8-cubic honeycomb

Last updated
8-cubic honeycomb
(no image)
Type Regular 8-honeycomb
Uniform 8-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,36,4}
{4,35,31,1}
t0,8{4,36,4}
{∞}(8)
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-face type {4,36}
7-face type {4,35}
6-face type {4,34}
5-face type {4,33}
4-face type {4,32}
Cell type {4,3}
Face type {4}
Face figure {4,3}
(octahedron)
Edge figure8 {4,3,3}
(16-cell)
Vertex figure256 {4,36}
(8-orthoplex)
Coxeter group [4,36,4]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

In geometry, the 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space.

Contents

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,36,4}. Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol {4,35,31,1}. The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(8).

The [4,36,4], CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, Coxeter group generates 511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical to the 8-cubic honeycomb.

The 8-cubic honeycomb can be alternated into the 8-demicubic honeycomb, replacing the 8-cubes with 8-demicubes, and the alternated gaps are filled by 8-orthoplex facets.

Quadrirectified 8-cubic honeycomb

A quadrirectified 8-cubic honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, contains all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D8* lattice. Facets can be identically colored from a doubled ×2, [[4,36,4]] symmetry, alternately colored from , [4,36,4] symmetry, three colors from , [4,35,31,1] symmetry, and 4 colors from , [31,1,34,31,1] symmetry.

See also

Related Research Articles

<span class="mw-page-title-main">Convex uniform honeycomb</span> Spatial tiling of convex uniform polyhedra

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

<span class="mw-page-title-main">Cubic honeycomb</span> Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation 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 called this honeycomb a cubille.

<span class="mw-page-title-main">Uniform 8-polytope</span>

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

<span class="mw-page-title-main">Uniform 7-polytope</span>

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

<span class="mw-page-title-main">Uniform 9-polytope</span>

In nine-dimensional geometry, a nine-dimensional polytope or 9-polytope is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.

<span class="mw-page-title-main">Uniform 6-polytope</span> Uniform 6-dimensional polytope

In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

<span class="mw-page-title-main">Tesseractic honeycomb</span>

In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations, represented by Schläfli symbol {4,3,3,4}, and constructed by a 4-dimensional packing of tesseract facets.

<span class="mw-page-title-main">16-cell honeycomb</span>

In four-dimensional Euclidean geometry, the 16-cell honeycomb is one of the three regular space-filling tessellations, represented by Schläfli symbol {3,3,4,3}, and constructed by a 4-dimensional packing of 16-cell facets, three around every face.

In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensional spaces with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group Rn for n ≥ 3.

The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.

The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation in Euclidean 6-space.

The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation in Euclidean 7-space.

The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.

<span class="mw-page-title-main">Uniform 5-polytope</span> Five-dimensional geometric shape

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

<span class="mw-page-title-main">Alternated hypercubic honeycomb</span>

In geometry, the alternated hypercube honeycomb is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group for n ≥ 4. A lower symmetry form can be created by removing another mirror on an order-4 peak.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

References

Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
E10Uniform 10-honeycomb{3[11]}δ11hδ11qδ11
En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21