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Regular convex polygons
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A polygon is any polytope of rank two. These are usually realized in two dimensions, as shapes bounded by straight line segments. Polygons can be convex or nonconvex. All polygons are orientable.

Polygons that aren't compounds consist of a single circuit of vertices and edges. They always have the same amount of vertices as edges. As such, polygons may be characterized by any of these numbers. A polygon with n  vertices or edges is called an n -gon. For small n , the number is often replaced by the appropriate Greek root. Trigons and tetragons are more commonly called triangles and quadrilaterals.

In Euclidean space[edit | edit source]

In Euclidean space the simplest possible non-degenerate polygon is a triangle. A digon cannot be realized in Euclidean space with curved or coinciding edges. Geometrically, the regular polygons are those with congruent edges and equal interior angles. There are infinitely many regular convex polygons, one for each number of sides starting from 3. This is unlike the higher-dimensional geometrically regular polytopes, of which there are only finitely many for each dimension.

The possible symmetries of a polygon include no symmetry (scalene triangle), central inversion symmetry (parallelogram), mirror symmetry (isosceles triangle), and dihedral symmetry (square). Polygonal symmetries can exist in higher dimensions, such as pyramidal symmetries, duoprismatic symmetries, and step prism symmetries.

Notably, an isogonal polygon in Euclidean space, even a skew one, can only have full dihedral symmetry. This is not to say that polygons with cyclic symmetry don't exist: an example is a pinwheel shape.

Properties[edit | edit source]

Restricting ourselves to finite polygons that are not compounds, a polygon is ordinary if no three edges meet at a point, either at endpoints or in their interiors.[1] A stronger condition is a simple polygon where no two edges intersect in their interiors. Simple polygons are precisely non-self-intersecting polygons, and their edges outline a simple Jordan curve.

A whisker is a part of a polygon where two or more edges perfectly overlap in a line segment of positive length (an example would be a polygon that backtracks at a vertex). The presence of a whisker may render a polygon degenerate depending on definition.

3D Euclidean space[edit | edit source]

A skew decagon. Its vertices and edges are drawn but its face is not, since it does not occupy a well defined space.

There also exist skew polygons, whose vertices lie in three dimensions or higher[note 1]. These skew polygons have no defined interior, however they do have straight edges. They appear as the Petrie polygons of polyhedra or as the faces of skew polyhedra. For example, the cube has a regular skew hexagon as its Petrie polygon.

Other spaces[edit | edit source]

Polygons in hyperbolic space behave very similarly to polygons in Euclidean space. However in spherical, toroidal, and real projective spaces lines can intersect at more than 1 point allowing the non-degenerate digons and monogons.

Abstract polygons[edit | edit source]

Abstractly there is exactly one polygon for every number greater than 1. Two abstract polygons with the same number of sides are isomorphic. Additionally all abstract polygons are regular and self-dual.

The monogon is not a valid abstract polytope since it violates the diamond condition.

Complexes and maniplexes[edit | edit source]

A monogon as a maniplex.

For rank 2, complexes and maniplexes are both equivalent to abstract polytopes. All rank 2 complexes are maniplexes[2], all rank 2 maniplexes are abstract polygons, all abstract polygons are complexes, etc.

However if the definition of a complex is loosened to allow multiple edges, the monogon is a valid complex (and maniplex). This interpretation of the monogon differs substantially from an abstract interpretation, having 2 flags instead of 1.

Notes[edit | edit source]

  1. The zigzag is often considered a skew apeirogon even though its vertices lie entirely in 2D space.

References[edit | edit source]

  1. Grunbaum and Shepard. "Rotation and winding numbers for planar polygons and curves."
  2. Wilson, Steve (2012). "Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators". Symmetry: 269. doi:10.3390/sym4020265.

External links[edit | edit source]