# Transitional 10-3 double gyrostep prism

Transitional 10-3 double gyrostep prism | |
---|---|

File:Transitional 10-3 double gyrostep prism.png | |

Rank | 4 |

Type | Isogonal |

Space | Spherical |

Elements | |

Cells | 10 tetragonal disphenoids, 10 chiral gyrobifastigia |

Faces | 40 isosceles triangles, 20 kites |

Edges | 20+40 |

Vertices | 20 |

Vertex figure | Tetragonal antiwedge |

Measures (based on unit decachoron) | |

Edge lengths | 4-valence (20): 1 |

3-valence (40): | |

Circumradius | |

Central density | 1 |

Related polytopes | |

Dual | Transitional 10-3 antibigyrochoron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | S_{2}(I_{2}(10)-3)×2I, order 40 |

Convex | Yes |

Nature | Tame |

The **transitional 10-3 double gyrostep prism** is a convex isogonal polychoron that consists of 10 chiral gyrobifastigia and 10 tetragonal disphenoids. 2 chiral gyrobifastigia and 4 tetragonal disphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal stretched 10-3 step prisms.

It can also be constructed as a subsymmetrical faceting of the decachoron, formed by removing a gyrochoron-symmetric subset of 10 vertices. The disphenoids are the decachoron's vertex figures, while the gyrobifastigium cells are subsymmetric facetings of the decachoron's truncated tetrahedron cells.

The ratio between the longest and shortest edges is 1: ≈ 1:1.73205.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a transitional 10-3 double gyrostep prism are given by:

- (
*a**sin(2π*k*/10),*a**cos(2π*k*/10),*b**sin(6π*k*/10),*b**cos(6π*k*/10)), - (
*b**sin(2π*k*/10),*b**cos(2π*k*/10), -*a**sin(6π*k*/10), -*a**cos(6π*k*/10)),

where *a* = , *b* = , and *k* is an integer from 0 to 9.

## External links[edit | edit source]

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