A regular square tiling.	A partial-filled cubic honeycomb Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation 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 regular octahedron....  in its regular form.
A checkboard square tiling	A partially-filled cubic honeycomb Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation 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 regular octahedron....  checkerboard.
An expanded square tiling	Expanded Expansion (geometry) In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...  cubic honeycomb

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a hypercubic honeycomb is a family of regular honeycombs (tessellation

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

s) in n-dimensions with the Schläfli symbols {4,3...3,4} and containing the symmetry of Coxeter group R_n (or B^~_n-1) for n>=3.

The tessellation is constructed from 4 n-hypercube

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, perpendicular to each other and of the same length.An...

s per ridge

Ridge (geometry)

In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a facet.By dimension, this corresponds to:*a vertex of a polygon;...

. The vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 is a cross-polytope

Cross-polytope

In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...

 {3...3,4}.

The hypercubic honeycombs are self-dual.

Coxeter named this family as δ_n+1 for an n-dimensional honeycomb.

Wythoff construction

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 :...

 classes by dimension

There are two general forms of the hypercube honeycombs, the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard

Checkerboard

A checkerboard or chequerboard is a board of chequered pattern on which English draughts is played. It is an 8×8 board and the 64 squares are of alternating dark and light color, often red and black....

.

A third form is generated by an expansion

Expansion (geometry)

In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...

 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 3 colors of cells around in vertex in 1:3:3:1 counts.

A more general class of honeycombs is called, orthotopic honeycombs, with identical topology, but allow each axial direction to have different edge lengths, for example with rectangle

Rectangle

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

 and cuboid

Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

 facets in 2 and 3 dimensions.

δn	Name	Schläfli symbols	Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s
			Orthotopic {∞}n (2m colors, m	Regular {4,3n-1,4} (1 color)	Checkerboard {4,3n-4,31,1} (2 colors)	Expanded Expansion (geometry) In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements... t0,n-1{4,3n-1,4} (n colors)
δ2	Apeirogon Apeirogon An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...	{∞}
δ3	Square tiling	{∞}2 {4,4}
δ4	Cubic honeycomb Cubic honeycomb The cubic honeycomb is the only regular space-filling tessellation 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 regular octahedron....	{∞}3 {4,3,4} {4,31,1}
δ5	Tesseractic honeycomb	{∞}4 {4,32,4} {4,3,31,1}
δ6	Penteractic honeycomb	{∞}5 {4,33,4} {4,32,31,1}
δ7	Hexeractic honeycomb	{∞}6 {4,34,4} {4,33,31,1}
δ8	Hepteractic honeycomb	{∞}7 {4,35,4} {4,34,31,1}
δ9	Octeractic honeycomb	{∞}8 {4,36,4} {4,35,31,1}
δ10	Enneractic honeycomb	{∞}9 {4,37,4} {4,36,31,1}
δ11	Dekeractic honeycomb	{∞}10 {4,38,4} {4,36,31,1}
δn	n-hypercubic honeycomb	{∞}n {4,3n-3,4} {4,3n-4,31,1}	...