# Pentagram

The **pentagram** is a non-convex polygon with 5 sides and the simplest star regular polygon. A regular pentagram has equal sides and equal angles.

Pentagram | |
---|---|

Rank | 2 |

Type | Regular |

Space | Spherical |

Notation | |

Bowers style acronym | Star |

Coxeter diagram | x5/2o () |

Schläfli symbol | {5/2} |

Elements | |

Edges | 5 |

Vertices | 5 |

Vertex figure | Dyad, length (√5–1)/2 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | |

Angle | 36° |

Central density | 2 |

Number of external pieces | 10 |

Level of complexity | 2 |

Related polytopes | |

Army | Peg, edge length |

Dual | Pentagram |

Conjugate | Pentagon |

Convex core | Pentagon |

Abstract & topological properties | |

Flag count | 10 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{2}, order 10 |

Convex | No |

Nature | Tame |

This is the only stellation of the pentagon. The only other polygons with a single non-compound stellation are the octagon, the decagon, and the dodecagon.

Pentagrams occur as faces in two of the four Kepler-Poinsot solids, namely the small stellated dodecahedron and great stellated dodecahedron.

## Vertex coordinatesEdit

Coordinates for the vertices of a regular pentagram of unit edge length, centered at the origin, are:

## RepresentationsEdit

A regular pentagram has the following Coxeter diagrams:

- x5/2o
- ß5o (as holosnub pentagon)

## In vertex figuresEdit

The regular pentagram appears as a vertex figure in two uniform polyhedra, namely the great icosahedron (with an edge length of 1) and the great dodecahedron (with an edge length of (1+√5)/2). Irregular pentagrams further appear as the vertex figures of some snub polyhedra.

## External linksEdit

- Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

- Klitzing, Richard. "Polygons"
- Nan Ma. "Pentagram {5/2}".

- Wikipedia Contributors. "Pentagram".
- Hartley, Michael. "{5}*10".