# Hexagonal-square antiprismatic duoprism

Hexagonal-square antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHasquap
Coxeter diagramx6o s2s8o
Elements
Tera6 square antiprismatic prisms, 8 triangular-hexagonal duoprisms, 2 square-hexagonal duoprisms
Cells48 triangular prisms, 12 cubes, 6 square antiprisms, 8+8 hexagonal prisms
Faces48 triangles, 12+48+48 squares, 8 hexagons
Edges48+48+48
Vertices48
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 2 (base trapezoid), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {12+{\sqrt {2}}}{8}}}\approx 1.29490}$
Hypervolume${\displaystyle {\frac {\sqrt {12+9{\sqrt {2}}}}{2}}\approx 2.48636}$
Diteral anglesThiddip–hip–thiddip: ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
Squappip–squap–squappip: 120°
Thiddip–hip–shiddip: = ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Thiddip–trip–squappip: 90°
Shiddip–cube–squappip: 90°
${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces16
Level of complexity40
Related polytopes
ArmyHasquap
RegimentHasquap
DualHexagonal-square antitegmatic duotegum
ConjugateHexagonal-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(8)×A1+, order 192
ConvexYes
NatureTame

The hexagonal-square antiprismatic duoprism or hasquap is a convex uniform duoprism that consists of 6 square antiprismatic prisms, 2 square-hexagonal duoprisms, and 8 triangular-hexagonal duoprisms. Each vertex joins 2 square antiprismatic prisms, 3 triangular-hexagonal duoprisms, and 1 square-hexagonal duoprism.

## Vertex coordinates

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

• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right).}$

## Representations

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

• x6o s2s8o (full symmetry; square antiprisms as alternated octagonal prisms)
• x6o s2s4s (square antiprisms as alternated ditetragonal prisms)
• x3x s2s8o (hexagons as ditrigons)
• x3x s2s4s