# Pentagonal-octagonal duoprism

(Redirected from Podip)
Pentagonal-octagonal duoprism Rank4
TypeUniform
SpaceSpherical
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$\sqrt{\frac{15+5\sqrt2+\sqrt5}{10}} ≈ 1.55908$ Hypervolume$\frac{\sqrt{75+50\sqrt2+30\sqrt5+20\sqrt{10}}}{2} ≈ 8.30720$ 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
Euler characteristic0
OrientableYes
Properties
SymmetryH2×I2(8), order 160
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:

• $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$ • $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{1\sqrt2}{2},\,±\frac12\right),$ • $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$ • $\left(±frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$ ## Representations

A pentagonal-octagonal duoprism has the following Coxeter diagrams:

• x5o x8o (full symmetry)
• x4x x5o (octagons as ditetragons)
• ofx xxx8ooo&#xt (octagonal axial)
• ofx xxx4xxx&#xt (ditetragonal axial)