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7-cubic honeycomb

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!bgcolor=#e7dcc3 colspan=2|7-cubic honeycomb

bgcolor=#ffffff align=center colspan=2|(no image)
bgcolor=#e7dcc3|TypeRegular 7-honeycomb
Uniform 7-honeycomb
bgcolor=#e7dcc3|FamilyHypercube honeycomb
bgcolor=#e7dcc3|Schläfli symbol{4,35,4}
{4,34,31,1}
{∞}(7)
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node_1|4|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|split1|nodes}}
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|4|node_1}}
bgcolor=#e7dcc3|7-face type{4,3,3,3,3,3}
bgcolor=#e7dcc3|6-face type{4,3,3,3,3}
bgcolor=#e7dcc3|5-face type{4,3,3,3}
bgcolor=#e7dcc3|4-face type{4,3,3}
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 figure128 {4,35}
(7-orthoplex)
bgcolor=#e7dcc3|Coxeter group[4,35,4]
bgcolor=#e7dcc3|Dualself-dual
bgcolor=#e7dcc3|Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive

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

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

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,35,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,34,31,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(7).

Related honeycombs

The [4,35,4], {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|4|node}}, Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb.

The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets.

= Quadritruncated 7-cubic honeycomb =

A quadritruncated 7-cubic honeycomb, {{CDD|branch_11|3ab|nodes|3ab|nodes|4a4b|nodes}}, contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D7* lattice. Facets can be identically colored from a doubled {\tilde{C}}_7×2, 4,35,4 symmetry, alternately colored from {\tilde{C}}_7, [4,35,4] symmetry, three colors from {\tilde{B}}_7, [4,34,31,1] symmetry, and 4 colors from {\tilde{D}}_7, [31,1,33,31,1] 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:8-polytopes

Category:Regular tessellations