Polygon

A polygon is a closed 2-dimensional shape constructed from line segments bound together in a closed loop. It can be considered a subset of a Euclidean plane. A polygon is composed of a single face, the individual line segments that bound the polygon are the edges of the polygon, and the points where two edges meet are the vertices. Polygons are 2 dimensional polytopes, which are generalizations of polygons and can have any arbitrary number of dimensions.

Polygons make up the boundary, the subfacets, of polyhedra and define their surcell volume.

There are many examples of generalizations of polygons. One example is series of the self-intersecting polygons, which allow for edges to intersect one another and includes the infinite series of star polygons (also known as polygrams). Another generalization is the family of complex polygons, which allow for edges to be comtela, and are subsets of the complex plane.

The boundary of a polygon can be considered a spherical tiling of a circle, a 1-dimensional hypersphere. Polygons with a finite number of sides degenerate in Euclidean space can be represented as a tiling of the circle. The only two polygons that can only be represented that way are the monogon and the digon. They do not have a defined interior. The other degenerate polygons, the apeirogon and long apeirogon (with $\aleph_0$ and $\aleph_1$ edges respectively), can only be represented as a tiling of a Euclidean line and long line, respectively.

Uniform polygons[]

Polygons whose sides and angles are all equal, known as regular polygons, are all uniform polygons. This means that the two terms can be used interchangeably.

The series of uniform/regular polygons is infinite. Below is a chart of uniform polygons up to 16 edges by Schläfli symbol. All uniform polygons have a one entry Schläfli symbol.


$\{0\}$	$\{1\}$	$\{2\}$	$\{3\}$	$\{4\}$	$\{5\}$	$\{\frac{5}{2}\}$	$\{6\}$	$\{7\}$	$\{\frac{7}{2}\}$	$\{\frac{7}{3}\}$	$\{8\}$	$\{\frac{8}{3}\}$	$\{9\}$	$\{\frac{9}{2}\}$	$\{\frac{9}{4}\}$	$\{10\}$	$\{\frac{10}{3}\}$	$\{11\}$	$\{\frac{11}{2}\}$	$\{\frac{11}{3}\}$	$\{\frac{11}{4}\}$	$\{\frac{11}{5}\}$	$\{12\}$	$\{\frac{12}{5}\}$	$\{13\}$	$\{\frac{13}{2}\}$	$\{\frac{13}{3}\}$	$\{\frac{13}{4}\}$	$\{\frac{13}{5}\}$	$\{\frac{13}{6}\}$	$\{14\}$	$\{\frac{14}{3}\}$	$\{\frac{14}{5}\}$	$\{15\}$	$\{\frac{15}{2}\}$	$\{\frac{15}{4}\}$	$\{\frac{15}{7}\}$	$\{16\}$	$\{\frac{16}{3}\}$	$\{\frac{16}{5}\}$	$\{\frac{16}{7}\}$	$\{17\}$	$\{\frac{17}{2}\}$	$\{\frac{17}{3}\}$	$\{\frac{17}{4}\}$	$\{\frac{17}{5}\}$	$\{\frac{17}{6}\}$	$\{\frac{17}{7}\}$	$\{\frac{17}{8}\}$	...	$\{\infty\}$	$\{x\}$	$\{\frac{\pi i}{\lambda}\}$
Zerogon	Monogon	Digon	Triangle	Square	Pentagon	Pentagram	Hexagon	Heptagon	Heptagram	Great heptagram	Octagon	Octagram	Enneagon	Enneagram	Great enneagram	Decagon	Decagram	Hendecagon	Small hendecagram	Hendecagram	Great hendecagram	Grand hendecagram	Dodecagon	Dodecagram	Tridecagon	Small tridecagram	Tridecagram	Medial tridecagram	Great tridecagram	Grand tridecagram	Tetradecagon	Tetradecagram	Great tetradecagram	Pentadecagon	Small pentadecagram	Pentadecagram	Great pentadecagram	Hexadecagon	Small hexadecagram	Hexadecagram	Great hexadecagram	Heptadecagon	Tiny heptadecagram	Small heptadecagram	Heptadecagram	Medial heptadecagram	Great heptadecagram	Giant heptadecagram	Grand heptadecagram	...	Apeirogon	Failed star polygon ($x$-gon)	Pseudogon ($\frac{\pi i}{\lambda}$-gon)

Polygon

Uniform polygons[]

See Also[]