Crossed polygon

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.

A crossed equilateral pentagon
The vertex figure of a snub icosidodecadodecahedron is a crossed hexagon.
A symmetric crossed decagon

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.[1]

Crossed quadrilateral

Crossed quadrilaterals are most common, including:

  • crossed parallelogram or antiparallelogram, a crossed quadrilateral with alternate edges of equal length.
  • crossed trapezoid' has two opposite parallel edges.
  • crossed rectangle, an antiparallelogram whose edges are two opposite sides and the two diagonals of a rectangle.
  • Crossed square, a crossed rectangle with two equal opposite sides and two diagonals of a square.

Crossed
square
Crossed
trapezoid
Crossed
parallelogram
Crossed
rectangles
Crossed quadrilaterals

See also

References
  1. Coxeter, H.S.M., M. S. Longuet-Higgins and J.C.P Miller, Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401–450.
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