geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the 8-cubic honeycomb or octeractic honeycomb is the only regular 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 ...

) in Euclidean 8-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 consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tili ...

of the plane and to the

cubic honeycomb The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 3-space made up of cube, cubic cells. It has 4 cubes around every edge, and 8 cubes around each verte ...

of 3-space, and the

tesseractic honeycomb In four-dimensional euclidean geometry, the tesseractic honeycomb is one of the three regular space-filling tessellations (or honeycombs), represented by Schläfli symbol , and consisting of a packing of tesseracts (4-hypercubes). Its vertex fi ...

of 4-space. 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 honeycomb. The most symmetric form is

regular Regular may refer to: Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instruments, tunings with equal intervals between the paired notes of successive open strings Other uses * Regular character, ...

, with

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

. Another form has two alternating

hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square ( ) and a cube ( ); the special case for is known as a ''tesseract''. It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel l ...

facets (like a

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

) with Schläfli symbol . The lowest symmetry Wythoff construction has 256 types of

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 around each vertex and a prismatic product Schläfli symbol ⁽⁸⁾.

Related honeycombs

The ,3⁶,4 , Coxeter group generates 511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical to the 8-cubic honeycomb. The ''8-cubic honeycomb'' can be alternated into the 8-demicubic honeycomb, replacing the 8-cubes with

8-demicube In geometry, a demiocteract or 8-demicube is a uniform 8-polytope, constructed from the 8-hypercube, octeract, with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes. E. L. Elt ...

s, and the alternated gaps are filled by

8-orthoplex In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 Vertex (geometry), vertices, 112 Edge (geometry), edges, 448 triangle Face (geometry), faces, 1120 tetrahedron Cell (mathematics), cells, 1792 5-cell ''4-faces'', 179 ...

facets.

Quadrirectified 8-cubic honeycomb

A quadrirectified 8-cubic honeycomb, , contains all trirectified 8-orthoplex 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 D₈^* lattice. Facets can be identically colored from a doubled

_8

×2, 4,3⁶,4 symmetry, alternately colored from

_8

, ,3⁶,4symmetry, three colors from

_8

, ,3⁵,3^1,1symmetry, and 4 colors from

_8

, ^1,1,3⁴,3^1,1symmetry.

References

* Coxeter, H.S.M. ''

Regular Polytopes ''Regular Polytopes'' is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a th ...

'', (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs * 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 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45 {{Honeycombs Honeycombs (geometry) 9-polytopes Regular tessellations

Related honeycombs

Quadrirectified 8-cubic honeycomb

See also

References