Cubic honeycomb

Cubic honeycomb
Rank	4
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Chon
Coxeter diagram	x4o3o4o ()
Schläfli symbol	{4,3,4}
Elements
Cells	N cubes
Faces	3N squares
Edges	3N
Vertices	N
Vertex figure	Octahedron, edge length √2
Measures (edge length 1)
Vertex density	$1$
Dual cell volume	$1$
Related polytopes
Army	Chon
Regiment	Chon
Dual	Cubic honeycomb
Petrie dual	Mucubic honeycomb
Conjugate	None
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes
Nature	Tame

For analogs in other dimensions, see Hypercubic honeycomb.

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

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

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)

Gallery

• 

  Wireframe

• 

  Wireframe

External links

• Klitzing, Richard. "chon".
• Wikipedia contributors. "Cubic honeycomb".
• Binnendyk, Eric. "Category 1: Primaries" (#1).