# Decagonal-square antiprismatic duoprism

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Decagonal-square antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDasquap
Coxeter diagramx10o s2s8o ()
Elements
Tera10 square antiprismatic prisms, 8 triangular-decagonal duoprisms, 2 square-decagonal duoprisms
Cells80 triangular prisms, 20 cubes, 10 square antiprisms, 8+8 decagonal prisms
Faces80 triangles, 20+80+80 squares, 8 decagons
Edges80+80+80
Vertices80
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, 2 (base trapezoid), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {16+{\sqrt {2}}+4{\sqrt {5}}}{8}}}\approx 1.81516}$
Hypervolume${\displaystyle {\frac {5{\sqrt {20+15{\sqrt {2}}+8{\sqrt {5}}+6{\sqrt {10}}}}}{6}}\approx 7.36336}$
Diteral anglesSquappip–squap–squappip: 144°
Tradedip–dip–tradedip: = ${\displaystyle \arccos \left({\frac {1-2{\sqrt {2}}}{3}}\right)\approx 127.55160^{\circ }}$
Tradedip–dip–squadedip: = ${\displaystyle \arccos \left({\frac {{\sqrt {3}}-{\sqrt {6}}}{3}}\right)\approx 103.83616^{\circ }}$
Tradedip–trip–squappip: 90°
Squadedip–cube–squappip: 90°
Height${\displaystyle {\frac {\sqrt[{4}]{8}}{2}}\approx 0.84090}$
Central density1
Number of external pieces20
Level of complexity40
Related polytopes
ArmyDasquap
RegimentDasquap
DualDecagonal-square antitegmatic duotegum
ConjugateDecagrammic-square antiprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(10)×I2(8)×A1+, order 320
ConvexYes
NatureTame

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

## Vertex coordinates

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

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,-{\frac {\sqrt[{4}]{8}}{4}}\right).}$

## Representations

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

• x10o s2s8o (full symmetry; square antiprisms as alternated octagonal prisms)
• x10o s2s4s () (square antiprisms as alternated ditetragonal prisms)
• x5x s2s8o () (decagons as dipentagons)
• x5x s2s4s ()