# Pentagonal-dodecahedral duoprism

Pentagonal-dodecahedral duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPedoe
Coxeter diagramx5o x5o3o
Elements
Tera12 pentagonal duoprisms, 5 dodecahedral prisms
Cells30+60 pentagonal prisms, 5 dodecahedra
Faces150 squares, 20+60 pentagons
Edges100+150
Vertices100
Vertex figureTriangular scalene, edge lengths (1+5)/2 (base triangle and top), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {65+19{\sqrt {5}}}{40}}}\approx 1.63925}$
Hypervolume${\displaystyle 5{\sqrt {\frac {445+199{\sqrt {5}}}{128}}}\approx 13.18422}$
Diteral anglesPedip–pip–pedip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{5}}\right)\approx 116.56505^{\circ }}$
Dope–doe–dope: 108°
Pedip–pip–dope: 90°
Central density1
Number of external pieces17
Level of complexity10
Related polytopes
ArmyPedoe
RegimentPedoe
DualPentagonal-icosahedral duotegum
ConjugatePentagrammic-great stellated dodecahedral duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH3×H2, order 1200
ConvexYes
NatureTame

The pentagonal-dodecahedral duoprism or pedoe is a convex uniform duoprism that consists of 5 dodecahedral prisms and 12 pentagonal duoprisms. Each vertex joins 2 dodecahedral prisms and 3 pentagonal duoprisms.

## Vertex coordinates

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

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

as well as all even permutations of the last three coordinates of:

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