Cubic 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

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.

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}