# Hexagonal-cuboctahedral duoprism

Hexagonal-cuboctahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHaco
Coxeter diagramx6o o4x3o ()
Elements
Tera8 triangular-hexagonal duoprisms, 6 square-hexagonal duoprisms, 6 cuboctahedral prisms
Cells48 triangular prisms, 36 cubes, 24 hexagonal prisms, 6 cuboctahedra
Faces48 triangles, 36+144 squares, 12 hexagons
Edges72+144
Vertices72
Vertex figureRectangular scalene, edge lengths 1, 2, 1, 2 (base rectangle), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {2}}\approx 1.41421}$
Hypervolume${\displaystyle {\frac {5{\sqrt {6}}}{2}}\approx 6.12372}$
Diteral anglesThiddip–hip–shiddip: ${\displaystyle \arccos \left(-{\frac {\sqrt {3}}{3}}\right)\approx 125.26439^{\circ }}$
Cope–co–cope: 120°
Thiddip–trip–cope: 90°
Shiddip–cube–cope: 90°
Central density1
Number of external pieces20
Level of complexity20
Related polytopes
ArmyHaco
RegimentHaco
DualHexagonal-rhombic dodecahedral duotegum
Abstract & topological properties
Flag count11520
Euler characteristic2
OrientableYes
Properties
SymmetryB3×G2, order 576
ConvexYes
NatureTame

The hexagonal-cuboctahedral duoprism or haco is a convex uniform duoprism that consists of 6 cuboctahedral prisms, 6 square-hexagonal duoprisms, and 8 triangular-hexagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-hexagonal duoprisms, and 2 square-hexagonal duoprisms.

## Vertex coordinates

The vertices of a hexagonal-cuboctahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

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

## Representations

A hexagonal-cuboctahedral duoprism has the following Coxeter diagrams:

• x6o o4x3o (full symmetry)
• x3x o4x3o (hexagons as ditrigons)
• x6o x3o3x
• x3x x3o3x
• oxx3xxo xxx3xxx&#xt (triangular-hexagonal duoprism || pseudo hexagonal duoprism || gyro triangular-hexagonal duoprism)