# Small rhombi-tetrahedral-octahedral honeycomb

Small rhombi-tetrahedral-octahedral honeycomb
Rank4
Typeuniform
SpaceEuclidean
Notation
Bowers style acronymSratoh
Coxeter diagramx4o3x2o3*b ()
Elements
Cells2N tetrahedra, N cubes, N small rhombicuboctahedra
Faces8N triangles, 6N+6N squares
Edges12N+12N
Vertices8N
Vertex figureTriangular frustum, edge lengths 1 (top) and 2 (sides and base)
Measures (edge length 1)
Vertex density${\displaystyle 80-56{\sqrt {2}}\approx 0.80404050711}$
Dual cell volume${\displaystyle {\frac {10+7{\sqrt {2}}}{16}}\approx 1.24371843354}$
Related polytopes
ArmySratoh
RegimentSratoh
DualApiculated triangular pyramidal honeycomb
ConjugateQuasirhombi-tetrahedral-octahedral honeycomb
Abstract & topological properties
OrientableYes
Properties
SymmetryS4
ConvexYes
NatureTame

The small rhombi-tetrahedral-octahedral honeycomb, or sratoh, also known as the runcic cubic honeycomb, is a convex uniform honeycomb. 1 tetrahedron, 1 cube, and 3 small rhombicuboctahedra join at each vertex of this honeycomb. It can be formed as an alternated faceting from the cubic honeycomb, alternating a subset of the cubes, or alternately as the birectification of the tetrahedral-octahedral honeycomb.

It can be subdivided into alternating truncated square tiling prism and truncated square tiling alterprism segments.

## Vertex coordinates

The vertices of a small rhombi-tetrahedral-octahedral honeycomb of edge length 1, centered at a small rhombicuboctahedron, are given by all permutations of:

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

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

## Representations

A small rhombated tetrahedral-octahedral honeycomb has the following Coxeter diagrams:

• x4o3x2o3*b () (full symmetry)
• s4o3o4x () (as runcic cubic honeycomb)
• qo3xx3oq3oo&#zx