# 7-2 step prism

7-2 step prism | |
---|---|

Rank | 4 |

Type | Isogonal |

Space | Spherical |

Elements | |

Cells | 7+7 phyllic disphenoids |

Faces | 14 scalene triangles, 7+7 isosceles triangles |

Edges | 7+7+7 |

Vertices | 7 |

Vertex figure | Bilaterally-symmetric bi-apiculated tetrahedron |

Measures (circumradius , based on a uniform duoprism) | |

Edge lengths | 5-valence (7): |

4-valence (7): | |

3-valence (7): | |

Central density | 1 |

Related polytopes | |

Dual | 7-2 gyrochoron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | S_{2}(I_{2}(7)-2), order 14 |

Convex | Yes |

Nature | Tame |

The **7-2 step prism** is a convex isogonal polychoron and a member of the step prism family. It has 14 phyllic disphenoids of two different types as cells, with 8 joining at each vertex. It can also be constructed as the 7-3 step prism.

It is the simplest step prism, excluding the pentachoron and the triangular duotegum, which are part of more specific families, as well as the only isogonal polychoron with 7 vertices. It is also the triangular funk tegum.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.34236.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 7-2 step prism inscribed in a heptagonal duoprism with base lengths *a* and *b* are given by:

- (
*a**sin(2π*k*/7),*a**cos(2π*k*/7),*b**sin(4π*k*/7),*b**cos(4π*k*/7)),

where *k* is an integer from 0 to 6.
If the edge length differences are to be minimized, the ratio of *a:b* must be equivalent to 1: ≈ 1:1.34236.

## Measures[edit | edit source]

The hypervolume of a 7-2 step prism inscribed in a heptagonal-heptagonal duoprism with base lengths *a* and *b* is given by:

where *ξ* ≈ 1.55622 is the largest real root of 884736*x*^{3}–1613472*x*–823543, equivalent to 49/(192cos(2π/7)-96)-49/96.

## Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

- Phyllic disphenoid (7): 7-2 step prism
- Scalene triangle (7): 7-2 step prism
- Scalene triangle (14): 14-2 step prism
- Edge (7): 7-2 step prism

## External links[edit | edit source]

- Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".