# Pentagonal-square antiprismatic duoprism

Pentagonal-square antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymPesquap
Coxeter diagramx5o s2s8o
Elements
Tera5 square antiprismatic prisms, 8 triangular-pentagonal duoprisms, 2 square-pentagonal duoprisms
Cells40 triangular prisms, 10 cubes, 5 square antiprisms, 8+8 pentagonal prisms
Faces40 triangles, 10+40+40 squares, 8 pentagons
Edges40+40+40
Vertices40
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 2 (base trapezoid), (1+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {40+5{\sqrt {2}}+4{\sqrt {5}}}{40}}}\approx 1.18338}$
Hypervolume${\displaystyle {\frac {\sqrt {5(20+15{\sqrt {2}}+8{\sqrt {5}}+6{\sqrt {10}})}}{12}}\approx 1.64650}$
Diteral anglesTrapedip–pip–trapedip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
Squappip–squap–squappip: 108°
Trapedip–pip–squipdip: = ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Trapedip–trip–squappip: 90°
Squipdip–cube–squappip: 90°
Height${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces15
Level of complexity40
Related polytopes
ArmyPesquap
RegimentPesquap
DualPentagonal-square antitegmatic duotegum
ConjugatePentagrammic-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryH2×I2(8)×A1+, order 160
ConvexYes
NatureTame

The pentagonal-square antiprismatic duoprism or pesquap is a convex uniform duoprism that consists of 5 square antiprismatic prisms, 2 square-pentagonal duoprisms, and 8 triangular-pentagonal duoprisms. Each vertex joins 2 square antiprismatic prisms, 3 triangular-pentagonal duoprisms, and 1 square-pentagonal duoprism.

## Vertex coordinates

The vertices of a pentagonal-square antiprismatic duoprism of edge length 1 are given by:

• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right).}$

## Representations

A pentagonal-square antiprismatic duoprism has the following Coxeter diagrams:

• x5o s2s8o (full symmetry; square antiprisms as alternated octagonal prisms)
• x5o s2s4s (square antiprisms as alternated ditetragonal prisms)