# Pentagonal-hexagonal antiprismatic duoprism

Pentagonal-hexagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPehap
Coxeter diagramx5o s2s12o
Elements
Tera5 hexagonal antiprismatic prisms, 12 triangular-pentagonal duoprisms, 2 pentagonal-hexagonal duoprisms
Cells60 triangular prisms, 12+12 pentagonal prisms, 10 hexagonal prisms, 5 hexagonal antiprisms
Faces60 triangles, 60+60 squares, 12 pentagons, 10 hexagons
Edges60+60+60
Vertices60
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 3 (base trapezoid), (1+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {25+5{\sqrt {3}}+2{\sqrt {5}}}{20}}}\approx 1.38080}$
Hypervolume${\displaystyle {\frac {\sqrt {10(5+5{\sqrt {3}}+2{\sqrt {5}}+2{\sqrt {15}})}}{4}}\approx 4.02169}$
Diteral anglesTrapedip–pip–trapedip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {3}}}{3}}\right)\approx 145.22189^{\circ }}$
Happip–hap–happip: 108°
Trapedip–pip–phiddip: = ${\displaystyle \arccos \left({\frac {3-2{\sqrt {3}}}{3}}\right)\approx 98.89943^{\circ }}$
Trapedip–trip–happip: 90°
Phiddip–hip–happip: 90°
Height${\displaystyle {\sqrt {{\sqrt {3}}-1}}\approx 0.85560}$
Central density1
Number of external pieces19
Level of complexity40
Related polytopes
ArmyPehap
RegimentPehap
DualPentagonal-hexagonal antitegmatic duotegum
ConjugatePentagrammic-hexagonal antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×I2(12)×A1+, order 240
ConvexYes
NatureTame

The pentagonal-hexagonal antiprismatic duoprism or pehap is a convex uniform duoprism that consists of 5 hexagonal antiprismatic prisms, 2 pentagonal-hexagonal duoprisms, and 12 triangular-pentagonal duoprisms. Each vertex joins 2 hexagonal antiprismatic prisms, 3 triangular-pentagonal duoprisms, and 1 pentagonal-hexagonal duoprism.

## Vertex coordinates

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

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

## Representations

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

• x5o s2s12o (full symmetry; hexagonal antiprisms as alternated dodecagonal prisms)
• x5o s2s6s (hexagonal antiprisms as alternated dihexagonal prisms)