# Heptagonal-decagonal duoprism

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Heptagonal-decagonal duoprism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Bowers style acronym | Hedadip |

Info | |

Coxeter diagram | x7o x10o |

Symmetry | I2(7)×I2(10), order 280 |

Army | Hedadip |

Regiment | Hedadip |

Elements | |

Vertex figure | Digonal disphenoid, edge lengths 2cos(π/7) (base 1), √(5+√5)/2 (base 2), and √2 (sides) |

Cells | 10 heptagonal prisms, 7 decagonal prisms |

Faces | 70 squares, 10 heptagons, 7 decagons |

Edges | 70+70 |

Vertices | 70 |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Hep–7–hep: 144° |

Dip–10–dip: | |

Hep–4–dip: 90° | |

Central density | 1 |

Euler characteristic | 0 |

Number of pieces | 17 |

Level of complexity | 6 |

Related polytopes | |

Dual | Heptagonal-decagonal duotegum |

Conjugates | Heptagonal-decagrammic duoprism, Heptagrammic-decagonal duoprism, Heptagrammic-decagrammic duoprism, Great heptagrammic-decagonal duoprism, Great heptagrammic-decagrammic duoprism |

Properties | |

Convex | Yes |

Orientable | Yes |

Nature | Tame |

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[edit | edit source]

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+2√5),
- (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+2√5),
- (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+2√5),
- (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+2√5),
- (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[edit | edit source]

A heptagonal-decagonal duoprism has the following Coxeter diagrams:

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

## External links[edit | edit source]

- Bowers, Jonathan. "Category A: Duoprisms".