# Pentagrammic retroprism

Pentagrammic retroprism | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Starp |

Coxeter diagram | s2s10/3o () |

Elements | |

Faces | 10 triangles, 2 pentagrams |

Edges | 10+10 |

Vertices | 10 |

Vertex figure | Crossed isosceles trapezoid, edge lengths 1, 1, 1, (√5–1)/2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–3: |

5/2–3: | |

Height | |

Central density | 3 |

Number of external pieces | 72 |

Level of complexity | 24 |

Related polytopes | |

Army | Pap, edge length |

Regiment | Starp |

Dual | Pentagrammic concave antitegum |

Conjugate | Pentagonal antiprism |

Convex core | Pentagonal antitegum |

Abstract & topological properties | |

Flag count | 80 |

Euler characteristic | 2 |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | (I_{2}(10)×A_{1})/2, order 20 |

Convex | No |

Nature | Tame |

The **pentagrammic retroprism**, or **starp**, also called the **pentagrammic crossed antiprism**, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagrams. Each vertex joins one pentagram and three triangles. As the name suggests, it is a crossed antiprism based on a pentagram, seen as a 5/3-gon rather than 5/2. This makes it the simplest uniform crossed antiprism.

Similar to how the pentagonal antiprism can be edge-inscribed into the regular icosahedron, the pentagrammic retroprism can be edge-inscribed into a great icosahedron. It can be constructed by diminishing two opposite vertices of the great icosahedron.

## Vertex coordinates[edit | edit source]

A pentagrammic retroprism of edge length 1 has vertex coordinates given by:

These coordinates are obtained by removing two opposite vertices from a great icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points along with their central inversions:

## Related polyhedra[edit | edit source]

Two non-prismatic uniform polyhedron compounds are composed of pentagrammic retroprisms:

There are also an infinite amount of prismatic uniform compounds that are the crossed antiprisms of compounds of pentagrams.

## External links[edit | edit source]

- Bowers, Jonathan. "Batch 2: Ike and Sissid Facetings" (#3 under sissid).

- Klitzing, Richard. "starp".
- Wikipedia contributors. "Pentagrammic crossed antiprism".
- McCooey, David. "Pentagrammic Crossed Antiprism"