# Pentagonal-octagrammic duoprism

Pentagonal-octagrammic duoprism
Rank4
TypeUniform
Notation
Bowers style acronymPistodip
Coxeter diagramx5o x8/3o ()
Elements
Cells8 pentagonal prisms, 5 octagrammic prisms
Faces40 squares, 8 pentagons, 5 octagrams
Edges40+40
Vertices40
Vertex figureDigonal disphenoid, edge lengths (1+5)/2 (base 1), 2–2 (base 2), 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {15-5{\sqrt {2}}+{\sqrt {5}}}{10}}}\approx 1.00822}$
Hypervolume${\displaystyle {\frac {\sqrt {75-50{\sqrt {2}}+30{\sqrt {5}}-20{\sqrt {10}}}}{2}}\approx 8.30720}$
Dichoral anglesStop–8/3–stop: 108°
Pip–4–stop: 90°
Pip–5–pip: 45°
Central density3
Number of external pieces21
Level of complexity12
Related polytopes
ArmySemi-uniform podip
RegimentPistodip
DualPentagonal-octagrammic duotegum
ConjugatesPentagonal-octagonal duoprism, Pentagrammic-octagonal duoprism, Pentagrammic-octagrammic duoprism
Abstract & topological properties
Flag count960
Euler characteristic0
OrientableYes
Properties
SymmetryH2×I2(8), order 160
ConvexNo
NatureTame

The pentagonal-octagrammic duoprism, also known as pistodip or the 5-8/3 duoprism, is a uniform duoprism that consists of 8 pentagonal prisms and 5 octagrammic prisms, with 2 of each at each vertex.

## Vertex coordinates

The coordinates of a pentagonal-octagrammic duoprism, centered at the origin and with edge length 1, are given by:

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

## Representations

A pentagonal-octagrammic duoprism has the following Coxeter diagrams: