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
Abstract & 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}