# Prismatorhombated cubic honeycomb

Prismatorhombated cubic honeycomb
Rank4
TypeUniform
SpaceEuclidean
Notation
Bowers style acronymPrich
Coxeter diagramx4x3o4x ()
Elements
Cells3N cubes, 3N octagonal prisms, N truncated cubes, N small rhombicuboctahedra
Faces8N triangles, 6N+12N+12N squares, 6N octagons
Edges12N+24N+24N
Vertices24N
Vertex figureIsosceles trapezoidal pyramid, base edge lengths 1, 2, 2, 2, side edge lengths 2, 2, 2+2, 2+2
Measures (edge length 1)
Vertex density${\displaystyle 60-42\sqrt2 \approx 0.60303038033}$
Dual cell volume${\displaystyle \frac{10+7\sqrt2}{12} \approx 1.65829124472}$
Related polytopes
ArmyPrich
RegimentPrich
DualKite pyramidal honeycomb
ConjugateQuasiprismatoquasirhombated cubic honeycomb
Topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes

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:

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

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

## 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}