Quadritruncated 7-cubic Honeycomb

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 . Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol ⁷.

Related honeycombs

The ,3⁵,4 , 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, , contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D₇^* lattice. Facets can be identically colored from a doubled $_7$×2, 4,3⁵,4 symmetry, alternately colored from $_7$, ,3⁵,4symmetry, three colors from $_7$, ,3⁴,3^1,1symmetry, and 4 colors from $_7$, ^1,1,3³,3^1,1symmetry.

See also

*List of regular polytopes

References

* Coxeter, H.S.M. '' Regular Polytopes'', (3rd edition, 1973), Dover edition, 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,

** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45
{{Honeycombs

Honeycombs (geometry)
8-polytopes
Regular tessellations