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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/77/Equilateral_pentagon-decatile3.svg/220px-Equilateral_pentagon-decatile3.svg.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/12/Snub_icosidodecadodecahedron_vertfig.png/220px-Snub_icosidodecadodecahedron_vertfig.png)
![](http://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Crossed-decagon.png/220px-Crossed-decagon.png)
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
editCrossed 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
editReferences
edit- ^ Coxeter, H.S.M., M. S. Longuet-Higgins and J.C.P Miller, Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401–450.