# Truncated cubic honeycomb

Truncated cubic honeycomb
Rank4
Typeuniform
SpaceEuclidean
Notation
Bowers style acronymTich
Coxeter diagramx4x3o4o ()
Elements
CellsN octahedra, N truncated cubes
Faces8N triangles, 3N octagons
Edges3N+12N
Vertices6N
Vertex figureSquare pyramid, edge lengths 1 (base) and 2+2 (sides)
Measures (edge length 1)
Vertex density${\displaystyle 30{\sqrt {2}}-42\approx 0.42641}$
Dual cell volume${\displaystyle {\frac {7+5{\sqrt {2}}}{6}}\approx 2.34518}$
Related polytopes
ArmyTich
RegimentTich
DualSquare pyramidal honeycomb
ConjugateQuasitruncated cubic honeycomb
Abstract & topological properties
OrientableYes
Properties
SymmetryR4
ConvexYes
NatureTame

The truncated cubic honeycomb, or tich, is a convex uniform honeycomb. 1 octahedron and 4 truncated cubes join at each vertex of this honeycomb. As the name suggests, it is the truncation of the cubic honeycomb.

## Vertex coordinates

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

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

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

## Representations

A truncated cubic honeycomb has the following Coxeter diagrams:

• x4x3o4o () (regular)
• x4x3o2o3*b () (S4 symmetry)
• wx4xo3ox4xw&#zx