# Bowtie

Bowtie | |
---|---|

Rank | 2 |

Type | Semi-uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Bowtie |

Elements | |

Edges | 2+2 |

Vertices | 4 |

Vertex figure | Dyad |

Measures (edge lengths a [short], b [long]) | |

Circumradius | |

Area | 0 |

Angle | |

Related polytopes | |

Army | Rect |

Dual | Infinite quadrilateral |

Conjugate | Bowtie |

Abstract & topological properties | |

Orientable | Yes |

Properties | |

Symmetry | K_{2}, order 4 |

Convex | No |

Nature | Tame |

The **bowtie**, or **crossed rectangle**, is a nonconvex semi-uniform quadrilateral with the same vertices as a rectangle but with two of the original sides removed and with the original's diagonals in place instead.

It is the only semi-uniform polygon with an even amount of sides that isn't the truncation of any other polygon. It is also unusual in that its sides with equal length don't go around its circumcircle in a consistent direction. It is the only semi-uniform polygon that does not have a set angle; its angle varies with its proportions.

Bowties are the simplest possible type of hemipolytope.

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of a bowtie with two sides of length *a* and two intersecting sides of length *b*, with *b* > *a*, are:

- (±
*a*/2, ±√*b*^{2}–*a*^{2}/2).

## In vertex figures[edit | edit source]

Many uniform hemipolyhedra have bowties as their vertex figure.

Name | Picture | Edge lengths |
---|---|---|

Tetrahemihexahedron | 1, √2 | |

Octahemioctahedron | 1, √3 | |

Cubohemioctahedron | √2, √3 | |

Small icosihemidodecahedron | 1, √(5+√5)/2 | |

Small dodecahemidodecahedron | (1+√5)/2, √(5+√5)/2 | |

Great icosihemidodecahedron | 1, √(5–√5)/2 | |

Great dodecahemidodecahedron | (√5–1)/2, √(5–√5)/2 | |

Small dodecahemicosahedron | (√5–1)/2, √3 | |

Great dodecahemicosahedron | (1+√5)/2, √3 |