Polygon

The segments of a closed polygonal chain are called its edges or sides.

The word polygon derives from the Greek adjective πολύς (polús) 'much', 'many' and γωνία (gōnía) 'corner' or 'angle'.

Polygons may be characterized by their convexity or type of non-convexity: The property of regularity may be defined in other ways: a polygon is regular if and only if it is both isogonal and isotoxal, or equivalently it is both cyclic and equilateral.

[4][5] The signed area depends on the ordering of the vertices and of the orientation of the plane.

[6] The area A of a simple polygon can also be computed if the lengths of the sides, a1, a2, ..., an and the exterior angles, θ1, θ2, ..., θn are known, from: The formula was described by Lopshits in 1963.

[7] If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points: the former number plus one-half the latter number, minus 1.

The lengths of the sides of a polygon do not in general determine its area.

The area of a regular n-gon can be expressed in terms of the radius R of its circumscribed circle (the unique circle passing through all vertices of the regular n-gon) as follows:[12][13] The area of a self-intersecting polygon can be defined in two different ways, giving different answers: Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are In these formulas, the signed value of area

For triangles (n = 3), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for n > 3.

Some of the more important include: The word polygon comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled".

Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon.

[17] Exceptions exist for side counts that are easily expressed in verbal form (e.g. 20 and 30), or are used by non-mathematicians.

Some special polygons also have their own names; for example the regular star pentagon is also known as the pentagram.

To construct the name of a polygon with more than 20 and fewer than 100 edges, combine the prefixes as follows.

[21] The "kai" term applies to 13-gons and higher and was used by Kepler, and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra,[25] though not all sources use it.

[40][41] The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century.

[42] In 1952, Geoffrey Colin Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.

[43] Polygons appear in rock formations, most commonly as the flat facets of crystals, where the angles between the sides depend on the type of mineral from which the crystal is made.

Regular hexagons can occur when the cooling of lava forms areas of tightly packed columns of basalt, which may be seen at the Giant's Causeway in Northern Ireland, or at the Devil's Postpile in California.

In biology, the surface of the wax honeycomb made by bees is an array of hexagons, and the sides and base of each cell are also polygons.

They are defined in a database, containing arrays of vertices (the coordinates of the geometrical vertices, as well as other attributes of the polygon, such as color, shading and texture), connectivity information, and materials.

Or, each vertex inside the square mesh connects four edges (lines).

The imaging system calls up the structure of polygons needed for the scene to be created from the database.

This is transferred to active memory and finally, to the display system (screen, TV monitors etc.)

Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation.

lies inside a simple polygon given by a sequence of line segments.