# 9-2 step prism

9-2 step prism | |
---|---|

Rank | 4 |

Type | Isogonal |

Elements | |

Cells | 9+9+9 phyllic disphenoids |

Faces | 18+18 scalene triangles, 9+9 isosceles triangles |

Edges | 9+9+9+9 |

Vertices | 9 |

Vertex figure | Ridge-triakis bi-apiculated tetrahedron |

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

Edge lengths | 7-valence (9): |

4-valence (9): | |

3-valence (9): | |

4-valence (9): | |

Central density | 1 |

Related polytopes | |

Dual | 9-2 gyrochoron |

Abstract & topological properties | |

Flag count | 648 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | S_{2}(I_{2}(9)-2), order 18 |

Flag orbits | 36 |

Convex | Yes |

Nature | Tame |

The **9-2 step prism** is a convex isogonal polychoron and a member of the step prism family. It has 27 phyllic disphenoids of three kinds as cells, with 12 joining at each vertex. It can also be constructed as the 9-4 step prism.

It is one of 3 isogonal polychora with 9 vertices, the others are the uniform triangular duoprism and the 9-3 step prism.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.56632.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 9-2 step prism inscribed in an enneagonal duoprism with base lengths *a* and *b* are given by:

- (
*a**sin(2π*k*/9),*a**cos(2π*k*/9),*b**sin(4π*k*/9),*b**cos(4π*k*/9)),

where *k* is an integer from 0 to 8.
If the edge length differences are to be minimized, the ratio of *a:b* must be equivalent to 1: ≈ 1:1.69688.

## Isogonal derivatives[edit | edit source]

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

- Phyllic disphenoid (9): 9-2 step prism
- Scalene triangle (9): 9-2 step prism
- Scalene triangle (19): 18-2 step prism
- Edge (9): 9-2 step prism

## External links[edit | edit source]

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