In
five-dimensional
A five-dimensional space is a space with five dimensions. In mathematics, a sequence of ''N'' numbers can represent a location in an ''N''-dimensional space. If interpreted physically, that is one more than the usual three spatial dimensions a ...
Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a 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). It is composed entirely of
omnitruncated 5-simplex facets.
The facets of all
omnitruncated simplectic honeycombs are called
permutahedra and can be positioned in ''n+1'' space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A5* lattice
The A lattice (also called A) is the union of six
A5 lattices, and is the dual
vertex arrangement to the ''omnitruncated 5-simplex honeycomb'', and therefore the
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 ...
of this lattice is an
omnitruncated 5-simplex.
:
∪
∪
∪
∪
∪
= dual of
Related polytopes and honeycombs
Projection by folding
The ''omnitruncated 5-simplex honeycomb'' can be projected into the 3-dimensional
omnitruncated cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a re ...
by a
geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space
vertex arrangement:
See also
Regular and uniform honeycombs in 5-space:
*
5-cube honeycomb
*
5-demicube honeycomb
The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform space-filling tessellation (or honeycomb) in Euclidean 5-space. It is constructed as an alternation of the regular 5-cube honeycomb.
It is the first tessellation in the demih ...
*
5-simplex honeycomb
In Five-dimensional space, five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb (geometry), honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified ...
Notes
References
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
* 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 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380-407, MR 2,10(1.9 Uniform space-fillings)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45{{Honeycombs
Honeycombs (geometry)
6-polytopes