# Hexagonal-hexagonal antiprismatic duoprism

Hexagonal-hexagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHahap
Coxeter diagramx6o s2s12o
Elements
Tera6 hexagonal antiprismatic prisms, 12 triangular-hexagonal duoprisms, 2 hexagonal duoprisms
Cells72 triangular prisms, 12+12+12 hexagonal prisms, 6 hexagonal antiprisms
Faces72 triangles, 72+72 squares, 12+12 hexagons
Edges72+72+72
Vertices72
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 3 (base trapezoid), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {7+{\sqrt {3}}}}{2}}\approx 1.47750}$
Hypervolume${\displaystyle {\frac {3{\sqrt {6+6{\sqrt {3}}}}}{2}}\approx 6.07311}$
Diteral anglesThiddip–hip–thiddip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
Happip–hap–happip: 120°
Thiddip–hip–hiddip: = ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Thiddip–trip–happip: 90°
Hiddip–hip–happip: 90°
HeightHiddip atop gyro hiddip: ≈ ${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces20
Level of complexity40
Related polytopes
ArmyHahap
RegimentHahap
DualHexagonal-hexagonal antitegmatic duotegum
ConjugateHexagonal-hexagonal antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(12)×A1+, order 288
ConvexYes
NatureTame

The hexagonal-hexagonal antiprismatic duoprism or hahap is a convex uniform duoprism that consists of 6 hexagonal antiprismatic prisms, 2 hexagonal duoprisms, and 12 triangular-hexagonal duoprisms. Each vertex joins 2 hexagonal antiprismatic prisms, 3 triangular-hexagonal duoprisms, and 1 hexagonal duoprism.

## Vertex coordinates

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

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

## Representations

A hexagonal-hexagonal antiprismatic duoprism has the following Coxeter diagrams:

• x6o s2s12o (full symmetry; hexagonal antiprisms as alternated dodecagonal prisms)
• x6o s2s6s (hexagonal antiprisms as alternated dihexagonal prisms)
• x3x s2s12o (hexagons as ditrigons)
• x3x s2s6s