Great rhombated cubic honeycomb

Great rhombated cubic honeycomb
Rank	4
Type	Uniform
Space	Euclidean
Notation
Bowers style acronym	Grich
Coxeter diagram	x4x3x4o ()
Elements
Cells	3N cubes, N truncated octahedra, N great rhombicuboctahedra
Faces	6N+12N squares, 8N hexagons, 3N octagons
Edges	12N+12N+24N
Vertices	24N
Vertex figure	Sphenoid, edge lengths √2 (3), √3 (2), and √2+√2 (1)
Measures (edge length 1)
Vertex density	$\frac{528\sqrt2-600}{343} \approx 0.42771067327$
Dual cell volume	$\frac{25+22\sqrt2}{24} \approx 2.33802909884$
Related polytopes
Army	Grich
Regiment	Grich
Dual	Sphenoidal honeycomb
Conjugate	Great quasirhombated cubic honeycomb
Topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes

The great rhombated cubic honeycomb, or grich, also known as the cantitruncated cubic honeycomb, is a convex uniform honeycomb. 1 truncated octahedron, 1 cube, and 2 great rhombicuboctahedra join at each vertex of this honeycomb. As the name suggests, it is the cantitruncation of the cubic honeycomb.

This honeycomb can be alternated into a snub rectified cubic honeycomb, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a cantic snub cubic honeycomb, which is also nonuniform.

Vertex coordinates

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

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

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

Representations

A great rhombated cubic honeycomb has the following Coxeter diagrams:

• x4x3x4o (regular)
• x3x3x *b4x (S4 symmetry)
• s4x3x4x (as alternated faceting)

Gallery

• 

  Wireframe

• 
• 

  The four cells meeting at each vertex

• 

  Orthogonal projection into Euclidean plane with V3 symmetry

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. "grich".

• Wikipedia Contributors. "Cantitruncated cubic honeycomb".
• Binnendyk, Eric. "Category 5: Greater Truncates" (#92).