In elementary geometry, a polygon /ˈpɒlɪɡɒn/ is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of the polygon is sometimes called its body. An n-gon is a polygon with n sides; for example, a triangle is a 3-gon. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions.
Contents
- Etymology
- Number of sides
- Convexity and non convexity
- Equality and symmetry
- Miscellaneous
- Properties and formulas
- Angles
- Simple polygons
- Self intersecting polygons
- Generalizations of polygons
- Naming polygons
- Constructing higher names
- History
- Polygons in nature
- Polygons in computer graphics
- References
The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not self-intersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and other self-intersecting polygons. These and other generalizations of polygons are described below.
Etymology
The word "polygon" derives from the Greek adjective πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". It has been suggested that γόνυ (gónu) "knee" may be the origin of “gon”.
Number of sides
Polygons are primarily classified by the number of sides. See table below.
Convexity and non-convexity
Polygons may be characterized by their convexity or type of non-convexity:
Equality and symmetry
Miscellaneous
Properties and formulas
Euclidean geometry is assumed throughout.
Angles
Any polygon has as many corners as it has sides. Each corner has several angles. The two most important ones are:
Simple polygons
For a non-self-intersecting (simple) polygon with n vertices xi, yi ( i = 1 to n), the signed area and the Cartesian coordinates of the centroid are given by:
where
To close the polygon, the first and last vertices are the same, i.e., xn, yn = x0, y0. The vertices must be ordered according to positive or negative orientation (counterclockwise or clockwise, respectively); if they are ordered negatively, the value given by the area formula will be negative but correct in absolute value, but when calculating
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.
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.
In every polygon with perimeter p and area A , the isoperimetric inequality
If any two simple polygons of equal area are given, then the first can be cut into polygonal pieces which can be reassembled to form the second polygon. This is the Bolyai–Gerwien theorem.
The area of a regular polygon is also given in terms of the radius r of its inscribed circle and its perimeter p by
This radius is also termed its apothem and is often represented as a.
The area of a regular n-gon with side s inscribed in a unit circle is
The area of a regular n-gon in terms of the radius R of its circumscribed circle and its perimeter p is given by
The area of a regular n-gon inscribed in a unit-radius circle, with side s and interior angle
The lengths of the sides of a polygon do not in general determine the area. However, if the polygon is cyclic the sides do determine the area.
Of all n-gons with given sides, the one with the largest area is cyclic. Of all n-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).
Self-intersecting polygons
The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer:
Generalizations of polygons
The idea of a polygon has been generalized in various ways. Some of the more important include:
Naming polygons
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. The triangle, quadrilateral and nonagon are exceptions.
Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.
Exceptions exist for side counts that are more 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.
Constructing higher names
To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows. The "kai" term applies to 13-gons and higher was used by Kepler, and advocated by John H. Conway for clarity to concatenated prefix numbers in the naming of quasiregular polyhedra.
History
Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, with the pentagram, a non-convex regular polygon (star polygon), appearing as early as the 7th century B.C. on a krater by Aristonothos, found at Caere and now in the Capitoline Museum.
The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine in the 14th century.
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.
Polygons in nature
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.
Polygons in computer graphics
In computer graphics, a polygon is a primitive used in modelling and rendering. 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.
Naming conventions differ from those of mathematicians:
Any surface is modelled as a tessellation called polygon mesh. If a square mesh has n + 1 points (vertices) per side, there are n squared squares in the mesh, or 2n squared triangles since there are two triangles in a square. There are (n + 1)2 / 2(n2) vertices per triangle. Where n is large, this approaches one half. 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.) so that the scene can be viewed. During this process, the imaging system renders polygons in correct perspective ready for transmission of the processed data to the display system. Although polygons are two-dimensional, through the system computer they are placed in a visual scene in the correct three-dimensional orientation.
In computer graphics and computational geometry, it is often necessary to determine whether a given point P = (x0,y0) lies inside a simple polygon given by a sequence of line segments. This is called the Point in polygon test.