# Hexagonal-pentagonal antiprismatic duoprism

Hexagonal-pentagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHapap
Coxeter diagramx6o s2s10o
Elements
Tera6 pentagonal antiprismatic prisms, 10 triangular-hexagonal duoprisms, 2 pentagonal-hexagonal duoprisms
Cells60 triangular prisms, 12 pentagonal prisms, 6 pentagonal antiprisms, 10+10 hexagonal prisms
Faces60 triangles, 60+60 squares, 12 pentagons, 10 hexagons
Edges60+60+60
Vertices60
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+5)/2 (base trapezoid), 3 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {13+{\sqrt {5}}}{8}}}\approx 1.38004}$
Hypervolume${\displaystyle {\frac {5{\sqrt {3}}+2{\sqrt {15}}}{4}}\approx 4.10156}$
Diteral anglesThiddip–hip–thiddip: = ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Pappip–pap–pappip: 120°
Thiddip–hip–phiddip: = ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Thiddip–trip–pappip: 90°
Phiddip–pip–pappip: 90°
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces18
Level of complexity40
Related polytopes
ArmyHapap
RegimentHapap
DualHexagonal-pentagonal antitegmatic duotegum
ConjugateHexagonal-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryG2×I2(10)×A1+, order 240
ConvexYes
NatureTame

The hexagonal-pentagonal antiprismatic duoprism or hapap is a convex uniform duoprism that consists of 6 pentagonal antiprismatic prisms, 2 pentagonal-hexagonal duoprisms, and 10 triangular-hexagonal duoprisms. Each vertex joins 2 pentagonal antiprismatic prisms, 3 triangular-hexagonal duoprisms, and 1 pentagonal-hexagonal duoprism.

## Vertex coordinates

The vertices of a hexagonal-pentagonal antiprismatic duoprism of edge length 1 are given by all central inversions of the last three coordinates of:

• ${\displaystyle \left(0,\,\pm 1,\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(0,\,\pm 1,\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right).}$

## Representations

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

• x6o s2s10o (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x6o s2s5s (pentagonal antiprisms as alternated dipentagonal prisms)
• x3x s2s10o (hexagons as ditrigons)
• x3x s2s5s