The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling
tessellation (or
honeycomb) in Euclidean 8-space. It is constructed as an
alternation of the regular
8-cubic honeycomb.
It is composed of two different types of
facets. The
8-cubes become alternated into
8-demicubes h and the alternated vertices create
8-orthoplex facets .
D8 lattice
The
vertex arrangement of the 8-demicubic honeycomb is the D
8 lattice. The 112 vertices of the
rectified 8-orthoplex
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being th ...
vertex figure of the ''8-demicubic honeycomb'' reflect the
kissing number 112 of this lattice. The best known is 240, from the
E8 lattice and the
521 honeycomb.
contains
as a subgroup of index 270. Both
and
can be seen as affine extensions of
from different nodes:
The D lattice (also called D) can be constructed by the union of two D8 lattices. This packing is only a lattice for even dimensions. The kissing number is 240. (2
n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). It is identical to the
E8 lattice. At 8-dimensions, the 240 contacts contain both the 2
7=128 from lower dimension contact progression (2
n-1), and 16*7=112 from higher dimensions (2n(n-1)).
: ∪ = .
The D lattice (also called D and C) can be constructed by the union of all four ''D8 lattices'': It is also the 7-dimensional
body centered cubic
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals.
There are three main varieties o ...
, the union of two
7-cube honeycombs in dual positions.
: ∪ ∪ ∪ = ∪ .
The
kissing number of the D lattice is 16 (''2n'' for n≥5). and its
Voronoi tessellation is a
quadrirectified 8-cubic honeycomb
The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean space, Euclidean 8-space.
It is analogous to the square tiling of the plane and to the cubic honeycomb o ...
, , containing 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 ...
Voronoi cell
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed t ...
, .
[Conway (1998), p. 466]
Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 256
8-demicube facets around each vertex.
See also
*
8-cubic honeycomb
*
Uniform polytope
Notes
References
*
Coxeter, H.S.M. ''
Regular Polytopes'', (3rd edition, 1973), Dover edition,
** pp. 154–156: Partial truncation or alternation, represented by ''h'' prefix: h=; h=, h=, ...
* 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*
N.W. Johnson: ''Geometries and Transformations'', (2018)
*
External links
{{Honeycombs
Honeycombs (geometry)
9-polytopes