Heptagonal-enneagonal duoprism Rank 4 Type Uniform Notation Bowers style acronym He en d ip Coxeter diagram x7o x9o (File:CDel node 1.png File:CDel 7.png File:CDel node.png File:CDel 2.png File:CDel node 1.png File:CDel 9.png File:CDel node.png ) Elements Cells 9 heptagonal prisms , 7 enneagonal prisms Faces 63 squares , 9 heptagons , 7 enneagons Edges 63+63 Vertices 63 Vertex figure Digonal disphenoid , edge lengths 2cos(π/7) (base 1), 2cos(π/9) (base 2), and √2 (sides)Measures (edge length 1) Circumradius
1
4
sin
2
π
7
+
1
4
sin
2
π
9
≈
1.86149
{\displaystyle {\sqrt {{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{9}}}}}}\approx 1.86149}
Hypervolume
63
16
tan
π
7
tan
π
9
≈
22.46421
{\displaystyle {\frac {63}{16\tan {\frac {\pi }{7}}\tan {\frac {\pi }{9}}}}\approx 22.46421}
Dichoral angles Hep–7–hep: 140° Ep–9–ep:
5
π
7
≈
128.57143
∘
{\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}
Hep–4–ep: 90° Central density 1 Number of external pieces 16 Level of complexity 6 Related polytopes Army Heendip Regiment Heendip Dual Heptagonal-enneagonal duotegum Conjugates Heptagonal-enneagrammic duoprism , Heptagonal-great enneagrammic duoprism , Heptagrammic-enneagonal duoprism , Heptagrammic-enneagrammic duoprism , Heptagrammic-great enneagrammic duoprism , Great heptagrammic-enneagonal duoprism , Great heptagrammic-enneagrammic duoprism , Great heptagrammic-great enneagrammic duoprism Abstract & topological properties Flag count1512 Euler characteristic 0 Orientable Yes Properties Symmetry I2 (7)×I2 (9) , order 252Flag orbits 6 Convex Yes Nature Tame
The heptagonal-enneagonal duoprism or heendip , also known as the 7-9 duoprism , is a uniform duoprism that consists of 7 enneagonal prisms and 9 heptagonal prisms , with two of each joining at each vertex.
The coordinates of a heptagonal-enneagonal duoprism, centered at the origin and with edge length 4sin(π/7)sin(π/9), are given by:
(
2
sin
π
9
,
0
,
2
sin
π
7
,
0
)
{\displaystyle \left(2\sin {\frac {\pi }{9}},0,2\sin {\frac {\pi }{7}},0\right)}
,
(
2
sin
π
9
,
0
,
2
sin
π
7
cos
(
j
π
9
)
,
±
2
sin
π
7
sin
(
j
π
9
)
)
{\displaystyle \left(2\sin {\frac {\pi }{9}},0,2\sin {\frac {\pi }{7}}\cos \left({\frac {j\pi }{9}}\right),\pm 2\sin {\frac {\pi }{7}}\sin \left({\frac {j\pi }{9}}\right)\right)}
,
(
2
sin
π
9
,
0
,
−
sin
π
7
,
±
3
sin
π
7
)
{\displaystyle \left(2\sin {\frac {\pi }{9}},0,-\sin {\frac {\pi }{7}},\pm {\sqrt {3}}\sin {\frac {\pi }{7}}\right)}
,
(
2
sin
π
9
cos
(
k
π
7
)
,
±
2
sin
π
9
sin
(
k
π
7
)
,
2
sin
π
7
,
0
)
{\displaystyle \left(2\sin {\frac {\pi }{9}}\cos \left({\frac {k\pi }{7}}\right),\pm 2\sin {\frac {\pi }{9}}\sin \left({\frac {k\pi }{7}}\right),2\sin {\frac {\pi }{7}},0\right)}
,
(
2
sin
π
9
cos
(
k
π
7
)
,
±
2
sin
π
9
sin
(
k
π
7
)
,
2
sin
π
7
cos
(
j
π
9
)
,
±
2
sin
π
7
sin
(
j
π
9
)
)
{\displaystyle \left(2\sin {\frac {\pi }{9}}\cos \left({\frac {k\pi }{7}}\right),\pm 2\sin {\frac {\pi }{9}}\sin \left({\frac {k\pi }{7}}\right),2\sin {\frac {\pi }{7}}\cos \left({\frac {j\pi }{9}}\right),\pm 2\sin {\frac {\pi }{7}}\sin \left({\frac {j\pi }{9}}\right)\right)}
,
(
2
sin
π
9
cos
(
k
π
7
)
,
±
2
sin
π
9
sin
(
k
π
7
)
,
−
sin
π
7
,
±
3
sin
π
7
)
{\displaystyle \left(2\sin {\frac {\pi }{9}}\cos \left({\frac {k\pi }{7}}\right),\pm 2\sin {\frac {\pi }{9}}\sin \left({\frac {k\pi }{7}}\right),-\sin {\frac {\pi }{7}},\pm {\sqrt {3}}\sin {\frac {\pi }{7}}\right)}
,
where j = 2, 4, 8 and k = 2, 4, 6.