Cubic 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
Dual cell volume
Related polytopes
Army	Chon
Regiment	Chon
Dual	Cubic honeycomb
Petrie dual	Mucubic honeycomb
Conjugate	None
Topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

Wireframe
Wireframe

Related polytopes[edit | edit source]

o4o3o4o truncations
Name	OBSA	Schläfli symbol
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	t_0,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t_0,1,3{4,3,4}
Prismatorhombated cubic honeycomb	prich	t_0,1,3{4,3,4}
Great prismated cubic honeycomb	gippich	t_0,1,2,3{4,3,4}