# Heptagonal-decagonal duoprism

Heptagonal-decagonal duoprism
Rank4
TypeUniform
Notation
Coxeter diagramx7o x10o ()
Elements
Cells10 heptagonal prisms, 7 decagonal prisms
Faces70 squares, 10 heptagons, 7 decagons
Edges70+70
Vertices70
Vertex figureDigonal disphenoid, edge lengths 2cos(π/7) (base 1), (5+5)/2 (base 2), and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {{\frac {3+{\sqrt {5}}}{2}}+{\frac {1}{4\sin ^{2}{\frac {\pi }{7}}}}}}\approx 1.98646}$
Hypervolume${\displaystyle {\frac {35{\sqrt {5+2{\sqrt {5}}}}}{8\tan {\frac {\pi }{7}}}}\approx 27.96008}$
Dichoral anglesHep–7–hep: 144°
Dip–10–dip: ${\displaystyle {\frac {5\pi }{7}}\approx 128.57143^{\circ }}$
Hep–4–dip: 90°
Central density1
Number of external pieces17
Level of complexity6
Related polytopes
DualHeptagonal-decagonal duotegum
ConjugatesHeptagonal-decagrammic duoprism, Heptagrammic-decagonal duoprism, Heptagrammic-decagrammic duoprism, Great heptagrammic-decagonal duoprism, Great heptagrammic-decagrammic duoprism
Abstract & topological properties
Flag count1680
Euler characteristic0
OrientableYes
Properties
SymmetryI2(7)×I2(10), order 280
ConvexYes
NatureTame

The heptagonal-decagonal duoprism or hedadip, also known as the 7-10 duoprism, is a uniform duoprism that consists of 7 decagonal prisms and 10 heptagonal prisms, with two of each joining at each vertex.

## Vertex coordinates

The coordinates of a heptagonal-decagonal duoprism, centered at the origin and with edge length 2sin(π/7), are given by:

• ${\displaystyle \left(1,0,\pm \sin {\frac {\pi }{7}},\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{7}}\right)}$,
• ${\displaystyle \left(1,0,\pm {\frac {3+{\sqrt {5}}}{2}}\sin {\frac {\pi }{7}},\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{7}}\right)}$,
• ${\displaystyle \left(1,0,\left(1+{\sqrt {5}}\right)\sin {\frac {\pi }{7}},0\right)}$,
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\pm \sin \left({\frac {j\pi }{7}}\right),\pm \sin {\frac {\pi }{7}},\pm {\sqrt {5+2{\sqrt {5}}}}\sin {\frac {\pi }{7}}\right)}$,
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\pm \sin \left({\frac {j\pi }{7}}\right),\pm {\frac {3+{\sqrt {5}}}{2}}\sin {\frac {\pi }{7}},\pm {\sqrt {\frac {5+{\sqrt {5}}}{2}}}\sin {\frac {\pi }{7}}\right)}$,
• ${\displaystyle \left(\cos \left({\frac {j\pi }{7}}\right),\pm \sin \left({\frac {j\pi }{7}}\right),\left(1+{\sqrt {5}}\right)\sin {\frac {\pi }{7}},0\right)}$,

where j = 2, 4, 6.

## Representations

A heptagonal-decagonal duoprism has the following Coxeter diagrams:

• x7o x10o () (full symmetry)
• x5x x7o () (decagons as dipentagons)