The 6-cubic honeycomb or hexeractic honeycomb is the only regular 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 A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen. beekeeping, Beekee ...

) in Euclidean 6-space. It is analogous to the

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of the s ...

of the plane and to the

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 ...

of 3-space.

Constructions

There are many different

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

s of this

. The most symmetric form is regular, 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

6-cube In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces. It has Schläfli symbol , being composed of 3 5-cubes around each 4-face. It ...

facets (like a

checkerboard A checkerboard (American English) or chequerboard (British English; see spelling differences) is a board of checkered pattern on which checkers (also known as English draughts) is played. Most commonly, it consists of 64 squares (8×8) of altern ...

) with Schläfli symbol . The lowest symmetry

has 64 types of facets around each vertex and a prismatic product Schläfli symbol ⁽⁶⁾.

Related honeycombs

The ,3⁴,4 , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb. The ''6-cubic honeycomb'' can be alternated into the

6-demicubic honeycomb The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 6-space. It is constructed as an Alternation (geometry), alternation of the regular 6-cube honeycomb. I ...

, replacing the 6-cubes with

6-demicube In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a ''6-cube'' ( hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elte i ...

s, and the alternated gaps are filled by

6-orthoplex In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell ''4-faces'', and 64 ''5-faces''. It has two constructed forms, the first being regular with ...

facets.

Trirectified 6-cubic honeycomb

A trirectified 6-cubic honeycomb, , contains all

birectified 6-orthoplex In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a Rectification (geometry), rectification of the regular 6-orthoplex. There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and th ...

facets and is the

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 ** ...

of the D₆^* lattice. Facets can be identically colored from a doubled

_6

×2, 4,3⁴,4 symmetry, alternately colored from

_6

, ,3⁴,4symmetry, three colors from

_6

, ,3³,3^1,1symmetry, and 4 colors from

_6

, ^1,1,3,3,3^1,1symmetry.

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, 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) 7-polytopes Regular tessellations

Constructions

Related honeycombs

Trirectified 6-cubic honeycomb

See also

References