# Pentagonal-pentagonal antiprismatic duoprism

Pentagonal-pentagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPepap
Coxeter diagramx5o s2s10o
Elements
Tera5 pentagonal antiprismatic prisms, 10 triangular-pentagonal duoprisms, 2 pentagonal duoprisms
Cells50 triangular prisms, 10+10+10 pentagonal prisms, 5 pentagonal antiprisms
Faces50 triangles, 50+50 squares, 10+10 pentagons
Edges50+50+50
Vertices50
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+5)/2 (base trapezoid), (1+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {45+9{\sqrt {5}}}{40}}}\approx 1.27598}$
Hypervolume${\displaystyle {\frac {5{\sqrt {85+38{\sqrt {5}}}}}{24}}\approx 2.71610}$
Diteral anglesTrapedip–pip–trapedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Pappip–pap–pappip: 108°
Trapedip–pip–pedip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Trapedip–trip–pappip: 90°
Pedip–pip–pappip: 90°
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces17
Level of complexity40
Related polytopes
ArmyPepap
RegimentPepap
DualPentagonal-pentagonal antitegmatic duotegum
ConjugatePentagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×I2(10)×A1+, order 200
ConvexYes
NatureTame

The pentagonal-pentagonal antiprismatic duoprism or pepap is a convex uniform duoprism that consists of 5 pentagonal antiprismatic prisms, 2 pentagonal duoprisms, and 10 triangular-pentagonal duoprisms. Each vertex joins 2 pentagonal antiprismatic prisms, 3 triangular-pentagonal duoprisms, and 1 pentagonal duoprism.

## Vertex coordinates

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

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

## Representations

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

• x5o s2s10o (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x5o s2s5s (pentagonal antiprisms as alternated dipentagonal prisms)