6-cubic honeycomb

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6-cubic honeycomb
(no image)
Type Regular 6-honeycomb
Uniform 6-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,34,4}
{4,33,31,1}
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 4.pngCDel node.png
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6-face type {4,34}
5-face type {4,33}
4-face type {4,3,3}
Cell type {4,3}
Face type {4}
Face figure {4,3}
(octahedron)
Edge figure8 {4,3,3}
(16-cell)
Vertex figure64 {4,34}
(6-orthoplex)
Coxeter group , [4,34,4]
, [4,33,31,1]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

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

Contents

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

Constructions

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,34,4}. Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol {4,33,31,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(6).

The [4,34,4], CDel node.pngCDel 4.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 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.

The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets.

Trirectified 6-cubic honeycomb

A trirectified 6-cubic honeycomb, CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a4b.pngCDel nodes.png, contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled ×2, [[4,34,4]] symmetry, alternately colored from , [4,34,4] symmetry, three colors from , [4,33,31,1] symmetry, and 4 colors from , [31,1,3,3,31,1] 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">Tetrahedral-octahedral honeycomb</span> Quasiregular space-filling tesselation

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

<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>

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.

The 5-demicube honeycomb is a uniform space-filling tessellation in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube 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 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation in Euclidean 7-space.

The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation in Euclidean 8-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.

<span class="mw-page-title-main">Order-6 cubic honeycomb</span>

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

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