Rectified cubic honeycomb

Rectified cubic honeycomb
Rank	4
Type	Uniform
Space	Euclidean
Notation
Bowers style acronym	Rich
Coxeter diagram	o4x3o4o ()
Elements
Cells	N octahedra, N cuboctahedra
Faces	8N triangles, 3N squares
Edges	12N
Vertices	3N
Vertex figure	Square prism, edge lengths 1 (base) and √2 (sides)
Measures (edge length 1)
Vertex density	$\frac{3\sqrt2}4 \approx 1.06066017180$
Dual cell volume	$\frac{2\sqrt2}3 \approx 0.94280904158$
Related polytopes
Army	Rich
Regiment	Rich
Dual	Joined cubic honeycomb
Conjugate	None
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes

The rectified cubic honeycomb, or rich, is a convex uniform honeycomb. 2 octahedra and 4 cuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the rectification of the cubic honeycomb. It is also the rectification of the tetrahedral-octahedral honeycomb.

Vertex coordinates

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

• $\left(\sqrt2i,\,±\frac{\sqrt2}{2}+\sqrt2j,\,±\frac{\sqrt2}{2}+\sqrt2k\right),$

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

Representations

A rectified cubic honeycomb has the following Coxeter diagrams:

• (full symmetry)
• (S₄ symmetry, as rectified tetrahedral-octahedral honeycomb)
• (S₄ symmetry)
• (P₄ symmetry, as rectified cyclotetrahedral honeycomb)
• (as alternated faceting)
• (as alternated faceting)
• qo4ox3xo4oq&#zx

Gallery

• 

  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. "rich".

• Wikipedia Contributors. "Rectified cubic honeycomb".
• Binnendyk, Eric. "Category 3: Rich regiment" (#22).