# Cubic honeycomb

Cubic honeycomb
Rank4
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymChon
Coxeter diagramx4o3o4o ()
Schläfli symbol{4,3,4}
Elements
CellsN cubes
Faces3N squares
Edges3N
VerticesN
Vertex figureOctahedron, edge length 2
Measures (edge length 1)
Vertex density${\displaystyle 1}$
Dual cell volume${\displaystyle 1}$
Related polytopes
ArmyChon
RegimentChon
DualCubic honeycomb
Petrie dualMucubic honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes
NatureTame

The cubic honeycomb, or chon, is the only regular honeycomb or tessellation of 3D Euclidean space. 8 cubes join at each vertex of this honeycomb. It is also the 3D hypercubic honeycomb.

This honeycomb can be alternated into a tetrahedral-octahedral honeycomb, which is uniform.

## Vertex coordinates

The vertices of a cubic honeycomb of edge length 1 are given by

• ${\displaystyle (i,j,k)}$ in which ${\displaystyle \{i,j,k\}\in \mathbb {Z} }$.

## Representations

A cubic honeycomb has the following Coxeter diagrams:

• x4o3o4o () (regular)
• x4o3o4x () (as expanded cubic honecyomb)
• x4o3o2o3*b () (S4 symmetry)
• xØo2x4o4o () (various square prismatic honeycombs)
• xØo2o4x4o ()
• xØo2x4o4x ()
• xØx2x4o4o ()
• xØx2o4x4o ()
• xØx2x4o4x ()
• xØo2xØo2xØo () (various apeirogonal triprismatic honeycombs)
• xØx2xØo2xØo ()
• xØx2xØx2xØo ()
• xØx2xØx2xØx ()
• qo3oo3oq3oo3*a&#zx (as hull of two alternate tetrahedral-octahedral honeycombs)