geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 of ge ...

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 coor ...

al spaces with the

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 mor ...

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 (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions ...

s per

ridge A ridge or a mountain ridge is a geographical feature consisting of a chain of mountains or hills that form a continuous elevated crest for an extended distance. The sides of the ridge slope away from the narrow top on either side. The line ...

. The

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 ...

is a cross-polytope The hypercubic honeycombs are self-dual.

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 to ...

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 faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...

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; 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 ...

. A third form is generated by an expansion 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, an orthotopeCoxeter, 1973 (also called a hyperrectangle or a box) is the generalization of a rectangle to higher dimensions. A necessary and sufficient condition is that it is congruent to the Cartesian product of intervals. If all ...

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

cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cu ...

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

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 th ...

, , ,
, , - , ,

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 ...

, , ,
, , - , , ''

4-cube 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 constructed by a 4-dimensional packing of tesseract facets. Its ve ...

'' , , ,
, , - , , ''

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 ...

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

8-cube honeycomb The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space. It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic ho ...

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

References

* Coxeter, H.S.M. '' Regular Polytopes'', (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