# Hexagonal-cubic duoprism

Hexagonal-cubic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHacube
Coxeter diagramx6o x4o3o ()
Elements
Tera6 tesseracts, 6 square-hexagonal duoprisms
Cells6+36 cubes, 12 hexagonal prisms
Faces36+72 squares, 8 hexagons
Edges48+72
Vertices48
Vertex figureTriangular scalene, edge lengths 3 (top), 2 (base triangle and sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7}}{2}}\approx 1.32288}$
Hypervolume${\displaystyle {\frac {3{\sqrt {3}}}{2}}\approx 2.59808}$
Diteral anglesTes–cube–tes: 120°
Tes–cube–shiddip: 90°
Shiddip–hip–shiddip: 90°
Height1
Central density1
Number of external pieces12
Level of complexity10
Related polytopes
ArmyHacube
RegimentHacube
DualHexagonal-octahedral duotegum
ConjugateHexagonal-cubic duoprism
Abstract & topological properties
Flag count5760
Euler characteristic2
OrientableYes
Properties
SymmetryB3×G2, order 576
ConvexYes
NatureTame

The hexagonal-cubic duoprism or hacube, also known as a square-hexagonal duoprismatic prism, is a convex uniform duoprism that consists of 6 tesseracts and 6 square-hexagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-hexagonal duoprisms. It is a duoprism based on a square and a hexagonal prism, which makes it a convex segmentoteron.

This polyteron can be alternated into a triangular-tetrahedral duoantiprism, although it cannot be made uniform.

## Vertex coordinates

The vertices of a hexagonal-cubic duoprism of edge length 1 are given by:

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

## Representations

A hexagonal-cubic duoprism has the following Coxeter diagrams:

• x6o x4o3o (full symmetry)
• x3x x4o3o (hexagons as ditrigons)
• x x4o x6o (square-hexagonal duoprismatic prism)
• x x3x x4o
• x x x x6o (hexagonal prismatic prismatic prism)
• x x x x3x
• xx4oo xx6oo&#x (square-hexagonal duoprism atop square-hexagonal duoprism)
• xx xx xx6oo&#x
• xx4oo xx3xx&#x
• xx xx xx3xx&#x