# 12-4 step prism

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12-4 step prism | |
---|---|

Rank | 4 |

Type | Isogonal |

Space | Spherical |

Info | |

Symmetry | I2(12)+×2×I, order 24 |

Elements | |

Vertex figure | Tetragonal antiwedge |

Cells | 12 phyllic disphenoids, 3 square antiprisms |

Faces | 12 isosceles triangles, 24 scalene triangles, 3 squares |

Edges | 12+12+12 |

Vertices | 12 |

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

Edge lengths | 3-valence (12): |

4-valence (12): | |

3-valence (12): 2 | |

Central density | 1 |

Euler characteristic | 0 |

Related polytopes | |

Dual | 12-4 gyrochoron |

Properties | |

Convex | Yes |

Orientable | Yes |

Nature | Tame |

The **12-4 step prism** is a convex isogonal polychoron and member of the step prism family. It has 3 chiral square antiprisms and 12 phyllic disphenoids as cells, with 4 disphenoids and 2 antiprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.16877.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a 12-4 step prism inscribed in a dodecagonal duoprism with base lengths *a* and *b* are given by:

- (
*a**sin(π*k*/6),*a**cos(π*k*/6),*b**sin(2π*k*/3),*b**cos(2π*k*/3)),

where *k* is an integer from 0 to 11.
If the edge length differences are to be minimized, the ratio of *a:b* must be equivalent to 1: ≈ 1:0.75984.

## Isogonal derivatives[edit | edit source]

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

- Phyllic disphenoid (12): 12-4 step prism
- Scalene triangle (12): 12-4 step prism
- Scalene triangle (24): 24-4 step prism
- Edge (12): 12-4 step prism

## External links[edit | edit source]

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