Decagonal-pentagonal antiprismatic duoprism

Decagonal-pentagonal antiprismatic duoprism
(OFF file)
Rank	5
Type	Uniform
Notation
Bowers style acronym	Dapap
Coxeter diagram	x10o s2s10o ()
Elements
Tera	10 pentagonal antiprismatic prisms, 10 triangular-decagonal duoprisms, 2 pentagonal-decagonal duoprisms
Cells	100 triangular prisms, 20 pentagonal prisms, 10 pentagonal antiprisms, 10+10 decagonal prisms
Faces	100 triangles, 100+100 squares, 20 pentagons, 10 decagons
Edges	100+100+100
Vertices	100
Vertex figure	Isosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+√5)/2 (base trapezoid), √(5+√5)/2 (top), √2 (side edges)
Measures (edge length 1)
Circumradius	${\sqrt {\frac {17+5{\sqrt {5}}}{8}}}\approx 1.87684$
Hypervolume	${\frac {5{\sqrt {425+190{\sqrt {5}}}}}{12}}\approx 12.14677$
Diteral angles	Pappip–pap–pappip: 144°
	Tradedip–dip–tradedip: = $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	Tradedip–dip–padedip: = $\arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }$
	Tradedip–trip–pappip: 90°
	Padedip–pip–pappip: 90°
Height	${\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065$
Central density	1
Number of external pieces	22
Level of complexity	40
Related polytopes
Army	Dapap
Regiment	Dapap
Dual	Decagonal-pentagonal antitegmatic duotegum
Conjugate	Decagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	I2(10)×I2(10)×A1+, order 400
Convex	Yes
Nature	Tame

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:

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

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