# Cubic honeycomb

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

Cubic honeycomb
Rank4
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymChon
Coxeter diagram
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
Topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes

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:

•         (regular)
•         (as expanded cubic honecyomb)
•       (S4 symmetry)
•           (various square prismatic honeycombs)
•
•
•
•
•
•             (various apeirogonal triprismatic honeycombs)
•
•
•
• qo3oo3oq3oo3*a&#zx (as hull of two alternate tetrahedral-octahedral honeycombs)

## Related polytopes

o4o3o4o truncations
Name OBSA Schläfli symbol CD diagram Picture
Cubic honeycomb chon {4,3,4}
Truncated cubic honeycomb tich t{4,3,4}
Rectified cubic honeycomb rich r{4,3,4}
Bitruncated cubic honeycomb batch 2t{4,3,4}
Rectified cubic honeycomb rich r{4,3,4}
Truncated cubic honeycomb tich t{4,3,4}
Cubic honeycomb chon {4,3,4}
Small rhombated cubic honeycomb srich rr{4,3,4}
Great rhombated cubic honeycomb grich tr{4,3,4}
Small rhombated cubic honeycomb srich rr{4,3,4}
Great rhombated cubic honeycomb grich tr{4,3,4}
Small prismated cubic honeycomb = Cubic honeycomb chon t0,3{4,3,4}
Prismatorhombated cubic honeycomb prich t0,1,3{4,3,4}
Prismatorhombated cubic honeycomb prich t0,1,3{4,3,4}
Great prismated cubic honeycomb gippich t0,1,2,3{4,3,4}