Crossed polygon

A crossed equilateral pentagon

The vertex figure of a snub icosidodecadodecahedron is a crossed hexagon.

A symmetric crossed decagon

A crossed polygon is a polygon in the plane with a turning number or density of zero, with the appearance of a figure 8, infinity symbol, or lemniscate curve.

Crossed polygons are related to star polygons which have turning numbers greater than 1.

The vertices with clockwise turning angles equal the vertices with counterclockwise turning angles. A crossed polygon will always have at least 2 edges or vertices intersecting or coinciding.

Any convex polygon with 4 or more sides can be remade into a crossed polygon by swapping the positions of two adjacent vertices.

Crossed polygons are common as vertex figures of uniform star polyhedra.

Crossed quadrilateral

Crossed quadrilaterals are most common, including:

• the crossed trapezoid or antiparallelogram, a crossed quadrilateral with alternate edges of equal length.
• the crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle, hence having two edges parallel.

Crossed square

Crossed trapezoids

Crossed rectangles

Crossed quadrilaterals

See also

• Skew polygon