# 20-6 step prism

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20-6 step prism | |
---|---|

File:20-6 step prism.png | |

Rank | 4 |

Type | Isogonal |

Space | Spherical |

Elements | |

Cells | 20+20+20+20 phyllic disphenoids, 10 rhombic disphenoids |

Faces | 40+40+40+40 scalene triangles, 20 isosceles triangles |

Edges | 10+20+20+20+20+20 |

Vertices | 20 |

Vertex figure | 11-vertex polyhedron with 18 triangular faces |

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

Edge lengths | 8-valence (20): |

6-valence (20): | |

4-valence (20): | |

4-valence (20): | |

3-valence (20): 2 | |

4-valence (10): 2 | |

Central density | 1 |

Related polytopes | |

Dual | 20-6 gyrochoron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | S_{2}(I_{2}(20)-6), order 40 |

Convex | Yes |

Nature | Tame |

The **20-6 step prism** is a convex isogonal polychoron and a member of the step prism family. It has 10 rhombic disphenoids and 80 phyllic disphenoids of four kinds as cells, with 18 (2 rhombic and 16 phyllic disphenoids) joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.82090.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 20-6 step prism inscribed in an icosagonal duoprism with base lengths *a* and *b* are given by:

- (
*a**sin(2π*k*/20),*a**cos(2π*k*/20),*b**sin(12π*k*/20),*b**cos(12π*k*/20)),

where *k* is an integer from 0 to 19.
If the edge length differences are to be minimized, the ratio of *a:b* must be equivalent to 1:1.

## External links[edit | edit source]

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