A crossed polygon is a polygon in the plane with a turning number or

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...

of zero, with the appearance of a figure 8, infinity symbol, or

lemniscate In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternative ...

curve. ''Crossed polygons'' are related to

star polygons In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

which have turning numbers greater than 1. The vertices with

clockwise Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite ...

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 In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...

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

Crossed quadrilateral

Crossed quadrilaterals are most common, including: *''crossed parallelogram'' or

antiparallelogram In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, sides in the lon ...

, 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.

References

{{reflist Types of polygons

Crossed quadrilateral

See also

References