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
Rank	4
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Chon
Coxeter diagram
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

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:

• (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)

Gallery

• 

  Wireframe

• 

  Wireframe

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}

External links

• Klitzing, Richard. "chon".

• Wikipedia Contributors. "Cubic honeycomb".
• Binnendyk, Eric. "Category 1: Primaries" (#1).