# 15-5 step prism

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15-5 step prism | |
---|---|

Rank | 4 |

Type | Isogonal |

Space | Spherical |

Elements | |

Cells | 15 phyllic disphenoids, 3 pentagonal gyroprisms |

Faces | 30 scalene triangles, 15 isosceles triangles, 3 pentagons |

Edges | 15+15+15 |

Vertices | 15 |

Vertex figure | Tetragonal antiwedge |

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

Edge lengths | 3-valence (15): |

4-valence (15): | |

3-valence (15): | |

Central density | 1 |

Related polytopes | |

Dual | 15-5 gyrochoron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | S_{2}(I_{2}(15)-5), order 30 |

Convex | Yes |

Nature | Tame |

The **15-5 step prism** is a convex isogonal polychoron and a member of the step prism family. It has 3 pentagonal gyroprisms and 15 phyllic disphenoids as cells, with 4 phyllic disphenoids and 2 pentagonal gyroprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.16349.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 15-5 step prism inscribed in a pentadecagonal duoprism with base lengths *a* and *b* are given by:

- (
*a**sin(2π*k*/15),*a**cos(2π*k*/15),*b**sin(2π*k*/3),*b**cos(2π*k*/3)),

where *k* is an integer from 0 to 14.
If the edge length differences are to be minimized, the ratio of *a:b* must be equivalent to 1: ≈ 1:0.63484.

## Isogonal derivatives[edit | edit source]

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

- Phyllic disphenoid (15): 15-5 step prism
- Scalene triangle (15): 15-5 step prism
- Scalene triangle (30): 30-5 step prism
- Edge (15): 15-5 step prism

## External links[edit | edit source]

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