In
five-dimensional Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, 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, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
(or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic cells built from beeswax by honey bees in their beehive, nests to contain their brood (eggs, larvae, and pupae) and stores of honey and pol ...
). It is composed entirely of
omnitruncated 5-simplex facets.
The facets of all
omnitruncated simplectic honeycomb In geometry an omnitruncated simplicial honeycomb or omnitruncated n-simplex honeycomb is an n-dimensional Uniform honeycomb, uniform tessellation, based on the symmetry of the _n affine Coxeter group. Each is composed of omnitruncation (geometry), ...
s 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
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
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. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...
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 by a
geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space
vertex arrangement
In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes.
For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...
:
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 Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. ...
Notes
References
*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
* 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 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