Hexagonal-cuboctahedral duoprism

Hexagonal-cuboctahedral duoprism
	(OFF file)
Rank	5
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Haco
Coxeter diagram	x6o o4x3o
Elements
Tera	8 triangular-hexagonal duoprisms, 6 square-hexagonal duoprisms, 6 cuboctahedral prisms
Cells	48 triangular prisms, 36 cubes, 24 hexagonal prisms, 6 cuboctahedra
Faces	48 triangles, 36+144 squares, 12 hexagons
Edges	72+144
Vertices	72
Vertex figure	Rectangular scalene, edge lengths 1, √2, 1, √2 (base rectangle), √3 (top), √2 (side edges)
Measures (edge length 1)
Circumradius
Hypervolume
Diteral angles	Thiddip–hip–shiddip:
	Cope–co–cope: 120°
	Thiddip–trip–cope: 90°
	Shiddip–cube–cope: 90°
Central density	1
Number of pieces	20
Level of complexity	20
Related polytopes
Army	Haco
Regiment	Haco
Dual	Hexagonal-rhombic dodecahedral duotegum
Conjugate	Hexagonal-cuboctahedral duoprism
Abstract properties
Euler characteristic	2
Topological properties
Orientable	Yes
Properties
Symmetry	B3×G2, order 576
Convex	Yes
Nature	Tame

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[edit | edit source]

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

$\left(0,\,±1,\,0,\,±\frac{\sqrt2}2,\,±\frac{\sqrt2}2\right),$
$\left(±\frac{\sqrt3}2,\,±\frac12,\,0,\,±\frac{\sqrt2}2,\,±\frac{\sqrt2}2\right).$

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)