# Hexagonal-truncated cubic duoprism

Hexagonal-truncated cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHatic
Coxeter diagramx6o x4x3o
Elements
Tera8 triangular-hexagonal duoprisms, 6 hexagonal-octagonal duoprisms, 6 truncated cubic prisms
Cells48 triangular prisms, 12+24 hexagonal prisms, 36 octagonal prisms, 6 truncated cubes
Faces48 triangles, 72+144 squares, 24 hexagons, 36 octagons
Edges72+144+144
Vertices144
Vertex figureDigonal disphenoidal pyramid, edge lengths 1, 2+2, 2+2 (base triangle), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {11+4{\sqrt {2}}}}{2}}\approx 2.04064}$
Hypervolume${\displaystyle 7{\frac {3{\sqrt {3}}+2{\sqrt {6}}}{2}}\approx 35.33296}$
Diteral anglesThiddip–hip–hodip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Ticcup–tic–ticcup: 120°
Thiddip–trip–ticcup: 90°
Hodip–op–ticcup: 90°
Hodip–hip–hodip: 90°
Central density1
Number of external pieces20
Level of complexity30
Related polytopes
ArmyHatic
RegimentHatic
DualHexagonal-triakis octahedral duotegum
ConjugateHexagonal-quasitruncated hexahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryB3×G2, order 576
ConvexYes
NatureTame

The hexagonal-truncated cubic duoprism or hatic is a convex uniform duoprism that consists of 6 truncated cubic prisms, 6 hexagonal-octagonal duoprisms and 8 triangular-hexagonal duoprisms. Each vertex joins 2 truncated cubic prisms, 1 triangular-hexagonal duoprism, and 2 hexagonal-octagonal duoprisms.

## Vertex coordinates

The vertices of a hexagonal-truncated cubic duoprism of edge length 1 are given by all permutations and sign changes of the last three coordinates of: The vertices of a hexagonal-truncated cubic duoprism of edge length 1 are given by all permutations of the last three coordinates of:

• ${\displaystyle \left(\pm 1,\,0,\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right).}$

## Representations

A hexagonal-truncated cubic duoprism has the following Coxeter diagrams:

• x6o x4x3o (full symmetry)
• x3x x4x3o (hexagons as ditrigons)