The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling
tessellation
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
(or
honeycomb) in Euclidean 6-space.
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
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
. Another form has two alternating
6-cube facets (like a
checkerboard
A checkerboard (American English) or chequerboard (British English; see spelling differences) is a board of checkered pattern on which checkers (also known as English draughts) is played. Most commonly, it consists of 64 squares (8×8) of altern ...
) with Schläfli symbol . The lowest symmetry
Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol
(6).
Related honeycombs
The
,34,4 , 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-demicube
In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' ( hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte i ...
s, and the alternated gaps are filled by
6-orthoplex facets.
Trirectified 6-cubic honeycomb
A trirectified 6-cubic honeycomb, , contains all
birectified 6-orthoplex facets and is the
Voronoi tessellation of the
D6* lattice. Facets can be identically colored from a doubled
×2,
4,3
4,4 symmetry, alternately colored from
,
,34,4symmetry, three colors from
,
,33,31,1symmetry, and 4 colors from
,
1,1,3,3,31,1symmetry.
See also
*
List of regular polytopes
This article lists the regular polytopes and regular polytope compounds in Euclidean geometry, Euclidean, spherical geometry, spherical and hyperbolic geometry, hyperbolic spaces.
The Schläfli symbol describes every regular tessellation of an ' ...
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
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
, 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)
7-polytopes
Regular tessellations