# Octagonal-pentagonal antiprismatic duoprism

Octagonal-pentagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymOpap
Coxeter diagramx8o s2s10o
Elements
Tera8 pentagonal antiprismatic prisms, 10 triangular-octagonal duoprisms, 2 pentagonal-octagonal duoprisms
Cells80 triangular prisms, 16 pentagonal prism]s, 8 pentagonal antiprisms, 10+10 octagonal prisms
Faces80 triangles, 80+80 squares, 16 pentagons, 10 octagons
Edges80+80+80
Vertices80
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+5)/2 (base trapezoid), 2+2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {13+4{\sqrt {2}}+{\sqrt {5}}}{8}}}\approx 1.61605}$
Hypervolume${\displaystyle {\frac {5+5{\sqrt {2}}+2{\sqrt {5}}+2{\sqrt {10}}}{3}}\approx 7.62259}$
Diteral anglesTodip–op–todip: = ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Pappip–pap–pappip: 135°
Todip–op–podip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Todip–trip–pappip: 90°
Podip–pip–pappip: 90°
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces20
Level of complexity40
Related polytopes
ArmyOpap
RegimentOpap
DualOctagonal-pentagonal antitegmatic duotegum
ConjugatesOctagrammic-pentagonal antiprismatic duoprism, octagonal-pentagrammic retroprismatic duoprism, octagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(8)×I2(10)×A1+, order 320
ConvexYes
NatureTame

The octagonal-pentagonal antiprismatic duoprism or opap is a convex uniform duoprism that consists of 8 pentagonal antiprismatic prisms, 2 pentagonal-octagonal duoprisms, and 10 triangular-octagonal duoprisms. Each vertex joins 2 pentagonal antiprismatic prisms, 3 triangular-octagonal duoprisms, and 1 pentagonal-octagonal duoprism.

## Vertex coordinates

The vertices of a octagonal-pentagonal antiprismatic duoprism of edge length 1 are given by all central inversions of the last three coordinates of:

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

## Representations

An octagonal-pentagonal antiprismatic duoprism has the following Coxeter diagrams:

• x8o s2s10o (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x8o s2s5s (pentagonal antiprisms as alternated dipentagonal prisms)
• x4x s2s10o (octagons as ditetragons)
• x4x s2s5s