# Heptagonal-pentagonal antiprismatic duoprism

Heptagonal-pentagonal antiprismatic duoprism
Rank5
TypeUniform
Notation
Bowers style acronymHepap
Coxeter diagramx7o s2s10o
Elements
Tera7 pentagonal antiprismatic prisms, 10 triangular-heptagonal duoprisms, 2 pentagonal-heptagonal duoprisms
Cells70 triangular prisms, 14 pentagonal prisms, 7 pentagonal antiprisms, 10+10 heptagonal prisms
Faces70 triangles, 70+70 squares, 14 pentagons, 10 heptagons
Edges70+70+70
Vertices70
Vertex figureIsosceles-trapezoidal scalene, edge lengths 1, 1, 1, (1+5)/2 (base trapezoid), 2cos(π/7) (top), 2 (side edges)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 1.49415}$
Hypervolume${\displaystyle 7{\frac {5+2{\sqrt {5}}}{24\tan {\frac {\pi }{7}}}}\approx 5.73682}$
Diteral anglesTheddip–hep–theddip: = ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Pappip–pap–pappip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Theddip–hep–pheddip: = ${\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}$
Theddip–trip–pappip: 90°
Pheddip–pip–pappip: 90°
Height${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}$
Central density1
Number of external pieces19
Level of complexity40
Related polytopes
ArmyHepap
RegimentHepap
DualHeptagonal-pentagonal antitegmatic duotegum
ConjugatesHeptagrammic-pentagonal antiprismatic duoprism, Great heptagrammic-pentagonal antiprismatic duoprism, heptagonal-pentagrammic retroprismatic duoprism, heptagrammic-pentagrammic retroprismatic duoprism, Great heptagrammic-pentagrammic retroprismatic duoprism
Abstract & topological properties
Euler characteristic2
OrientableYes
Properties
SymmetryI2(7)×I2(10)×A1+, order 280
ConvexYes
NatureTame

The heptagonal-pentagonal antiprismatic duoprism or hepap is a convex uniform duoprism that consists of 7 pentagonal antiprismatic prisms, 2 pentagonal-heptagonal duoprisms, and 10 triangular-heptagonal duoprisms. Each vertex joins 2 pentagonal antiprismatic prisms, 3 triangular-heptagonal duoprisms, and 1 pentagonal-heptagonal duoprism.

## Vertex coordinates

The vertices of a heptagonal-pentagonal antiprismatic duoprism of edge length 2sin(π/7) are given by all central inversions of the last three coordinates of:

• ${\displaystyle \left(1,\,0,\,0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,0,\,2{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm {\frac {(1+{\sqrt {5}})\sin {\frac {\pi }{7}}}{2}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(1,\,0,\,\pm \sin {\frac {\pi }{7}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{7}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}}\right),}$
• ${\displaystyle \left(\cos {\frac {j\pi }{7}},\,\pm \sin {\frac {j\pi }{7}},\,\pm \sin {\frac {\pi }{7}},\,-{\sqrt {\frac {5-2{\sqrt {5}}}{5}}}\sin {\frac {\pi }{7}},\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\sin {\frac {\pi }{7}}\right),}$

where j = 2, 4, 6.

## Representations

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

• x7o s2s10o (full symmetry; pentagonal antiprisms as alternated decagonal prisms)
• x7o s2s5s (pentagonal antiprisms as alternated dipentagonal prisms)