Decagonal-pentagonal antiprismatic duoprism

Decagonal-pentagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymDapap
Coxeter diagramx10o s2s10o ()
Elements
Tera10 pentagonal antiprismatic prisms, 10 triangular-decagonal duoprisms, 2 pentagonal-decagonal duoprisms
Cells100 triangular prisms, 20 pentagonal prisms, 10 pentagonal antiprisms, 10+10 decagonal prisms
Faces100 triangles, 100+100 squares, 20 pentagons, 10 decagons
Edges100+100+100
Vertices100
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+5)/2 (base trapezoid), (5+5)/2 (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {17+5{\sqrt {5}}}{8}}}\approx 1.87684}$
Hypervolume${\displaystyle {\frac {5{\sqrt {425+190{\sqrt {5}}}}}{12}}\approx 12.14677}$
Diteral anglesPappip–pap–pappip: 144°
Tradedip–dip–tradedip: = ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Tradedip–dip–padedip: = ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces22
Level of complexity40
Related polytopes
ArmyDapap
RegimentDapap
DualDecagonal-pentagonal antitegmatic duotegum
ConjugateDecagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(10)×I2(10)×A1+, order 400
ConvexYes
NatureTame

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

Vertex coordinates

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

• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5-{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}}\right).}$

Representations

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

• x10o s2s10o (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x10o s2s5s (pentagonal antiprisms as alternated dipentagonal prisms)
• x5x s2s10o (decagons as dipentagons)
• x5x s2s5s