Heptagonal-pentagonal antiprismatic duoprism Rank 5 Type Uniform Notation Bowers style acronym Hepap Coxeter diagram x7o s2s10o Elements Tera 7 pentagonal antiprismatic prisms , 10 triangular-heptagonal duoprisms , 2 pentagonal-heptagonal duoprisms Cells 70 triangular prisms , 14 pentagonal prisms , 7 pentagonal antiprisms , 10+10 heptagonal prisms Faces 70 triangles , 70+70 squares , 14 pentagons , 10 heptagons Edges 70+70+70 Vertices 70 Vertex figure Isosceles-trapezoidal scalene , edge lengths 1, 1, 1, (1+√5 )/2 (base trapezoid), 2cos(π/7) (top), √2 (side edges)Measures (edge length 1) Circumradius
5
+
5
+
2
sin
2
π
7
8
≈
1.49415
{\displaystyle {\sqrt {\frac {5+{\sqrt {5}}+{\frac {2}{\sin ^{2}{\frac {\pi }{7}}}}}{8}}}\approx 1.49415}
Hypervolume
7
5
+
2
5
24
tan
π
7
≈
5.73682
{\displaystyle 7{\frac {5+2{\sqrt {5}}}{24\tan {\frac {\pi }{7}}}}\approx 5.73682}
Diteral angles Theddip–hep–theddip: =
arccos
(
−
5
3
)
≈
138.18969
∘
{\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}
Pappip–pap–pappip:
5
π
7
≈
128.57143
∘
{\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}
Theddip–hep–pheddip: =
arccos
(
−
5
−
2
5
15
)
≈
100.81232
∘
{\displaystyle \arccos \left(-{\sqrt {\frac {5-2{\sqrt {5}}}{15}}}\right)\approx 100.81232^{\circ }}
Theddip–trip–pappip: 90° Pheddip–pip–pappip: 90° Height
5
+
5
10
≈
0.85065
{\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{10}}}\approx 0.85065}
Central density 1 Number of external pieces 19 Level of complexity 40 Related polytopes Army Hepap Regiment Hepap Dual Heptagonal-pentagonal antitegmatic duotegum Conjugates Heptagrammic-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 characteristic 2 Orientable Yes Properties Symmetry I2 (7)×I2 (10)×A1 + , order 280Convex Yes Nature Tame
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.
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:
(
1
,
0
,
0
,
2
5
+
5
10
sin
π
7
,
5
+
5
10
sin
π
7
)
,
{\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),}
(
cos
j
π
7
,
±
sin
j
π
7
,
0
,
2
5
+
5
10
sin
π
7
,
5
+
5
10
sin
π
7
)
,
{\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),}
(
1
,
0
,
±
(
1
+
5
)
sin
π
7
2
,
5
−
5
10
sin
π
7
,
5
+
5
10
sin
π
7
)
,
{\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),}
(
cos
j
π
7
,
±
sin
j
π
7
,
±
(
1
+
5
)
sin
π
7
2
,
5
−
5
10
sin
π
7
,
5
+
5
10
sin
π
7
)
,
{\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),}
(
1
,
0
,
±
sin
π
7
,
−
5
−
2
5
5
sin
π
7
,
5
+
5
10
sin
π
7
)
,
{\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),}
(
cos
j
π
7
,
±
sin
j
π
7
,
±
sin
π
7
,
−
5
−
2
5
5
sin
π
7
,
5
+
5
10
sin
π
7
)
,
{\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.
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) 
