Cubic honeycomb

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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 {\tilde{C}}_3, [4,3,4]
Dual self-dual
Properties vertex-transitive, quasiregular

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.

Cartesian coordinates[edit]

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[edit]

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[edit]

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 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 Hexagonal latticeR.svg Lattic simple 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[edit]

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

Related polytopes and honeycombs[edit]

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.

{p,3,4}
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name {3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
{5,3,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel split1.pngCDel nodes.png
{6,3,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1.pngCDel branch 11.pngCDel uaub.pngCDel nodes.png
{7,3,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 7.pngCDel node.pngCDel split1.pngCDel nodes.png
{8,3,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1-44.pngCDel branch 11.pngCDel label4.pngCDel uaub.pngCDel nodes.png
... {∞,3,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node.pngCDel ultra.pngCDel node 1.pngCDel split1-ii.pngCDel branch 11.pngCDel labelinfin.pngCDel uaub.pngCDel nodes.png
Image Stereographic polytope 16cell.png Cubic honeycomb.png H3 534 CC center.png H3 634 FC boundary.png
Cells Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 237-1.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 238-1.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 23i-1.png
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png

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

{4,3,p}
Space S3 E3 H3
Form Finite Affine Compact Paracompact Noncompact
Name
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
{4,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
{4,3,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{4,3,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel ultra.pngCDel node.pngCDel split1.pngCDel branch.pngCDel uaub.pngCDel nodes 11.png
{4,3,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{4,3,8}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
... {4,3,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Image Stereographic polytope 8cell.png Cubic honeycomb.png H3 435 CC center.png H3 436 CC center.png
Vertex
figure
CDel node 1.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
8-cell verf.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cubic honeycomb verf.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
Order-5 cubic honeycomb verf.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.png
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.png
H2 tiling 237-4.png
{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 238-4.png
{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2 tiling 23i-4.png
{3,∞}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
{p,3,p}
Space S3 Euclidean H3
Form Finite Affine Compact Paracompact Noncompact
Name
CDel node 1.pngCDel p.pngCDel node.pngCDel 3.pngCDel node.pngCDel p.pngCDel node.png
{3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
{4,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.pngCDel 2.pngCDel labelinfin.pngCDel branch 10.png
{5,3,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
{6,3,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.png
CDel node 1.pngCDel splitplit1u.pngCDel branch4u 11.pngCDel uabc.pngCDel branch4u.pngCDel splitplit2u.pngCDel node.png
{7,3,7}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
{8,3,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel split1.pngCDel branch.pngCDel label4.png
... {∞,3,∞}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel split1.pngCDel branch.pngCDel labelinfin.png
Image Stereographic polytope 5cell.png Cubic honeycomb.png H3 535 CC center.png H3 636 FC boundary.png
Cells Tetrahedron.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Hexahedron.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Dodecahedron.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 237-1.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 238-1.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 23i-1.png
{∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
Vertex
figure
5-cell verf.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cubic honeycomb verf.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes.png
Order-5 dodecahedral honeycomb verf.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 63-t2.png
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.png
H2 tiling 237-4.png
{3,7}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 238-4.png
{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel label4.png
H2 tiling 23i-4.png
{3,∞}
CDel node 1.pngCDel 3.pngCDel node.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel branch.pngCDel labelinfin.png

Related Euclidean tessellations[edit]

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.

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

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

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.

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

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

This honeycomb is one of five distinct uniform honeycombs[2] constructed by the {\tilde{A}}_3 Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

Space
group
Fibrifold Square
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
F43m
(216)
1o:2 a1 [3[4]] CDel node.pngCDel split1.pngCDel nodes.pngCDel split2.pngCDel node.png ×1 (None)
Fd3m
(227)
2+:2 p2 [[3[4]]] CDel branch 11.pngCDel 3ab.pngCDel branch.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node h1.png
×2 CDel branch 11.pngCDel 3ab.pngCDel branch.png 3
Fm3m
(225)
2:2 d2 <[3[4]]>
↔ [4,3,31,1]
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png
CDel node.pngCDel 4.pngCDel node c3.pngCDel split1.pngCDel nodeab c1-2.png
×2 CDel node.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node.png 1,CDel node 1.pngCDel split1.pngCDel nodes 10luru.pngCDel split2.pngCDel node 1.png 2
Pm3m
(221)
4:2 d4 [2[3[4]]]
↔ [4,3,4]
CDel node c1.pngCDel split1.pngCDel nodeab c2.pngCDel split2.pngCDel node c1.png
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png
×4 CDel node.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node.png 4
Im3m
(229)
8o:2 r8 [4[3[4]]]
↔ [[4,3,4]]
CDel branch c1.pngCDel 3ab.pngCDel branch c1.png
CDel branch c1.pngCDel 4a4b.pngCDel nodes.png
×8 CDel branch 11.pngCDel 3ab.pngCDel branch 11.png 5, CDel branch hh.pngCDel 3ab.pngCDel branch hh.png (*)



Rectified cubic honeycomb[edit]

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{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 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 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 {\tilde{C}}_3, [4,3,4]
Dual oblate octahedrille
(Square bipyramidal honeycomb)
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 oblate octahedrille.

Rectified cubic tiling.pngHC A3-P3.png

Symmetry[edit]

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]
{\tilde{C}}_3
[1+,4,3,4]
[4,31,1], {\tilde{B}}_3
[4,3,4,1+]
[4,31,1], {\tilde{B}}_3
[1+,4,3,4,1+]
[3[4]], {\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[edit]

Truncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol t{4,3,4} or t0,1{4,3,4}
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 {\tilde{C}}_3, [4,3,4]
Dual Pyramidille
(Hexakis cubic honeycomb)
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

Symmetry[edit]

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], {\tilde{B}}_3 [4,3,4], {\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



Bitruncated cubic honeycomb[edit]

Bitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol 2t{4,3,4} or t1,2{4,3,4}
Coxeter diagram CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel branch 11.pngCDel 4a4b.pngCDel nodes.png

CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png = CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node h0.png

Cell type (4.6.6)
Face types square {4}
hexagon {6}
Edge figure isosceles triangle {3}
Vertex figure Bitruncated cubic honeycomb verf2.png
(disphenoid tetrahedron)
Space group
Fibrifold notation
Coxeter notation
Im3m (229)
8o:2
[[4,3,4]]
Coxeter group {\tilde{C}}_3, [4,3,4]
Dual Oblate tetrahedrille
Disphenoid tetrahedral honeycomb
Properties isogonal, isotoxal, isochoric

The bitruncated cubic honeycomb or bitruncated cubic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic and catoptric tessellation list, with its dual called an oblate tetrahedrille, also called a disphenoid tetrahedral honeycomb. Although a regular tetrahedron can not tessellate space alone, this dual has identical disphenoid tetrahedron cells with isosceles triangle faces.

It can be realized as the Voronoi tessellation of the body-centred cubic lattice. Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) is the optimal soap bubble foam. However, the Weaire–Phelan structure is a less symmetrical, but more efficient, foam of soap bubbles.

Bitruncated cubic tiling.png HC-A4.png

Symmetry[edit]

The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the {\tilde{A}}_3 Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction.

Five uniform colorings by cell
Space group Im3m (229) Pm3m (221) Fm3m (225) F43m (216) Fd3m (227)
Fibrifold 8o:2 4:2 2:2 1o:2 2+:2
Coxeter group {\tilde{C}}_3×2
[[4,3,4]]
=[4[3[4]]]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c1.pngCDel 4.pngCDel node.png = CDel branch c1.pngCDel 3ab.pngCDel branch c1.png
{\tilde{C}}_3
[4,3,4]
=[2[3[4]]]
CDel node.pngCDel 4.pngCDel node c1.pngCDel 3.pngCDel node c2.pngCDel 4.pngCDel node.png = CDel branch c1-2.pngCDel 3ab.pngCDel branch c2-1.png
{\tilde{B}}_3
[4,31,1]
=<[3[4]]>
CDel nodeab c1-2.pngCDel split2.pngCDel node c3.pngCDel 4.pngCDel node.png = CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c3.png
{\tilde{A}}_3
[3[4]]
 
CDel node c3.pngCDel split1.pngCDel nodeab c1-2.pngCDel split2.pngCDel node c4.png
{\tilde{A}}_3×2
[[3[4]]]
=[[3[4]]]
CDel branch c1.pngCDel 3ab.pngCDel branch c2.png
Coxeter diagram CDel branch 11.pngCDel 4a4b.pngCDel nodes.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png CDel branch 11.pngCDel 3ab.pngCDel branch 11.png
truncated octahedra 1
Uniform polyhedron-43-t12.svg
1:1
Uniform polyhedron-43-t12.svg:Uniform polyhedron-43-t12.svg
2:1:1
Uniform polyhedron-43-t12.svg:Uniform polyhedron-43-t12.svg:Uniform polyhedron-33-t012.png
1:1:1:1
Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png
1:1
Uniform polyhedron-33-t012.png:Uniform polyhedron-33-t012.png
Vertex figure Bitruncated cubic honeycomb verf2.png Bitruncated cubic honeycomb verf.png Cantitruncated alternate cubic honeycomb verf.png Omnitruncated 3-simplex honeycomb verf.png Omnitruncated 3-simplex honeycomb verf2.png
Vertex
figure
symmetry
[2+,4]
(order 8)
[2]
(order 4)
[ ]
(order 2)
[ ]+
(order 1)
[2]+
(order 2)
Image
Colored by
cell
Bitruncated Cubic Honeycomb1.svg Bitruncated Cubic Honeycomb.svg Bitruncated cubic honeycomb3.png Bitruncated cubic honeycomb2.png Bitruncated Cubic Honeycomb1.svg

Projection by folding[edit]

The bitruncated cubic honeycomb can be orthogonally projected into the planar truncated square tiling by a geometric folding operation that maps two pairs of mirrors into each other. The projection of the bitruncated cubic honeycomb creating two offset copies of the truncated square tiling vertex arrangement of the plane:

Coxeter
group
{\tilde{A}}_3 {\tilde{C}}_2
Coxeter
diagram
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel split2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png
Graph Bitruncated Cubic Honeycomb flat.png
Bitruncated cubic honeycomb
Uniform tiling 44-t012.png
Truncated square tiling



Alternated bitruncated cubic honeycomb[edit]

Alternated bitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol 2s{4,3,4}
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 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], {\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[edit]

Cantellated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol rr{4,3,4} or t0,2{4,3,4}
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], {\tilde{C}}_3
Dual quarter oblate octahedrille
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.png HC A5-A3-P2.png

Images[edit]

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.

Symmetry[edit]

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], {\tilde{C}}_3
=<[4,31,1]>
[4,31,1], {\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



Cantitruncated cubic honeycomb[edit]

Cantitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol tr{4,3,4} or t0,1,2{4,3,4}
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], {\tilde{C}}_3
Space group
Fibrifold notation
Pm3m (221)
4:2
Dual triangular pyramidille
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[edit]

Four cells exist around each vertex:

2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png

Symmetry[edit]

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], {\tilde{C}}_3
=<[4,31,1]>
[4,31,1], {\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



Alternated cantitruncated cubic honeycomb[edit]

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], {\tilde{B}}_3
Dual square quarter pyramidille
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



Runcic cantitruncated cubic honeycomb[edit]

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], {\tilde{C}}_3
Dual
Properties vertex-transitive

The runcic cantitruncated cubic honeycom or runcic 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[edit]

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], {\tilde{C}}_3
Space group
Fibrifold notation
Pm3m (221)
4:2
Dual square quarter pyramidille
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


Omnitruncated cubic honeycomb[edit]

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], {\tilde{C}}_3
Dual eighth pyramidille
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

Symmetry[edit]

Cells can be shown in two different symmetries. The Coxeter diagram form has two colors of truncated cuboctahedra and octahedral 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 octahedral prism cells.

Two uniform colorings
Symmetry {\tilde{C}}_3, [4,3,4] {\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



Alternated omnitruncated cubic honeycomb[edit]

Alternated omnitruncated cubic honeycomb
Type 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]]+
Properties vertex-transitive

A alternated omnitruncated cubic honeycomb or full 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.



Truncated square prismatic honeycomb[edit]

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
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[edit]

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
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[edit]

References[edit]

  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.
  • Richard Klitzing, 3D Euclidean Honeycombs, x4o3o4o - chon - O1
  • Uniform Honeycombs in 3-Space: 01-Chon