Small rhombated cubic honeycomb

Small rhombated cubic honeycomb
Rank	4
Type	uniform
Space	Euclidean
Notation
Bowers style acronym	Srich
Coxeter diagram	x4o3x4o ()
Elements
Cells	3N cubes, N cuboctahedra, N small rhombicuboctahedra
Faces	8N triangles, 3N+6N+12N squares
Edges	12N+24N
Vertices	12N
Vertex figure	Rectangular wedge, edge lengths 1 (two edges of base) and √2 (remaining edges)
Measures (edge length 1)
Vertex density
Dual cell volume
Related polytopes
Army	Srich
Regiment	Srich
Dual	Notch honeycomb
Conjugate	Quasirhombated cubic honeycomb
Topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes

The small rhombated cubic honeycomb, or srich, also known as the cantellated cubic honeycomb, is a convex uniform honeycomb. 1 cuboctahedron, 2 small rhombicuboctahedra, and 2 cubes join at each vertex of this honeycomb. As the name suggests, it is the cantellation of the cubic honeycomb.

Vertex coordinates[edit | edit source]

The vertices of a small rhombated cubic honeycomb of edge length 1 are given by all permutations of:

$\left(±\frac12+(1+\sqrt2)i,\,±\frac12+(1+\sqrt2)j,\,±\frac{1+\sqrt2}{2}+(1+\sqrt2)k\right),$

Where i, j, and k range over the integers.

A small rhombated cubic honeycomb has the following Coxeter diagrams:

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}