# Heptagonal-decagonal duoprism

Heptagonal-decagonal duoprism
Rank4
TypeUniform
SpaceSpherical
Info
Coxeter diagramx7o x10o
SymmetryI2(7)×I2(10), order 280
Elements
Vertex figureDigonal disphenoid, edge lengths 2cos(π/7) (base 1), (5+5)/2 (base 2), and 2 (sides)
Cells10 heptagonal prisms, 7 decagonal prisms
Faces70 squares, 10 heptagons, 7 decagons
Edges70+70
Vertices70
Measures (edge length 1)
Circumradius${\displaystyle \sqrt{\frac{3+\sqrt5}{2}+\frac{1}{4\sin^2\frac\pi7}} ≈ 1.98646}$
Hypervolume${\displaystyle \frac{35\sqrt{5+2\sqrt5}}{8\tan\frac\pi7} ≈ 27.96008}$
Dichoral anglesHep–7–hep: 144°
Dip–10–dip: ${\displaystyle \frac{5\pi}{7} ≈ 128.57143°}$
Hep–4–dip: 90°
Central density1
Euler characteristic0
Number of 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
Properties
ConvexYes
OrientableYes
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:

• (1, 0, ±sin(π/7), ±sin(π/7)5+25),
• (1, 0, ±sin(π/7)(3+5)/2, ±sin(π/7)(5+5)/2),
• (1, 0, ±sin(π/7)(1+5), 0),
• (cos(2π/7), ±sin(2π/7), ±sin(π/7), ±sin(π/7)5+25),
• (cos(2π/7), ±sin(2π/7), ±sin(π/7)(3+5)/2, ±sin(π/7)(5+5)/2),
• (cos(2π/7), ±sin(2π/7), ±sin(π/7)(1+5), 0),
• (cos(4π/7), ±sin(4π/7), ±sin(π/7), ±sin(π/7)5+25),
• (cos(4π/7), ±sin(4π/7), ±sin(π/7)(3+5)/2, ±sin(π/7)(5+5)/2),
• (cos(4π/7), ±sin(4π/7), ±sin(π/7)(1+5), 0),
• (cos(6π/7), ±sin(6π/7), ±sin(π/7), ±sin(π/7)5+25),
• (cos(6π/7), ±sin(6π/7), ±sin(π/7)(3+5)/2, ±sin(π/7)(5+5)/2),
• (cos(6π/7), ±sin(6π/7), ±sin(π/7)(1+5), 0).

## Representations

A heptagonal-decagonal duoprism has the following Coxeter diagrams:

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