polygon reduction technique
   HOME

TheInfoList



OR:

In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, a polygon () is a plane
figure Figure may refer to: General *A shape, drawing, depiction, or geometric configuration *Figure (wood), wood appearance *Figure (music), distinguished from musical motif *Noise figure, in telecommunication *Dance figure, an elementary dance pattern ...
that is described by a finite number of straight line segments connected to form a closed ''
polygonal chain In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments co ...
'' (or ''polygonal circuit''). The bounded plane
region In geography, regions, otherwise referred to as zones, lands or territories, are areas that are broadly divided by physical characteristics ( physical geography), human impact characteristics ( human geography), and the interaction of humanity an ...
, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
is a 3-gon. A
simple polygon In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If ...
is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
in any number of dimensions. There are many more generalizations of polygons defined for different purposes.


Etymology

The word ''polygon'' derives from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
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''.


Classification


Number of sides

Polygons are primarily classified by the number of sides. See the table below.


Convexity and intersection

Polygons may be characterized by their convexity or type of non-convexity: *
Convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. As a consequence, all its interior angles are less than 180°. Equivalently, any line segment with endpoints on the boundary passes through only interior points between its endpoints. This condition is true for polygons in any geometry, not just Euclidean. * Non-convex: a line may be found which meets its boundary more than twice. Equivalently, there exists a line segment between two boundary points that passes outside the polygon. *
Simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
: the boundary of the polygon does not cross itself. All convex polygons are simple. *
Concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set * The concavity of a ...
: Non-convex and simple. There is at least one interior angle greater than 180°. * Star-shaped: the whole interior is visible from at least one point, without crossing any edge. The polygon must be simple, and may be convex or concave. All convex polygons are star-shaped. * Self-intersecting: the boundary of the polygon crosses itself. The term ''complex'' is sometimes used in contrast to ''simple'', but this usage risks confusion with the idea of a ''
complex polygon The term ''complex polygon'' can mean two different things: * In geometry, a polygon in the unitary plane, which has two complex dimensions. * In computer graphics, a polygon whose boundary is not simple. Geometry In geometry, a complex polygo ...
'' as one which exists in the complex
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
plane consisting of two
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
dimensions. * Star polygon: a polygon which self-intersects in a regular way. A polygon cannot be both a star and star-shaped.


Equality and symmetry

* Equiangular: all corner angles are equal. *
Equilateral In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
: all edges are of the same length. * Regular: both equilateral and equiangular. *
Cyclic Cycle, cycles, or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in s ...
: all corners lie on a single
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
, called the
circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
. *
Tangential In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
: all sides are tangent to an inscribed circle. * Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular. * Isotoxal or
edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given t ...
: all sides lie within the same symmetry orbit. The polygon is also equilateral and tangential. 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. A non-convex regular polygon is called a ''regular star polygon''.


Miscellaneous

* Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees. * Monotone with respect to a given line ''L'': every line orthogonal to L intersects the polygon not more than twice.


Properties and formulas

Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
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: *
Interior angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
– The sum of the interior angles of a simple ''n''-gon is
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s or degrees. This is because any simple ''n''-gon ( having ''n'' sides ) can be considered to be made up of triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular ''n''-gon is \left(1-\tfrac\right)\pi radians or 180-\tfrac degrees. The interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra: for a regular \tfrac-gon (a ''p''-gon with central density ''q''), each interior angle is \tfrac radians or \tfrac degrees. * Exterior angle – The exterior angle is the supplementary angle to the interior angle. Tracing around a convex ''n''-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an ''n''-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple ''d'' of 360°, e.g. 720° for a
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
and 0° for an angular "eight" or antiparallelogram, where ''d'' is the
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
or
turning number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...
of the polygon. See also
orbit (dynamics) In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynami ...
.


Area

In this section, the vertices of the polygon under consideration are taken to be (x_0, y_0), (x_1, y_1), \ldots, (x_, y_) in order. For convenience in some formulas, the notation will also be used.


Simple polygons

If the polygon is non-self-intersecting (that is,
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
), the signed
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
is :A = \frac \sum_^( x_i y_ - x_ y_i) \quad \text x_=x_ \text y_n=y_, or, using
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
s :16 A^ = \sum_^ \sum_^ \begin Q_ & Q_ \\ Q_ & Q_ \end , where Q_ is the squared distance between (x_i, y_i) and (x_j, y_j). The signed area depends on the ordering of the vertices and of the orientation of the plane. Commonly, the positive orientation is defined by the (counterclockwise) rotation that maps the positive -axis to the positive -axis. If the vertices are ordered counterclockwise (that is, according to positive orientation), the signed area is positive; otherwise, it is negative. In either case, the area formula is correct in absolute value. This is commonly called the '' shoelace formula'' or ''surveyor's formula''. The area ''A'' of a simple polygon can also be computed if the lengths of the sides, ''a''1, ''a''2, ..., ''an'' and the exterior angles, ''θ''1, ''θ''2, ..., ''θn'' are known, from: :\beginA = \frac12 ( a_1 _2 \sin(\theta_1) + a_3 \sin(\theta_1 + \theta_2) + \cdots + a_ \sin(\theta_1 + \theta_2 + \cdots + \theta_)\\ + a_2 _3 \sin(\theta_2) + a_4 \sin(\theta_2 + \theta_3) + \cdots + a_ \sin(\theta_2 + \cdots + \theta_)\\ + \cdots + a_ _ \sin(\theta_)). \end 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 p^2 > 4\pi A holds. For any two simple polygons of equal area, the Bolyai–Gerwien theorem asserts that the first can be cut into polygonal pieces which can be reassembled to form the second polygon. The lengths of the sides of a polygon do not in general determine its area. However, if the polygon is simple and cyclic then the sides ''do'' determine the area. Of all ''n''-gons with given side lengths, 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).


Regular polygons

Many specialized formulas apply to the areas of
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s. The area of a regular polygon is given in terms of the radius ''r'' of its inscribed circle and its perimeter ''p'' by :A = \tfrac \cdot p \cdot r. This radius is also termed its
apothem The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...
and is often represented as ''a''. The area of a regular ''n''-gon in terms of the radius ''R'' of its
circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...
can be expressed trigonometrically as: :A = R^2 \cdot \frac \cdot \sin \frac = R^2 \cdot n \cdot \sin \frac \cdot \cos \frac The area of a regular ''n''-gon inscribed in a unit-radius circle, with side ''s'' and interior angle \alpha, can also be expressed trigonometrically as: :A = \frac\cot \frac = \frac\cot\frac = n \cdot \sin \frac \cdot \cos \frac.


Self-intersecting

The area of a self-intersecting polygon can be defined in two different ways, giving different answers: * Using the formulas for simple polygons, we allow that particular regions within the polygon may have their area multiplied by a factor which we call the ''density'' of the region. For example, the central convex pentagon in the center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure. * Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon or to the area of one or more simple polygons having the same outline as the self-intersecting one. In the case of the cross-quadrilateral, it is treated as two simple triangles.


Centroid

Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are :C_x = \frac \sum_^ (x_i + x_) (x_i y_ - x_ y_i), :C_y = \frac \sum_^ (y_i + y_) (x_i y_ - x_ y_i). In these formulas, the signed value of area A must be used. For
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
s (), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for . The
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
of the vertex set of a polygon with vertices has the coordinates :c_x=\frac 1n \sum_^x_i, :c_y=\frac 1n \sum_^y_i.


Generalizations

The idea of a polygon has been generalized in various ways. Some of the more important include: * A spherical polygon is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows the
digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...
, a polygon having only two sides and two corners, which is impossible in a flat plane. Spherical polygons play an important role in
cartography Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
(map making) and in Wythoff's construction of the
uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...
. * A
skew polygon Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do not ...
does not lie in a flat plane, but zigzags in three (or more) dimensions. The
Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a ...
s of the regular polytopes are well known examples. * An apeirogon is an infinite sequence of sides and angles, which is not closed but has no ends because it extends indefinitely in both directions. * A skew apeirogon is an infinite sequence of sides and angles that do not lie in a flat plane. * A
complex polygon The term ''complex polygon'' can mean two different things: * In geometry, a polygon in the unitary plane, which has two complex dimensions. * In computer graphics, a polygon whose boundary is not simple. Geometry In geometry, a complex polygo ...
is a configuration analogous to an ordinary polygon, which exists in the complex plane of two
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
and two imaginary dimensions. * An abstract polygon is an algebraic
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
representing the various elements (sides, vertices, etc.) and their connectivity. A real geometric polygon is said to be a ''realization'' of the associated abstract polygon. Depending on the mapping, all the generalizations described here can be realized. * A
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on ...
is a three-dimensional solid bounded by flat polygonal faces, analogous to a polygon in two dimensions. The corresponding shapes in four or higher dimensions are called
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
s. (In other conventions, the words ''polyhedron'' and ''polytope'' are used in any dimension, with the distinction between the two that a polytope is necessarily bounded.)


Naming

The word ''polygon'' comes from
Late Latin Late Latin ( la, Latinitas serior) is the scholarly name for the form of Literary Latin of late antiquity.Roberts (1996), p. 537. English dictionary definitions of Late Latin date this period from the , and continuing into the 7th century in t ...
''polygōnum'' (a noun), from
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
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 Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
-derived
numerical prefix Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cyc ...
with the suffix ''-gon'', e.g. '' pentagon'', '' dodecagon''. The
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...
,
quadrilateral In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
and
nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ''nonagon'' is a prefix hybrid formation, from Latin (''nonus'', "ninth" + ''gonon''), used equivalently, attested already in the 16th century in French ''nonogo ...
are exceptions. Beyond decagons (10-sided) and dodecagons (12-sided), mathematicians generally use numerical notation, for example 17-gon and 257-gon.Mathworld 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 A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
. 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 and was used by
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
, and advocated by John H. Conway for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra, though not all sources use it.


History

Polygons have been known since ancient times. The
regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s were known to the ancient Greeks, with the
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
, a non-convex regular polygon ( star polygon), appearing as early as the 7th century B.C. on a
krater A krater or crater ( grc-gre, , ''kratēr'', literally "mixing vessel") was a large two-handled shape of vase in Ancient Greek pottery and metalwork, mostly used for the mixing of wine with water. Form and function At a Greek symposium, krat ...
by
Aristophanes Aristophanes (; grc, Ἀριστοφάνης, ; c. 446 – c. 386 BC), son of Philippus, of the deme Kydathenaion ( la, Cydathenaeum), was a comic playwright or comedy-writer of ancient Athens and a poet of Old Attic Comedy. Eleven of his for ...
, found at
Caere : Caere (also Caisra and Cisra) is the Latin name given by the Romans to one of the larger cities of southern Etruria, the modern Cerveteri, approximately 50–60 kilometres north-northwest of Rome. To the Etruscans it was known as Cisra, t ...
and now in the
Capitoline Museum The Capitoline Museums (Italian: ''Musei Capitolini'') are a group of art and archaeological museums in Piazza del Campidoglio, on top of the Capitoline Hill in Rome, Italy. The historic seats of the museums are Palazzo dei Conservatori and Pal ...
. 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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
dimension is accompanied by an imaginary one, to create complex polygons.


In nature

Polygons appear in rock formations, most commonly as the flat facets of
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
s, 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 Lava is molten or partially molten rock (magma) that has been expelled from the interior of a terrestrial planet (such as Earth) or a moon onto its surface. Lava may be erupted at a volcano or through a fracture in the crust, on land or un ...
forms areas of tightly packed columns of
basalt Basalt (; ) is an aphanitic (fine-grained) extrusive igneous rock formed from the rapid cooling of low-viscosity lava rich in magnesium and iron (mafic lava) exposed at or very near the surface of a rocky planet or moon. More than 90 ...
, which may be seen at the
Giant's Causeway The Giant's Causeway is an area of about 40,000 interlocking basalt columns, the result of an ancient volcanic fissure eruption. It is located in County Antrim on the north coast of Northern Ireland, about three miles (5 km) northeast of ...
in
Northern Ireland Northern Ireland ( ga, Tuaisceart Éireann ; sco, label= Ulster-Scots, Norlin Airlann) is a part of the United Kingdom, situated in the north-east of the island of Ireland, that is variously described as a country, province or region. Nort ...
, or at the Devil's Postpile in
California California is a state in the Western United States, located along the Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the most populous U.S. state and the 3rd largest by area. It is also the m ...
. In
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
, the surface of the wax
honeycomb A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of honey ...
made by bees is an array of
hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...
s, and the sides and base of each cell are also polygons.


Computer graphics

In
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, 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 Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geolog ...
. Any surface is modelled as a tessellation called
polygon mesh In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangles ( triangle mesh), quadrilaterals (quads), or other simple convex p ...
. If a square mesh has points (vertices) per side, there are ''n'' squared squares in the mesh, or 2''n'' squared triangles since there are two triangles in a square. There are 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=(x_0,y_0) lies inside a simple polygon given by a sequence of line segments. This is called the
point in polygon In computational geometry, the point-in-polygon (PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon. It is a special case of point location problems and finds applications in areas that dea ...
test.


See also

* Boolean operations on polygons *
Complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...
*
Constructible polygon In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinite ...
*
Cyclic polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...
* Geometric shape * Golygon *
List of polygons In geometry, a polygon is traditionally a plane (mathematics), plane Shape, figure that is bounded by a finite chain of straight line segments closing in a loop to form a Polygonal chain, closed chain. These segments are called its ''edges'' or ...
*
Polyform In recreational mathematics, a polyform is a plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle ...
* Polygon soup * Polygon triangulation * Precision polygon * Spirolateral *
Synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass ...
*
Tiling Tiling may refer to: *The physical act of laying tiles * Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester T ...
* Tiling puzzle


References


Bibliography

* Coxeter, H.S.M.; ''
Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', Methuen and Co., 1948 (3rd Edition, Dover, 1973). * Cromwell, P.; ''Polyhedra'', CUP hbk (1997), pbk. (1999). * Grünbaum, B.; Are your polyhedra the same as my polyhedra? ''Discrete and comput. geom: the Goodman-Pollack festschrift'', ed. Aronov et al. Springer (2003) pp. 461–488.''
pdf


Notes


External links

*

with Greek Numerical Prefixes

with interactive animation
How to draw monochrome orthogonal polygons on screens
by Herbert Glarner
comp.graphics.algorithms Frequently Asked Questions
solutions to mathematical problems computing 2D and 3D polygons

compares capabilities, speed and numerical robustness

Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons {{Authority control Euclidean plane geometry