# Pentagonal-octagonal duoprism

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Pentagonal-octagonal duoprism
Rank4
TypeUniform
Notation
Bowers style acronymPodip
Coxeter diagramx5o x8o ()
Elements
Cells8 pentagonal prisms, 5 octagonal prisms
Faces40 squares, 8 pentagons, 5 octagons
Edges40+40
Vertices40
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1), 2+2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15+5{\sqrt {2}}+{\sqrt {5}}}{10}}}\approx 1.55908}$
Hypervolume${\displaystyle {\frac {\sqrt {75+50{\sqrt {2}}+30{\sqrt {5}}+20{\sqrt {10}}}}{2}}\approx 8.30720}$
Dichoral anglesPip–5–pip: 135°
Op–8–op: 108°
Pip–4–op: 90°
Central density1
Number of external pieces13
Level of complexity6
Related polytopes
ArmyPodip
RegimentPodip
DualPentagonal-octagonal duotegum
ConjugatesPentagonal-octagrammic duoprism,
Pentagrammic-octagonal duoprism,
Pentagrammic-octagrammic duoprism
Abstract & topological properties
Flag count960
Euler characteristic0
OrientableYes
Properties
SymmetryH2×I2(8), order 160
Flag orbits6
ConvexYes
NatureTame

The pentagonal-octagonal duoprism or podip, also known as the 5-8 duoprism, is a uniform duoprism that consists of 5 octagonal prisms and 8 pentagonal prisms, with two of each joining at each vertex.

## Vertex coordinates

Coordinates for the vertices of a pentagonal-octagonal duoprism with edge length 1 are given by:

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

## Representations

A pentagonal-octagonal duoprism has the following Coxeter diagrams:

• x5o x8o () (full symmetry)
• x4x x5o () (B2×H2 symmetry, octagons as ditetragons)
• ofx xxx8ooo&#xt (I2(8)×A1 axial)
• ofx xxx4xxx&#xt (B2×A1 symmetry, ditetragonal axial)