Pentagonal-octagonal duoprism

Pentagonal-octagonal duoprism
(OFF file)
Rank	4
Type	Uniform
Notation
Bowers style acronym	Podip
Coxeter diagram	x5o x8o ()
Elements
Cells	8 pentagonal prisms, 5 octagonal prisms
Faces	40 squares, 8 pentagons, 5 octagons
Edges	40+40
Vertices	40
Vertex figure	Digonal disphenoid, edge lengths (1+√5)/2 (base 1), √2+√2 (base 2), and √2 (sides)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {15+5{\sqrt {2}}+{\sqrt {5}}}{10}}}\approx 1.55908$
Hypervolume	${\frac {\sqrt {75+50{\sqrt {2}}+30{\sqrt {5}}+20{\sqrt {10}}}}{2}}\approx 8.30720$
Dichoral angles	Pip–5–pip: 135°
	Op–8–op: 108°
	Pip–4–op: 90°
Central density	1
Number of external pieces	13
Level of complexity	6
Related polytopes
Army	Podip
Regiment	Podip
Dual	Pentagonal-octagonal duotegum
Conjugates	Pentagonal-octagrammic duoprism, Pentagrammic-octagonal duoprism, Pentagrammic-octagrammic duoprism
Abstract & topological properties
Flag count	960
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	H2×I2(8), order 160
Flag orbits	6
Convex	Yes
Nature	Tame

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.

Gallery

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  Net

Vertex coordinates

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

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

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)

External links

• Bowers, Jonathan. "Category A: Duoprisms".

• Klitzing, Richard. "podip".