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	$60-42{\sqrt {2}}\approx 0.60303$
Dual cell volume	${\frac {10+7{\sqrt {2}}}{12}}\approx 1.65829$
Related polytopes
Army	Prich
Regiment	Prich
Dual	Kite pyramidal honeycomb
Conjugate	Quasiprismatoquasirhombated cubic honeycomb
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	R4
Convex	Yes
Nature	Tame

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:

• $\left(\pm {\frac {1}{2}}+(2+{\sqrt {2}})i,\,\pm {\frac {1+{\sqrt {2}}}{2}}+(2+{\sqrt {2}})j,\,\pm {\frac {1+{\sqrt {2}}}{2}}+(2+{\sqrt {2}})k\right)$,

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

Gallery

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  Wireframe

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  Orthogonal projection into Euclidean plane with V₃ symmetry

External links

• Klitzing, Richard. "prich".
• Wikipedia contributors. "Runcitruncated cubic honeycomb".
• Binnendyk, Eric. "Category 6: Prismatorhombates" (#110).