# Pentagonal duoprismatic prism

Pentagonal duoprismatic prism
Rank5
TypeUniform
Notation
Bowers style acronymPepip
Coxeter diagramx x5o x5o
Elements
Tera10 square-pentagonal duoprisms, 2 pentagonal duoprisms
Cells25 cubes, 10+20 pentagonal prisms
Faces50+50 squares, 20 pentagons
Edges25+100
Vertices50
Vertex figureTetragonal disphenoidal pyramid, edge lengths (1+5)/2 (disphenoid bases) and 2 (remaining edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {25+4{\sqrt {5}}}{20}}}\approx 1.30277}$
Hypervolume${\displaystyle {\frac {5(5+2{\sqrt {5}})}{16}}\approx 2.96004}$
Diteral anglesSquipdip–pip–squipdip: 108°
Squipdip–cube–squipdip: 90°
Pedip–pip–squipdip: 90°
Height1
Central density1
Number of external pieces12
Level of complexity15
Related polytopes
ArmyPepip
RegimentPepip
DualPentagonal duotegmatic tegum
ConjugatePentagrammic duoprismatic prism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2≀S2×A1, order 400
ConvexYes
NatureTame

The pentagonal duoprismatic prism or pepip, also known as the pentagonal-pentagonal prismatic duoprism, is a convex uniform duoprism that consists of 2 pentagonal duoprisms and 10 square-pentagonal duoprisms. Each vertex joins 4 square-pentagonal duoprisms and 1 pentagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates

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

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

## Representations

A pentagonal duoprismatic prism has the following Coxeter diagrams:

• x x5o x5o (full symmetry)
• xx5oo xx5oo&#x (pentagonal duoprism atop pentagonal duoprism)