Truncated tetrahedral-octahedral honeycomb

Truncated tetrahedral-octahedral honeycomb
Rank4
Typeuniform
SpaceEuclidean
Notation
Bowers style acronymTatoh
Coxeter diagram
Elements
Cells2N truncated tetrahedra, N cuboctahedra, N truncated octahedra
Faces8N triangles, 6N squares, 8N hexagons
Edges6N+24N
Vertices12N
Vertex figureRectangular pyramid, base edge lengths 1 and 2, side edge lengths 3
Measures (edge length 1)
Vertex density${\displaystyle {\frac {4{\sqrt {2}}}{9}}\approx 0.62853936105}$
Dual cell volume${\displaystyle {\frac {9{\sqrt {2}}}{8}}\approx 1.59099025767}$
Related polytopes
ArmyTatoh
RegimentTatoh
DualRhombic pyramidal honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryS4
ConvexYes
NatureTame

The truncated tetrahedral-octahedral honeycomb, or tatoh, also known as the cantic cubic honeycomb, is a convex uniform honeycomb. 1 cuboctahedron, 2 truncated octahedra, and 2 truncated tetrahedra join at each vertex of this honeycomb. As one of its names suggests, it can be formed by truncation of the tetrahedral-octahedral honeycomb, or as an alternated faceting from the small rhombated cubic honeycomb.

Vertex coordinates

The vertices of a truncated tetrahedral-octahedral honeycomb of edge length 1 are given by all permutations of:

• ${\displaystyle \left(\pm {\sqrt {2}}+3{\sqrt {2}}i,\,\pm {\frac {\sqrt {2}}{2}}+3{\sqrt {2}}j,\,3{\sqrt {2}}k\right),}$
• ${\displaystyle \left(\pm {\frac {3{\sqrt {2}}}{2}}+3{\sqrt {2}}i,\,\pm {\frac {\sqrt {2}}{2}}+3{\sqrt {2}}j,\,\pm {\frac {\sqrt {2}}{2}}+3{\sqrt {2}}k\right),}$

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

Representations

A truncated tetrahedral-octahedral honeycomb has the following Coxeter diagrams:

• (full symmetry)
• (P4 symmetry, as truncated cyclotetrahedral honeycomb)
• (as cantic cubic honeycomb)