In
geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling
tessellation (or
honeycomb) in
Euclidean 5-space. Four
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 meet at each cubic cell, and it is more explicitly called an ''order-4
penteractic honeycomb''.
It is analogous to the
square tiling of the plane and to the
cubic honeycomb of
3-space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
, and the
tesseractic honeycomb of
4-space.
Constructions
There are many different
Wythoff constructions of this honeycomb. The most symmetric form is
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
, 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
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, ...
facets (like a checkerboard) with Schläfli symbol . The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol
5.
Related polytopes and honeycombs
The
,33,4 , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The
expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.
The ''5-cubic honeycomb'' can be
alternated into the
5-demicubic 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 dem ...
, replacing the 5-cubes with
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, and the alternated gaps are filled by
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 with ...
facets.
It is also related to the regular
6-cube which exists in 6-space with ''3'' ''5''-cubes on each cell. This could be considered as a tessellation on the
5-sphere, an ''order-3 penteractic honeycomb'', .
Tritruncated 5-cubic honeycomb
A tritruncated 5-cubic honeycomb, , contains all
bitruncated 5-orthoplex facets and is the
Voronoi tessellation of the
D5* lattice. Facets can be identically colored from a doubled
×2,
4,3
3,4 symmetry, alternately colored from
,
,33,4symmetry, three colors from
,
,3,3,31,1symmetry, and 4 colors from
,
1,1,3,31,1symmetry.
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 ' ...
Regular and uniform honeycombs in 5-space:
*
5-demicubic 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 dem ...
*
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 ...
*
Truncated 5-simplex honeycomb
In five-dimensional Euclidean geometry, the cyclotruncated 5-simplex honeycomb or cyclotruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed of 5-simplex, truncated 5-simplex, and bitruncated 5-simplex fac ...
*
Omnitruncated 5-simplex honeycomb
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)
6-polytopes
Regular tessellations