# Enneagonal-pentagonal antiprismatic duoprism

Enneagonal-pentagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymEpap
Coxeter diagramx9o s2s10o ()
Elements
Tera9 pentagonal antiprismatic prisms, 10 triangular-enneagonal duoprisms, 2 pentagonal-enneagonal duoprisms
Cells90 triangular prisms, 18 pentagonal prisms, 9 pentagonal antiprisms, 10+10 enneagonal prisms
Faces90 triangles, 90+90 squares, 18 pentagons, 10 enneagons
Edges90+90+90
Vertices90
Vertex figureIsosceles trapezoidal scalene, edge lengths 1, 1, 1, (1+5)/2 (base trapezoid), 2cos(π/9) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{9}}}}}{8}}}\approx 1.74404}$
Hypervolume${\displaystyle 3{\frac {5+2{\sqrt {5}}}{8\tan {\frac {\pi }{9}}}}\approx 9.75918}$
Diteral anglesPappip–pap–pappip: 140°
Tedip–ep–tedip: = ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Tedip–ep–peendip: = ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Tedip–trip–pappip: 90°
Peendip–pip–pappip: 90°
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces21
Level of complexity40
Related polytopes
ArmyEpap
RegimentEpap
DualEnneagonal-pentagonal antitegmatic duotegum
ConjugatesEnneagrammic-pentagonal antiprismatic duoprism, Great enneagrammic-pentagonal antiprismatic duoprism, enneagonal-pentagrammic retroprismatic duoprism, enneagrammic-pentagrammic retroprismatic duoprism, Great enneagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(9)×I2(10)×A1+, order 360
ConvexYes
NatureTame

The enneagonal-pentagonal antiprismatic duoprism or epap is a convex uniform duoprism that consists of 9 pentagonal antiprismatic prisms, 2 pentagonal-enneagonal duoprisms, and 10 triangular-enneagonal duoprisms. Each vertex joins 2 pentagonal antiprismatic prisms, 3 triangular-enneagonal duoprisms, and 1 pentagonal-enneagonal duoprism.

## Vertex coordinates

The vertices of an enneagonal-pentagonal antiprismatic duoprism of edge length 2sin(π/9) are given by all central inversions of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{9}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{9}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{9}},\,\pm \sin {\frac {j\pi }{9}},\,\pm \sin {\frac {\pi }{9}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$
• ${\displaystyle \left(-{\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm \sin {\frac {\pi }{9}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{9}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{9}}\right),}$

where j = 2, 4, 8.

## Representations

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

• x9o s2s10o () (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x9o s2s5s () (pentagonal antiprisms as alternated dipentagonal prisms)