# Stephanoid

n-gonal stephanoid | |
---|---|

Rank | 3 |

Type | Noble |

Elements | |

Faces | 2n butterflies |

Edges | 2n+2n |

Vertices | 2n |

Vertex figure | Butterfly |

Measures | |

Volume | |

Central density | 0 |

Related polytopes | |

Dual | n-gonal stephanoid |

Convex hull | n-gonal prism or antiprism |

Abstract & topological properties | |

Flag count | 16n |

Euler characteristic | 0 |

Orientable | Yes |

Genus | 1 |

Properties | |

Convex | No |

Nature | Tame |

A **stephanoid** or **crown polyhedron** is a noble polyhedron whose faces are butterflies and which has dihedral symmetry. Their convex hulls are prisms or antiprisms. They are self-dual.

There is a stephanoid with -gonal dihedral symmetry for every pair and where the faces have vertices steps apart on one base and steps apart on the other base, where and (those cases are degenerate). This gives distinct -gonal stephanoids, although if , , and share a common factor, the resulting stephanoid is a compound. If is even, the convex hull is a prism, else it is an antiprism.

## In vertex figures[edit | edit source]

Square stephanoids appear as the vertex figures of sirc and girc. Pentagonal (1,3)-stephanoids appear as the vertex figures of sriphi, mriphi, griphi, and graphi. Non-noble variants of pentagonal and hexagonal stephanoids appear as the vertex figures of sidpaxhi, gidpaxhi, and toditdy.