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	$\frac{b}{2}$
Area	0
Angle	$\arccos\left(\frac{a}{b}\right)$
Related polytopes
Army	Rect
Dual	Infinite quadrilateral
Conjugate	Bowtie
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	K2, 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

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, ±√b²–a²/2).

In vertex figures

Many uniform hemipolyhedra have bowties as their vertex figure.

Bowties in vertex figures

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