Type	Regular honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,3,4}
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...
Cell type	{4,3} Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
Face type	{4} Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
Vertex figure	(octahedron Octahedron In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.... )
Cells/edge	{4,3}4
Faces/edge	44
Cells/vertex	{4,3}8
Faces/vertex	412
Edges/vertex	6
Euler characteristic Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...	0
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	, [4,3,4]
Dual	self-dual
Properties	vertex-transitive Vertex-transitive In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...

The cubic honeycomb is the only regular space-filling 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...

 (or honeycomb

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....

) in Euclidean 3-space, made up of cubic

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

 cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its 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 regular octahedron

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

.

It is a self-dual tessellation with Schläfli symbol {4,3,4}. It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane.

It is one of 28 uniform honeycombs

Convex uniform honeycomb

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...

 using convex uniform polyhedral

Uniform polyhedron

A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

 cells.

Related polytopes and tesellations

It is related to the regular 4-polytope tesseract

Tesseract

In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...

, Schläfli symbol {4,3,3}, which exists in 4-space, and only has 3 cubes around each edge. It's also related to the order-5 cubic honeycomb, Schläfli symbol {4,3,5}, of hyperbolic space

Hyperbolic space

In mathematics, hyperbolic space is a type of non-Euclidean geometry. Whereas spherical geometry has a constant positive curvature, hyperbolic geometry has a negative curvature: every point in hyperbolic space is a saddle point...

 with 5 cubes around each edge.

Uniform colorings

There is a large number of uniform coloring

Uniform coloring

In geometry, a uniform coloring is a property of a uniform figure that is colored to be vertex-transitive...

s, derived from different symmetries. These include:

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...	Schläfli symbol	Partial honeycomb	Colors by letters
	{4,3,4}		1: aaaa/aaaa
	{4,4}xt{∞}		2: aaaa/bbbb
	t1{4,4}x{∞}		2: abba/abba
	{4,31,1}		2: abba/baab
	t{∞}xt{∞}x{∞}		4: abcd/abcd
	t0,3{4,3,4}		4: abbc/bccd
	t{∞}xt{∞}xt{∞}		8: abcd/efgh

See also

• List of regular polytopes
• Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge