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 ...

, a hypercubic honeycomb is a family of regular honeycombs (

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 ...

s) in -

dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...

al spaces with the

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 ...

s and containing the symmetry of

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

(or ) for . The tessellation is constructed from 4 -

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 ...

s per

ridge A ridge is a long, narrow, elevated geomorphologic landform, structural feature, or a combination of both separated from the surrounding terrain by steep sides. The sides of a ridge slope away from a narrow top, the crest or ridgecrest, wi ...

. The

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

is a

cross-polytope In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular, convex polytope that exists in ''n''- dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a reg ...

The hypercubic honeycombs are

self-dual In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involution ...

Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

named this family as for an -dimensional honeycomb.

Wythoff construction classes by dimension

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 ...

is a method for constructing a

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruence (geometry), congruent. Uniform po ...

or plane tiling. The two general forms of the hypercube honeycombs are the ''regular'' form with identical hypercubic facets and one ''semiregular'', with alternating hypercube 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 ...

. A third form is generated by an

expansion Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansi ...

operation applied to the regular form, creating facets in place of all lower-dimensional elements. For example, an ''expanded cubic honeycomb'' has cubic cells centered on the original cubes, on the original faces, on the original edges, on the original vertices, creating 4 colors of cells around in vertex in 1:3:3:1 counts. The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each of the three axial directions may have different edge lengths. The facets are

hyperrectangle In geometry, a hyperrectangle (also called a box, hyperbox, k-cell or orthotopeCoxeter, 1973), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient cond ...

s, also called orthotopes; in 2 and 3 dimensions the orthotopes are

rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...

s and

cuboid In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...

s respectively. }
(2 colors) , - , ,

Apeirogon In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an in ...

, , , , , - , ,

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 ...

, , ,
, , - , ,

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 ...

, , ,
, , - , , '' 4-cube honeycomb'' , , ,
, , - , , '' 5-cube honeycomb'' , , ,
, , - , , ''

6-cube honeycomb The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean 6-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space. Cons ...

'' , , ,
, , - , , '' 7-cube honeycomb'' , , ,
, , - , , ''

8-cube honeycomb In geometry, the 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Euclidean space, Euclidean 8-space. It is analogous to the square tiling of the plane and to the cubi ...

'' , , ,
, , - , , -''hypercubic honeycomb'' , , colspan=2, ...

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, *# pp. 122–123. (The lattice of hypercubes γ_n form the ''cubic honeycombs'', δ_n+1) *# pp. 154–156: Partial truncation or alternation, represented by ''h'' prefix: h=; h=, h= *# p. 296, Table II: Regular honeycombs, δ_n+1 {{Honeycombs Honeycombs (geometry) Polytopes Regular tessellations

Wythoff construction classes by dimension

See also

References