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8-cubic honeycomb#Quadrirectified 8-cubic honeycomb

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!bgcolor=#e7dcc3 colspan=2|8-cubic honeycomb

bgcolor=#ffffff align=center colspan=2|(no image)
bgcolor=#e7dcc3|TypeRegular 8-honeycomb
Uniform 8-honeycomb
bgcolor=#e7dcc3|FamilyHypercube honeycomb
bgcolor=#e7dcc3|Schläfli symbol{4,36,4}
{4,35,31,1}
t0,8{4,36,4}
{∞}(8)
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|split1|nodes}}
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node_1}}
{{CDD|node_1|4|node|3|node|3|node|4|node|2|node_1|4|node|3|node|3|node|4|node}}
bgcolor=#e7dcc3|8-face type{4,36}
bgcolor=#e7dcc3|7-face type{4,35}
bgcolor=#e7dcc3|6-face type{4,34}
bgcolor=#e7dcc3|5-face type{4,33}
bgcolor=#e7dcc3|4-face type{4,32}
bgcolor=#e7dcc3|Cell type{4,3}
bgcolor=#e7dcc3|Face type{4}
bgcolor=#e7dcc3|Face figure{4,3}
(octahedron)
bgcolor=#e7dcc3|Edge figure8 {4,3,3}
(16-cell)
bgcolor=#e7dcc3|Vertex figure256 {4,36}
(8-orthoplex)
bgcolor=#e7dcc3|Coxeter group[4,36,4]
bgcolor=#e7dcc3|Dualself-dual
bgcolor=#e7dcc3|Propertiesvertex-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.

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).

Related honeycombs

The [4,36,4], {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}, 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, {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|4a4b|nodes}}, contains all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D8* lattice. Facets can be identically colored from a doubled {\tilde{C}}_8×2, 4,36,4 symmetry, alternately colored from {\tilde{C}}_8, [4,36,4] symmetry, three colors from {\tilde{B}}_8, [4,35,31,1] symmetry, and 4 colors from {\tilde{D}}_8, [31,1,34,31,1] symmetry.

See also

References

  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p. 296, Table II: Regular honeycombs
  • 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

{{Honeycombs}}

Category:Honeycombs (geometry)

Category:9-polytopes

Category:Regular tessellations