The 5-demicube honeycomb (or demipenteractic honeycomb) is a uniform 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 of ge ...

(or

honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey t ...

) in Euclidean 5-space. It is constructed as an alternation of the regular

5-cube honeycomb In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an ''order-4 penteractic h ...

. It is the first tessellation in the demihypercube honeycomb family which, with all the next ones, is not regular, being composed of two different types of

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

facet Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cu ...

s. The

5-cube In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces. It is represented by Schläfli symbol or , constructed as 3 tesseracts ...

s become alternated into

5-demicube In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' ( penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...

s h and the alternated vertices create

5-orthoplex In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces. It has two constructed forms, the first being regular wi ...

facets.

D5 lattice

The

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 ...

of the 5-demicubic honeycomb is the D₅ lattice which is the densest known

sphere packing In geometry, a sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three- dimensional Euclidean space. However, sphere pack ...

in 5 dimensions. The 40 vertices of the

rectified 5-orthoplex In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex. There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and t ...

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

of the ''5-demicubic honeycomb'' reflect the

kissing number In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...

40 of this lattice. The D packing (also called D) can be constructed by the union of two D₅ lattices. The analogous packings form lattices only in even dimensions. The kissing number is 2⁴=16 (2^n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8). : ∪ The D lattice (also called D and C) can be constructed by the union of all four 5-demicubic lattices: It is also the 5-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 ...

, the union of two 5-cube honeycombs in dual positions. : ∪ ∪ ∪ = ∪ . The

of the D lattice is 10 (''2n'' for n≥5) and its

Voronoi tessellation Voronoi or Voronoy is a Slavic masculine surname; its feminine counterpart is Voronaya. It may refer to *Georgy Voronoy (1868–1908), Russian and Ukrainian mathematician **Voronoi diagram ** Weighted Voronoi diagram ** Voronoi deformation density * ...

is a

tritruncated 5-cubic honeycomb In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an ''order-4 penteractic hone ...

, , containing all bitruncated 5-orthoplex,

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 ...

s.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 32

facets around each vertex.

Related honeycombs

References

* Coxeter, H.S.M. ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

'', (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*

External links

{{DEFAULTSORT:Demipenteractic Honeycomb Honeycombs (geometry) 6-polytopes

D5 lattice

Symmetry constructions

Related honeycombs

See also

References

External links