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
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
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, ...
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 256 types of
facets around each vertex and a prismatic product Schläfli symbol
8.
Related honeycombs
The
,36,4 , 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, , contains all
trirectified 8-orthoplex
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a Rectification (geometry), rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and ...
facets and is the
Voronoi tessellation of the
D8* lattice. Facets can be identically colored from a doubled
×2,
4,3
6,4 symmetry, alternately colored from
,
,36,4symmetry, three colors from
,
,35,31,1symmetry, and 4 colors from
,
1,1,34,31,1symmetry.
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)
9-polytopes
Regular tessellations