# Pentagonal-hexagonal duoprismatic prism

Pentagonal-hexagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymPehip
Coxeter diagramx x5o x6o
Elements
Tera6 square-pentagonal duoprisms, 5 square-hexagonal duoprisms, 2 pentagonal-hexagonal duoprisms
Cells30 cubes,5+10 hexagonal prisms, 6+12 pentagonal prisms
Faces30+30+60 squares, 10 hexagons, 12 pentagons
Edges30+60+60
Vertices60
Vertex figureDigonal disphenoidal pyramid, edge lengths (1+5)/2 (disphenoid base 1), 3 (disphenoid base 2), 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {35+2{\sqrt {5}}}{20}}}\approx 1.40485}$
Hypervolume${\displaystyle {\frac {3{\sqrt {75+30{\sqrt {5}}}}}{8}}\approx 4.46993}$
Diteral anglesSquipdip–pip–squipdip: 120°
Shiddip–hip–shiddip: 108°
Shiddip–cube–squipdip: 90°
Phiddip–pip–squipdip: 90°
Shiddip–hip–phiddip: 90°
Height1
Central density1
Number of external pieces13
Level of complexity30
Related polytopes
ArmyPehip
RegimentPehip
DualPentagonal-hexagonal duotegmatic tegum
ConjugatePentagrammic-hexagonal duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×G2×A1, order 240
ConvexYes
NatureTame

The pentagonal-hexagonal duoprismatic prism or pehip, also known as the pentagonal-hexagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal-hexagonal duoprisms, 5 square-hexagonal duoprisms, and 6 square-pentagonal duoprisms. Each vertex joins 2 square-pentagonal duoprisms, 2 square-hexagonal duoprisms, and 1 pentagonal-hexagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a pentagonal-hexagonal duoprismatic prism of edge length 1 are given by:

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

## Representations

A pentagonal-hexagonal duoprismatic prism has the following Coxeter diagrams:

• x x5o x6o (full symmetry)
• x x5o x3x (hexagons as ditrigons)
• xx5oo xx6oo&#x (pentagonal-hexagonal duoprism atop pentagonal-hexagonal duoprism)
• xx5oo xx3xx&#x