# 8-2 step prism

8-2 step prism | |
---|---|

Rank | 4 |

Type | Isogonal |

Elements | |

Cells | 8+8 phyllic disphenoids, 4 rhombic disphenoids |

Faces | 16+16 scalene triangles, 8 isosceles triangles |

Edges | 4+8+8+8 |

Vertices | 8 |

Vertex figure | Ridge-triakis notch |

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

Edge lengths | 6-valence (8): |

4-valence (4): 2 | |

4-valence (8): | |

3-valence (8): | |

Central density | 1 |

Related polytopes | |

Dual | 8-2 gyrochoron |

Abstract & topological properties | |

Flag count | 480 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | S_{2}(I_{2}(8)-2), order 16 |

Flag orbits | 30 |

Convex | Yes |

Nature | Tame |

The **8-2 step prism** is a convex isogonal polychoron and a step prism. It has 4 rhombic disphenoids and 16 phyllic disphenoids of two kinds as cells, with a total of 10 (2 rhombic and 8 phyllic disphenoids) joining at each vertex. It is one of 3 isogonal polychora with 8 vertices, and the only one not to be uniform (the other 2 are the hexadecachoron and tetrahedral prism).

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.48563.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of an 8-2 step prism inscribed in an octagonal duoprism with base lengths *a* and *b* are given by:

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

where *k* is an integer from 0 to 7.
If the edge length differences are to be minimized, the ratio of *a:b* must be equivalent to 1: ≈ 1:1.30656.

## Isogonal derivatives[edit | edit source]

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

- Phyllic disphenoid (8):
**8-2 step prism** - Scalene triangle (8):
**8-2 step prism** - Scalene triangle (16): 16-2 step prism
- Edge (8):
**8-2 step prism**

## External links[edit | edit source]

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