Prismatorhombated cubic honeycomb

Prismatorhombated cubic honeycomb
Rank	4
Type	Uniform
Space	Euclidean
Notation
Bowers style acronym	Prich
Coxeter diagram	x4x3o4x ()
Elements
Cells	3N cubes, 3N octagonal prisms, N truncated cubes, N small rhombicuboctahedra
Faces	8N triangles, 6N+12N+12N squares, 6N octagons
Edges	12N+24N+24N
Vertices	24N
Vertex figure	Isosceles trapezoidal pyramid, base edge lengths 1, √2, √2, √2, side edge lengths √2, √2, √2+√2, √2+√2
Measures (edge length 1)
Vertex density	$60-42\sqrt2 \approx 0.60303038033$
Dual cell volume	$\frac{10+7\sqrt2}{12} \approx 1.65829124472$
Related polytopes
Army	Prich
Regiment	Prich
Dual	Kite pyramidal honeycomb
Conjugate	Quasiprismatoquasirhombated cubic honeycomb
Topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes

The prismatorhombated cubic honeycomb, or prich, also known as the runcitruncated cubic honeycomb, is a convex uniform honeycomb. 1 small rhombicuboctahedron, 1 truncated cube, 1 cube, and 2 octagonal prisms join at each vertex of this honeycomb. As the name suggests, it is the runcitruncation of the cubic honeycomb.

This honeycomb can be subsymmetrically faceted into a runcicantic snub cubic honeycomb, although it cannot be made scaliform.

Vertex coordinates

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

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

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

Gallery

• 

  Wireframe

• 
• 

  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. "prich".

• Wikipedia Contributors. "Runcitruncated cubic honeycomb".
• Binnendyk, Eric. "Category 6: Prismatorhombates" (#110).