Cubic honeycomb























































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
C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}, [4,3,4]
Dual
self-dual
Cell: Cubic full domain.png
Properties
vertex-transitive, regular

The cubic honeycomb or cubic cellulation is the only 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






  • 1 Cartesian coordinates


  • 2 Related honeycombs


  • 3 Isometries of simple cubic lattices


  • 4 Uniform colorings


    • 4.1 Projections




  • 5 Related polytopes and honeycombs


  • 6 Related Euclidean tessellations


    • 6.1 Rectified cubic honeycomb


      • 6.1.1 Projections


      • 6.1.2 Symmetry




    • 6.2 Truncated cubic honeycomb


      • 6.2.1 Projections


      • 6.2.2 Symmetry




    • 6.3 Alternated bitruncated cubic honeycomb


    • 6.4 Cantellated cubic honeycomb


      • 6.4.1 Images


      • 6.4.2 Projections


      • 6.4.3 Symmetry


      • 6.4.4 Quarter oblate octahedrille




    • 6.5 Cantitruncated cubic honeycomb


      • 6.5.1 Images


      • 6.5.2 Projections


      • 6.5.3 Symmetry


      • 6.5.4 Triangular pyramidille


      • 6.5.5 Related polyhedra and honeycombs




    • 6.6 Alternated cantitruncated cubic honeycomb


    • 6.7 Runcic cantitruncated cubic honeycomb


    • 6.8 Runcitruncated cubic honeycomb


      • 6.8.1 Projections


      • 6.8.2 Related skew apeirohedron


      • 6.8.3 Square quarter pyramidille




    • 6.9 Omnitruncated cubic honeycomb


      • 6.9.1 Projections


      • 6.9.2 Symmetry


      • 6.9.3 Related polyhedra




    • 6.10 Alternated omnitruncated cubic honeycomb


      • 6.10.1 Dual alternated omnitruncated cubic honeycomb




    • 6.11 Truncated square prismatic honeycomb


    • 6.12 Snub square prismatic honeycomb




  • 7 See also


  • 8 References





Cartesian coordinates





Simple cubic


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 honeycombs


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 lattices


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

Monoclinic.svg

Orthorhombic.svg

Tetragonal.svg

Rhombohedral.svg

Cubic.svg

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
-

CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png

CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
-

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png


Uniform colorings


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)

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png = CDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4.pngCDel node.png
{4,3,4}

Partial cubic honeycomb.png
1: aaaa/aaaa
[4,31,1] = [4,3,4,1+]
Fm3m (225)

CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
{4,31,1}

Bicolor cubic honeycomb.png
2: abba/baab
[4,3,4]
Pm3m (221)

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
t0,3{4,3,4}

Runcinated cubic honeycomb.png
4: abbc/bccd
[[4,3,4]]
Pm3m (229)

CDel branch.pngCDel 4a4b.pngCDel nodes 11.png
t0,3{4,3,4}

4: abbb/bbba
[4,3,4,2,∞]

CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png
or CDel labelinfin.pngCDel branch 11.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
{4,4}×t{∞}

Square prismatic honeycomb.png
2: aaaa/bbbb
[4,3,4,2,∞]

CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
t1{4,4}×{∞}

Square prismatic 2-color honeycomb.png
2: abba/abba
[∞,2,∞,2,∞]

CDel labelinfin.pngCDel branch 11.pngCDel 2.pngCDel labelinfin.pngCDel branch 11.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
t{∞}×t{∞}×{∞}

Square 4-color prismatic honeycomb.png
4: abcd/abcd
[∞,2,∞,2,∞] = [4,(3,4)*]

CDel labelinfin.pngCDel branch 11.pngCDel 2.pngCDel labelinfin.pngCDel branch 11.pngCDel 2.pngCDel labelinfin.pngCDel branch 11.png = CDel node 1.pngCDel 4.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel 4g.pngCDel node g.png
t{∞}×t{∞}×t{∞}

Cubic 8-color honeycomb.png
8: abcd/efgh


Projections


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

Cubic honeycomb-2.png

Cubic honeycomb-1.png

Cubic honeycomb-3.png
Frame

Cubic honeycomb-2b.png

Cubic honeycomb-1b.png

Cubic honeycomb-3b.png


Related polytopes and honeycombs


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 tessellations


The [4,3,4], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, 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], CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png, 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 A~3{displaystyle {tilde {A}}_{3}}{tilde {A}}_{3} Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:






































































Rectified cubic honeycomb








































Rectified cubic honeycomb
Type
Uniform honeycomb
Cells
Octahedron Octahedron.svg
Cuboctahedron Cuboctahedron.svg
Schläfli symbol r{4,3,4} or t1{4,3,4}
r{4,31,1}
2r{4,31,1}
r{3[4]}
Coxeter diagrams
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png = CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
CDel node.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png = CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node 1.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png = CDel node h0.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
Vertex figure
Rectified cubic honeycomb verf.png
Cuboid

Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group
C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}, [4,3,4]
Dual
oblate octahedrille
Cell: Cubic square bipyramid.png
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.


Rectified cubic tiling.pngHC A3-P3.png



Projections


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

Rectified cubic honeycomb-2.png

Rectified cubic honeycomb-1.png

Rectified cubic honeycomb-3.png
Frame

Rectified cubic honeycomb-2b.png

Rectified cubic honeycomb-1b.png

Rectified cubic honeycomb-3b.png


Symmetry


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]
C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
[1+,4,3,4]
[4,31,1], B~3{displaystyle {tilde {B}}_{3}}{tilde {B}}_{3}
[4,3,4,1+]
[4,31,1], B~3{displaystyle {tilde {B}}_{3}}{tilde {B}}_{3}
[1+,4,3,4,1+]
[3[4]], A~3{displaystyle {tilde {A}}_{3}}{tilde {A}}_{3}
Space group Pm3m
(221)
Fm3m
(225)
Fm3m
(225)
F43m
(216)
Coloring

Rectified cubic honeycomb.png

Rectified cubic honeycomb4.png

Rectified cubic honeycomb3.png

Rectified cubic honeycomb2.png

Coxeter
diagram

CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png

CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png

CDel node.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png

CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png

CDel node 1.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node 1.png

Vertex figure

Rectified cubic honeycomb verf.png

Rectified alternate cubic honeycomb verf.png

Cantellated alternate cubic honeycomb verf.png

T02 quarter cubic honeycomb verf.png
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 CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png, and symbol s3{2,6,3}, with coxeter notation symmetry [2+,6,3].



Runcic snub 263 honeycomb.png.





Truncated cubic honeycomb












































Truncated cubic honeycomb
Type
Uniform honeycomb
Schläfli symbol t{4,3,4} or t0,1{4,3,4}
t{4,31,1}
Coxeter diagrams
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png
Cell type
3.8.8, {3,4}
Face type
{3}, {4}, {8}
Vertex figure
Truncated cubic honeycomb verf.png
Isosceles square pyramid

Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group
C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}, [4,3,4]
Dual
Pyramidille
Cell: Cubic square pyramid.png
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.


Truncated cubic tiling.pngHC A2-P3.png



Projections


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

Truncated cubic honeycomb-2.png

Truncated cubic honeycomb-1.png

Truncated cubic honeycomb-3.png
Frame

Truncated cubic honeycomb-2b.png

Truncated cubic honeycomb-1b.png

Truncated cubic honeycomb-3b.png


Symmetry


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], B~3{displaystyle {tilde {B}}_{3}}{tilde {B}}_{3}
[4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
=<[4,31,1]>
Space group Fm3m Pm3m
Coloring

Truncated cubic honeycomb2.png

Truncated cubic honeycomb.png

Coxeter diagram

CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes.png = CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h0.png

CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

Vertex figure

Bicantellated alternate cubic honeycomb verf.png

Truncated cubic honeycomb verf.png





Alternated bitruncated cubic honeycomb
































Alternated bitruncated cubic honeycomb
Type
Nonuniform honeycomb
Schläfli symbol 2s{4,3,4}
2s{4,31,1}
sr{3[4]}
Coxeter diagrams
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png = CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png
CDel nodes hh.pngCDel split2.pngCDel node h.pngCDel 4.pngCDel node.png = CDel node h0.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.png = CDel node h0.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png
Cells
tetrahedron
icosahedron
Vertex figure
Alternated bitruncated cubic honeycomb verf.png
Coxeter group [4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
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: CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png, CDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png, and CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.png. 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

CDel branch hh.pngCDel 4a4b.pngCDel nodes.png

CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png

CDel node.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png

CDel branch hh.pngCDel 3ab.pngCDel branch hh.png

CDel node h.pngCDel split1.pngCDel nodes hh.pngCDel split2.pngCDel node h.png
Order
double
full
half
quarter
double
quarter





Cantellated cubic honeycomb








































Cantellated cubic honeycomb
Type
Uniform honeycomb
Schläfli symbol rr{4,3,4} or t0,2{4,3,4}
rr{4,31,1}
Coxeter diagram
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png = CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
Cells
rr{4,3} Uniform polyhedron-43-t02.png
r{4,3} Uniform polyhedron-43-t1.png
{4,3} Uniform polyhedron-43-t0.png
Vertex figure
Cantellated cubic honeycomb verf.png
(Wedge)

Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group [4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
Dual
quarter oblate octahedrille
Cell: Quarter oblate octahedrille cell.png
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.



Cantellated cubic tiling.pngHC A5-A3-P2.png


Images






Cantellated cubic honeycomb.png

Perovskite.jpg
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.


Projections


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

Cantellated cubic honeycomb-2.png

Cantellated cubic honeycomb-1.png

Cantellated cubic honeycomb-3.png
Frame

Cantellated cubic honeycomb-2b.png

Cantellated cubic honeycomb-1b.png

Cantellated cubic honeycomb-3b.png


Symmetry


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], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
=<[4,31,1]>
[4,31,1], B~3{displaystyle {tilde {B}}_{3}}{tilde {B}}_{3}
Space group Pm3m Fm3m

Coxeter diagram

CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png

CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes 11.png
Coloring

Cantellated cubic honeycomb.png

Cantellated cubic honeycomb2.png

Vertex figure

Cantellated cubic honeycomb verf.png

Runcicantellated alternate cubic honeycomb verf.png
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1





Quarter oblate octahedrille


The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png, 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.


Quarter oblate octahedrille cell.png


Cantitruncated cubic honeycomb




































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
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png = CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png
Vertex figure
Cantitruncated cubic honeycomb verf.pngOmnitruncated alternated cubic honeycomb verf.png
(Irreg. tetrahedron)
Coxeter group [4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}

Space group
Fibrifold notation
Pm3m (221)
4:2
Dual
triangular pyramidille
Cells: Triangular pyramidille cell1.png
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.



Cantitruncated cubic tiling.png HC A6-A4-P2.png


Images


Four cells exist around each vertex:


2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png


Projections


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

Cantitruncated cubic honeycomb-2.png

Cantitruncated cubic honeycomb-1.png

Cantitruncated cubic honeycomb-3.png
Frame

Cantitruncated cubic honeycomb-2b.png

Cantitruncated cubic honeycomb-1b.png

Cantitruncated cubic honeycomb-3b.png


Symmetry


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], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
=<[4,31,1]>
[4,31,1], B~3{displaystyle {tilde {B}}_{3}}{tilde {B}}_{3}
Space group Pm3m (221) Fm3m (225)
Fibrifold 4:2 2:2
Coloring

Cantitruncated Cubic Honeycomb.svg

Cantitruncated Cubic Honeycomb2.svg

Coxeter diagram

CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png

CDel node 1.pngCDel 4.pngCDel node 1.pngCDel split1.pngCDel nodes 11.png

Vertex figure

Cantitruncated cubic honeycomb verf.png

Omnitruncated alternated cubic honeycomb verf.png
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1


Triangular pyramidille


The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png. This honeycomb cells represents the fundamental domains of B~3{displaystyle {tilde {B}}_{3}}{tilde {B}}_{3} 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.


Triangular pyramidille cell1.png


Related polyhedra and honeycombs


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

Cantitruncated cubic honeycomb apeirohedron 4466.png

Omnitruncated cubic honeycomb apeirohedron 4466.png


Alternated cantitruncated cubic honeycomb




































Alternated cantitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol sr{4,3,4}
sr{4,31,1}
Coxeter diagrams
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png = CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png
Cells
tetrahedron
pseudoicosahedron
snub cube
Vertex figure
Alternated cantitruncated cubic honeycomb verf.png
Coxeter group [4,31,1], B~3{displaystyle {tilde {B}}_{3}}{tilde {B}}_{3}
Dual
CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
Cell: Althalfcell-frame.png
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 CDel node h.pngCDel 4.pngCDel node h.pngCDel split1.pngCDel nodes hh.png or CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png.






Alternated cantitruncated cubic honeycomb.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png

Althalfcell-honeycomb-cube3x3x3.png
CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png





Runcic cantitruncated cubic honeycomb




































Runcic cantitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol sr3{4,3,4}
Coxeter diagrams
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png
Cells
rhombicuboctahedron
snub cube
cube
Vertex figure
Coxeter group [4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
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 CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node 1.png.






Runcitruncated cubic honeycomb








































Runcitruncated cubic honeycomb
Type
Uniform honeycomb
Schläfli symbol t0,1,3{4,3,4}
Coxeter diagrams
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Cells
rhombicuboctahedron
truncated cube
octagonal prism
cube
Vertex figure
Runcitruncated cubic honeycomb verf.png
(Trapezoidal pyramid)
Coxeter group [4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}

Space group
Fibrifold notation
Pm3m (221)
4:2
Dual
square quarter pyramidille
Cell Square quarter pyramidille cell.png
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, CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png 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.


Runcitruncated cubic tiling.png HC A5-A2-P2-Pr8.png Runcitruncated cubic honeycomb.jpg



Projections


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

Runcitruncated cubic honeycomb-2.png

Runcitruncated cubic honeycomb-1.png

Runcitruncated cubic honeycomb-3.png
Frame

Runcitruncated cubic honeycomb-2b.png

Runcitruncated cubic honeycomb-1b.png

Runcitruncated cubic honeycomb-3b.png


Related skew apeirohedron


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.


Skew polyhedron 34444.png


Square quarter pyramidille


The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node f1.png. Faces exist in 3 of 4 hyperplanes of the [4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3} 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.


Square quarter pyramidille cell.png


Omnitruncated cubic honeycomb




































Omnitruncated cubic honeycomb
Type
Uniform honeycomb
Schläfli symbol t0,1,2,3{4,3,4}
Coxeter diagram
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Vertex figure
Omnitruncated cubic honeycomb verf.png
Phyllic disphenoid

Space group
Fibrifold notation
Coxeter notation

Im3m (229)
8o:2
[[4,3,4]]
Coxeter group [4,3,4], C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}
Dual
eighth pyramidille
Cell Fundamental tetrahedron1.png
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.



Omnitruncated cubic tiling.png HC A6-Pr8.png


Projections


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

Omnitruncated cubic honeycomb-2.png

Omnitruncated cubic honeycomb-1.png

Omnitruncated cubic honeycomb-3.png
Frame

Omnitruncated cubic honeycomb-2b.png

Omnitruncated cubic honeycomb-1b.png

Omnitruncated cubic honeycomb-3b.png


Symmetry


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

C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}, [4,3,4]

C~3{displaystyle {tilde {C}}_{3}}{tilde {C}}_{3}×2, [[4,3,4]]
Space group Pm3m (221) Im3m (229)
Fibrifold 4:2 8o:2
Coloring

Omnitruncated cubic honeycomb1.png

Omnitruncated cubic honeycomb2.png

Coxeter diagram

CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.png

CDel branch 11.pngCDel 4a4b.pngCDel nodes 11.png

Vertex figure

Omnitruncated cubic honeycomb verf.png

Omnitruncated cubic honeycomb verf2.png





Related polyhedra


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
Omnitruncated cubic honeycomb skew1 verf.png
4.8.4.8
Omnitruncated cubic honeycomb skew2 verf.png

Omnitruncated cubic honeycomb apeirohedron 4446.png

Skew polyhedron 4848.png


Alternated omnitruncated cubic honeycomb




































Alternated omnitruncated cubic honeycomb
Type Alternated uniform honeycomb
Schläfli symbol ht0,1,2,3{4,3,4}
Coxeter diagram
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
Cells
snub cube
square antiprism
tetrahedron
Vertex figure
Snub cubic honeycomb verf.png
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: CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png 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 honeycomb




























Dual alternated omnitruncated cubic honeycomb
Type Dual alternated uniform honeycomb
Schläfli symbol dht0,1,2,3{4,3,4}
Coxeter diagram
CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Cells
Altbasetet-solid.png
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.


Eighth pyramidille cell.png

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


Altbasetet-24-in-cube.png

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


















Cell views

Altbasetet net.png
Net

Altbasetet.png

Altbasetet-frame1.png

Altbasetet-frame2.png

Altbasetet-frame3.png

Altbasetet-frame4.png

Altbasetet-frame5.png

Altbasetet-frame7.png


Truncated square prismatic honeycomb




























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
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Coxeter group [4,4,2,∞]
Dual
Tetrakis square prismatic tiling
Cell: Cubic half domain.png
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.


Truncated square prismatic honeycomb.png


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


It is one of 28 convex uniform honeycombs.






Snub square prismatic honeycomb




























Snub square prismatic honeycomb
Type
Uniform honeycomb
Schläfli symbol s{4,4}×{∞}
sr{4,4}×{∞}
Coxeter-Dynkin diagram
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
Coxeter group [4+,4,2,∞]
[(4,4)+,2,∞]
Dual
Cairo pentagonal prismatic honeycomb
Cell: Snub square prismatic honeycomb dual cell.png
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.


Snub square prismatic honeycomb.png


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


It is one of 28 convex uniform honeycombs.




See also







  • Architectonic and catoptric tessellation

  • Alternated cubic honeycomb

  • List of regular polytopes


  • Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge

  • voxel



References





  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, .mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    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

Space

Family

A~n−1{displaystyle {tilde {A}}_{n-1}}{tilde {A}}_{n-1}

C~n−1{displaystyle {tilde {C}}_{n-1}}{tilde {C}}_{n-1}

B~n−1{displaystyle {tilde {B}}_{n-1}}{tilde {B}}_{n-1}

D~n−1{displaystyle {tilde {D}}_{n-1}}{tilde {D}}_{n-1}

G~2{displaystyle {tilde {G}}_{2}}{tilde {G}}_{2} / F~4{displaystyle {tilde {F}}_{4}}{tilde {F}}_{4} / E~n−1{displaystyle {tilde {E}}_{n-1}}{tilde {E}}_{n-1}
E2

Uniform tiling

{3[3]}

δ3

3

3

Hexagonal
E3

Uniform convex honeycomb

{3[4]}

δ4

4

4

E4

Uniform 4-honeycomb

{3[5]}

δ5

5

5

24-cell honeycomb
E5

Uniform 5-honeycomb

{3[6]}

δ6

6

6

E6

Uniform 6-honeycomb

{3[7]}

δ7

7

7

222
E7

Uniform 7-honeycomb

{3[8]}

δ8

8

8

133 • 331
E8

Uniform 8-honeycomb

{3[9]}

δ9

9

9

152 • 251 • 521
E9

Uniform 9-honeycomb
{3[10]}

δ10

10

10

En-1
Uniform (n-1)-honeycomb

{3[n]}

δn

n

n

1k2 • 2k1 • k21



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