# 10-2 step prism

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10-2 step prism | |
---|---|

Rank | 4 |

Type | Isogonal |

Elements | |

Cells | 10+10+10 phyllic disphenoids, 5 rhombic disphenoids |

Faces | 20+20+20 scalene triangles, 10 isosceles triangles |

Edges | 5+10+10+10+10 |

Vertices | 10 |

Vertex figure | Ridge-ditriakis notch |

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

Edge lengths | 8-valence (10): |

4-valence (5): 2 | |

3-valence (10): | |

4-valence (10): | |

4-valence (10): | |

Central density | 1 |

Related polytopes | |

Dual | 10-2 gyrochoron |

Abstract & topological properties | |

Flag count | 840 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | S_{2}(I_{2}(10)-2), order 20 |

Flag orbits | 42 |

Convex | Yes |

Nature | Tame |

The **10-2 step prism** is a convex isogonal polychoron and a step prism. It has 5 rhombic disphenoids and 30 phyllic disphenoids of three kinds as cells, with 14 (2 rhombic and 12 phyllic disphenoids) joining at each vertex.

It is one of 4 isogonal polychora with 10 vertices, and the only one to have only step prism symmetry in its highest symmetry form.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.73855.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 10-2 step prism inscribed in a decagonal duoprism with base lengths *a* and *b* are given by:

- (
*a**sin(π*k*/5),*a**cos(π*k*/5),*b**sin(2π*k*/5),*b**cos(2π*k*/5)),

where *k* is an integer from 0 to 9.
If the edge length differences are to be minimized, the ratio of *a:b* must be equivalent to 1: ≈ 1:1.61803.

## External links[edit | edit source]

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