        
      =        
Cells tetrahedron
icosahedron
Vertex figure  
Coxeter group [4,3,4],  
Properties vertex-transitive

The alternated bitruncated cubic honeycomb or bisnub cubic honeycomb can be creating regular icosahedron from the truncated octahedra with irregular tetrahedral cells created in the gaps. There are three constructions from three related Coxeter diagrams:        ,      , and      . These have symmetry [4,3+,4], [4,(31,1)+] and [3[4]]+ respectively. The first and last symmetry can be doubled as [[4,3+,4]] and [[3[4]]]+.

This honeycomb is represented in the boron atoms of the α-rhombihedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.[3]

Five uniform colorings
Space group I3 (204) Pm3 (200) Fm3 (202) Fd3 (203) F23 (196)
Fibrifold 8−o 4 2 2o+ 1o
Coxeter group [[4,3+,4]] [4,3+,4] [4,(31,1)+] [[3[4]]]+ [3[4]]+
Coxeter diagram                            
Order double full half quarter
double
quarter

Cantellated cubic honeycombEdit

Cantellated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol rr{4,3,4} or t0,2{4,3,4}
rr{4,31,1}
Coxeter diagram        
      =        
Cells rr{4,3}  
r{4,3}  
{4,3}  
Vertex figure  
(Wedge)
Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group [4,3,4],  
Dual quarter oblate octahedrille
Cell:  
Properties vertex-transitive

The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3.

John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.

   

ImagesEdit

   
It is closely related to the perovskite structure, shown here with cubic symmetry, with atoms placed at the center of the cells of this honeycomb.

ProjectionsEdit

The cantellated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid      
Frame      

SymmetryEdit

There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells.

Vertex uniform colorings by cell
Construction Truncated cubic honeycomb Bicantellated alternate cubic
Coxeter group [4,3,4],  
=<[4,31,1]>
[4,31,1],  
Space group Pm3m Fm3m
Coxeter diagram              
Coloring    
Vertex figure    
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1

Quarter oblate octahedrilleEdit

The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram        , containing faces from two of four hyperplanes of the cubic [4,3,4] fundamental domain.

It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, and 2 vertices.

 

Cantitruncated cubic honeycombEdit

Cantitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol tr{4,3,4} or t0,1,2{4,3,4}
tr{4,31,1}
Coxeter diagram        
      =        
Vertex figure   
(Irreg. tetrahedron)
Coxeter group [4,3,4],  
Space group
Fibrifold notation
Pm3m (221)
4:2
Dual triangular pyramidille
Cells:  
Properties vertex-transitive

The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3.

John Horton Conway calls this honeycomb a n-tCO-trille, and its dual triangular pyramidille.

   

ImagesEdit

Four cells exist around each vertex:

 

ProjectionsEdit

The cantitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid      
Frame      

SymmetryEdit

Cells can be shown in two different symmetries. The linear Coxeter diagram form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating.

Construction Cantitruncated cubic Omnitruncated alternate cubic
Coxeter group [4,3,4],  
=<[4,31,1]>
[4,31,1],  
Space group Pm3m (221) Fm3m (225)
Fibrifold 4:2 2:2
Coloring    
Coxeter diagram              
Vertex figure    
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1

Triangular pyramidilleEdit

The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram,        . This honeycomb cells represents the fundamental domains of   symmetry.

A cell can be as 1/24 of a translational cube with vertices positioned: taking two corner, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.

 

Related polyhedra and honeycombsEdit

It is related to a skew apeirohedron with vertex configuration 4.4.6.6, with the octagons and some of the squares removed. It can be seen as constructed by augmenting truncated cuboctahedral cells, or by augmenting alternated truncated octahedra and cubes.

Two views
   

Alternated cantitruncated cubic honeycombEdit

Alternated cantitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol sr{4,3,4}
sr{4,31,1}
Coxeter diagrams        
      =        
Cells tetrahedron
pseudoicosahedron
snub cube
Vertex figure  
Coxeter group [4,31,1],  
Dual        
Cell:  
Properties vertex-transitive

The alternated cantitruncated cubic honeycomb or snub rectified cubic honeycomb contains three types of cells: snub cubes, icosahedra (snub tetrahedron), and tetrahedra. In addition the gaps created at the alternated vertices form tetrahedral cells.
Although it is not uniform, constructionally it can be given as Coxeter diagrams       or        .

 
       
 
       

Runcic cantitruncated cubic honeycombEdit

Runcic cantitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol sr3{4,3,4}
Coxeter diagrams        
Cells rhombicuboctahedron
snub cube
cube
Vertex figure
Coxeter group [4,3,4],  
Dual
Properties vertex-transitive

The runcic cantitruncated cubic honeycomb or runcic cantitruncated cubic cellulation contains cells: snub cubes, rhombicuboctahedrons, and cubes. In addition the gaps created at the alternated vertices form an irregular cell.
Although it is not uniform, constructionally it can be given as Coxeter diagram        .


Runcitruncated cubic honeycombEdit

Runcitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol t0,1,3{4,3,4}
Coxeter diagrams        
Cells rhombicuboctahedron
truncated cube
octagonal prism
cube
Vertex figure  
(Trapezoidal pyramid)
Coxeter group [4,3,4],  
Space group
Fibrifold notation
Pm3m (221)
4:2
Dual square quarter pyramidille
Cell  
Properties vertex-transitive

The runcitruncated cubic honeycomb or runcitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3.

Its name is derived from its Coxeter diagram,         with three ringed nodes representing 3 active mirrors in the Wythoff construction from its relation to the regular cubic honeycomb.

John Horton Conway calls this honeycomb a 1-RCO-trille, and its dual square quarter pyramidille.

     

ProjectionsEdit

The runcitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid      
Frame      

Related skew apeirohedronEdit

A related uniform skew apeirohedron exists with the same vertex arrangement, but some of the square and all of the octagons removed. It can be seen as truncated tetrahedra and truncated cubes augmented together.

 

Square quarter pyramidilleEdit

The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram        . Faces exist in 3 of 4 hyperplanes of the [4,3,4],   Coxeter group.

Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one mid-edge point, two face centers, and the cube center.

 

Omnitruncated cubic honeycombEdit

Omnitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol t0,1,2,3{4,3,4}
Coxeter diagram        
Vertex figure  
Phyllic disphenoid
Space group
Fibrifold notation
Coxeter notation
Im3m (229)
8o:2
[[4,3,4]]
Coxeter group [4,3,4],  
Dual eighth pyramidille
Cell  
Properties vertex-transitive

The omnitruncated cubic honeycomb or omnitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cuboctahedra and octagonal prisms in a ratio of 1:3.

John Horton Conway calls this honeycomb a b-tCO-trille, and its dual eighth pyramidille.

   

ProjectionsEdit

The omnitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Orthogonal projections
Symmetry p6m (*632) p4m (*442) pmm (*2222)
Solid      
Frame      

SymmetryEdit

Cells can be shown in two different symmetries. The Coxeter diagram form has two colors of truncated cuboctahedra and octagonal prisms. The symmetry can be doubled by relating the first and last branches of the Coxeter diagram, which can be shown with one color for all the truncated cuboctahedral and octagonal prism cells.

Two uniform colorings
Symmetry  , [4,3,4]  ×2, [[4,3,4]]
Space group Pm3m (221) Im3m (229)
Fibrifold 4:2 8o:2
Coloring    
Coxeter diagram            
Vertex figure    

Related polyhedraEdit

Two related uniform skew apeirohedron exist with the same vertex arrangement. The first has octagons removed, and vertex configuration 4.4.4.6. It can be seen as truncated cuboctahedra and octagonal prisms augmented together. The second can be seen as augmented octagonal prisms.

4.4.4.6
 
4.8.4.8
 
   

Alternated omnitruncated cubic honeycombEdit

Alternated omnitruncated cubic honeycomb
Type Alternated uniform honeycomb
Schläfli symbol ht0,1,2,3{4,3,4}
Coxeter diagram        
Cells snub cube
square antiprism
tetrahedron
Vertex figure  
Symmetry [[4,3,4]]+
Dual Phyllic disphenoidal honeycomb
Properties vertex-transitive

A alternated omnitruncated cubic honeycomb orfull snub cubic honeycomb can be constructed by alternation of the omnitruncated cubic honeycomb, although it can not be made uniform, but it can be given Coxeter diagram:         and has symmetry [[4,3,4]]+. It makes snub cubes from the truncated cuboctahedra, square antiprisms from the octagonal prisms and with new tetrahedral cells created in the gaps.

Dual alternated omnitruncated cubic honeycombEdit

Dual alternated omnitruncated cubic honeycomb
Type Dual alternated uniform honeycomb
Schläfli symbol dht0,1,2,3{4,3,4}
Coxeter diagram        
Cells  
Symmetry [[4,3,4]]+
Properties Cell-transitive

A dual alternated omnitruncated cubic honeycomb is a space-filling honeycomb constructed as the dual of the alternated omnitruncated cubic honeycomb.

Cells can be seen from tetrahedra as 1/48 of a cube, augmented by new center point of adjacent tetrahedra.

 

24 cells fit around a vertex, making a chiral octahedral symmetry that can be stacked in all 3-dimensions:

 

Individual cells have 2-fold rotational symmetry. In 2D orthogonal projection, this looks like a mirror symmetry.

Cell views
 
Net
     
       

Truncated square prismatic honeycombEdit

Truncated square prismatic honeycomb
Type Uniform honeycomb
Schläfli symbol t{4,4}×{∞} or t0,1,3{4,4,2,∞}
tr{4,4}×{∞} or t0,1,2,3{4,4,∞}
Coxeter-Dynkin diagram          
         
Coxeter group [4,4,2,∞]
Dual Tetrakis square prismatic tiling
Cell:  
Properties vertex-transitive

The truncated square prismatic honeycomb or tomo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octagonal prisms and cubes in a ratio of 1:1.

 

It is constructed from a truncated square tiling extruded into prisms.

It is one of 28 convex uniform honeycombs.


Snub square prismatic honeycombEdit

Snub square prismatic honeycomb
Type Uniform honeycomb
Schläfli symbol s{4,4}×{∞}
sr{4,4}×{∞}
Coxeter-Dynkin diagram          
         
Coxeter group [4+,4,2,∞]
[(4,4)+,2,∞]
Dual Cairo pentagonal prismatic honeycomb
Cell:  
Properties vertex-transitive

The snub square prismatic honeycomb or simo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.

 

It is constructed from a snub square tiling extruded into prisms.

It is one of 28 convex uniform honeycombs.

See alsoEdit

ReferencesEdit

  1. ^ 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).
  2. ^ [1], A000029 6-1 cases, skipping one with zero marks
  3. ^ Williams, 1979, p 199, Figure 5-38.
  • 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)
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  • George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
  • Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
  • 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 [2]
    • (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, 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 (1905) 75–129.
  • Klitzing, Richard. "3D Euclidean Honeycombs x4o3o4o - chon - O1".
  • Uniform Honeycombs in 3-Space: 01-Chon
Fundamental convex regular and uniform honeycombs in dimensions 2-9
          /   /  
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21