![Assorted polygons](https://upload.wikimedia.org/wikipedia/commons/1/1f/Assorted_polygons.svg)
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 c ...
, a polygon () is a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
figure that is described by a finite number of straight
line segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between ...
s connected to form a closed ''
polygonal chain'' (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 and t ...
, the bounding circuit, or the two together, may be called a polygon.
The segments of a polygonal circuit are called its ''
edges
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
'' 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 Edge (geometry), edges and three Vertex (geometry), 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, an ...
is a 3-gon.
A
simple polygon 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 polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
s and other
self-intersecting polygons
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edges cross each other. They contrast with simple polygons, whose edges never cross.
Some types of self-intersecting polygons are:
*the crossed qu ...
.
A polygon is a 2-dimensional example of the more general
polytope in any number of dimensions. There are many more
generalizations of polygons defined for different purposes.
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''.
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: 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
In ca ...
: Non-convex and simple. There is at least one interior angle greater than 180°.
*
Star-shaped
In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defini ...
: 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'' as one which exists in the complex
Hilbert plane consisting of two
complex dimensions.
*
Star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
: 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: 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 soc ...
: 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 const ...
, called the
circumcircle.
*
Tangential: all sides are tangent to an
inscribed circle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
.
* Isogonal or
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
: all corners lie within the same
symmetry orbit
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
. The polygon is also cyclic and equiangular.
* Isotoxal or
edge-transitive: all sides lie within the same
symmetry orbit
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
. 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
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
''.
Miscellaneous
*
Rectilinear: the polygon's sides meet at right angles, i.e. all its interior angles are 90 or 270 degrees.
*
Monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
with respect to a given line ''L'': every line
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
to L intersects the polygon not more than twice.
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:
*
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 that c ...
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
radians or
degrees. The interior angles of regular
star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
s were first studied by Poinsot, in the same paper in which he describes the four
regular star polyhedra: for a regular
-gon (a ''p''-gon with central density ''q''), each interior angle is
radians or
degrees.
*
Exterior 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 exterior angle is the
supplementary angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles are ...
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
Turn may refer to:
Arts and entertainment
Dance and sports
* Turn (dance and gymnastics), rotation of the body
* Turn (swimming), reversing direction at the end of a pool
* Turn (professional wrestling), a transition between face and heel
* 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 and 0° for an angular "eight" or
antiparallelogram
In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, sides in the lon ...
, 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 of the polygon. See also
orbit (dynamics).
Area
![Polygon vertex labels](https://upload.wikimedia.org/wikipedia/commons/4/42/Polygon_vertex_labels.svg)
In this section, the vertices of the polygon under consideration are taken to be
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
A shape or figure is a graphics, graphical representation of an obje ...
is
:
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 and ...
s
:
where
is the squared distance between
and
The signed area depends on the ordering of the vertices and of the
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
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
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
. This is commonly called the ''
shoelace formula
The shoelace formula, shoelace algorithm, or shoelace method (also known as Gauss's area formula and the surveyor's formula) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian c ...
'' 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, ..., ''a
n'' and the
exterior 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 ...
s, ''θ''
1, ''θ''
2, ..., ''θ
n'' 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
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 18 ...
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
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
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 polygons.
The area of a regular polygon is given in terms of the radius ''r'' of its
inscribed circle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
and its perimeter ''p'' by
:
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 can be expressed trigonometrically as:
:
The area of a regular ''n''-gon inscribed in a unit-radius circle, with side ''s'' and interior angle
can also be expressed trigonometrically as:
:
Self-intersecting
The area of a
self-intersecting polygon
Self-intersecting polygons, crossed polygons, or self-crossing polygons are polygons some of whose edge (geometry), edges cross each other. They contrast with simple polygons, whose edges never cross.
Some types of self-intersecting polygons are: ...
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
:
:
In these formulas, the signed value of area
must be used.
For
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...
s (), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for . The
centroid of the vertex set of a polygon with vertices has the coordinates
:
:
Generalizations
The idea of a polygon has been generalized in various ways. Some of the more important include:
* A
spherical polygon
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
is a circuit of arcs of great circles (sides) and vertices on the surface of a sphere. It allows the
digon, 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 im ...
(map making) and in
Wythoff's construction
In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscope, kaleidoscopic construction.
Constructio ...
of the
uniform polyhedra.
* 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 polygons of the regular polytopes are well known examples.
* An
apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
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
In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lin ...
is an infinite sequence of sides and angles that do not lie in a flat plane.
* A
complex polygon is a
configuration
Configuration or configurations may refer to:
Computing
* Computer configuration or system configuration
* Configuration file, a software file used to configure the initial settings for a computer program
* Configurator, also known as choice board ...
analogous to an ordinary polygon, which exists in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
of two
real 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 (mathematics), set. A poset consists of a set toget ...
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 th ...
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
polytopes. (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 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
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
'', ''
dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
Regular dodecagon
A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational sym ...
''. The
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...
,
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.
[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
A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
is also known as the
pentagram.
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
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
for clarity of concatenated prefix numbers in the naming of quasiregular polyhedra
In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular polygon, regular faces, which alternate around each vertex (geometry), vertex. They are vertex-transitive and edge-transitive, hence a step closer ...
, though not all sources use it.
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
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...
), appearing as early as the 7th century B.C. on a krater by Aristophanes
Aristophanes (; grc, Ἀριστοφάνης, ; c. 446 – c. 386 BC), son of Philippus, of the deme
In Ancient Greece, a deme or ( grc, δῆμος, plural: demoi, δημοι) was a suburb or a subdivision of Athens and other city-states ...
, found at Caere 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 Palazz ...
.
The first known systematic study of non-convex polygons in general was made by Thomas Bradwardine
Thomas Bradwardine (c. 1300 – 26 August 1349) was an English cleric, scholar, mathematician, physicist, courtier and, very briefly, Archbishop of Canterbury. As a celebrated scholastic philosopher and doctor of theology, he is often call ...
in the 14th century.
In 1952, Geoffrey Colin Shephard
Geoffrey Colin Shephard is a mathematician who works on convex geometry and reflection groups. He asked Shephard's problem on the volumes of projected convex bodies, posed another problem on polyhedral nets, proved the Shephard–Todd theorem in ...
generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons
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 polygon ...
.
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, macros ...
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 aphanite, 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 planetary surface, surface of a terrestrial ...
, which may be seen at the Giant's Causeway 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
Devils Postpile National Monument is a U.S. National Monument located near Mammoth Mountain in Eastern California. The monument protects Devils Postpile, an unusual rock formation of columnar basalt, “all closely and perfectly fitted together ...
in California
California is a U.S. state, state in the Western United States, located along the West Coast of the United States, Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the List of states and territori ...
.
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 made by bee
Bees are winged insects closely related to wasps and ants, known for their roles in pollination and, in the case of the best-known bee species, the western honey bee, for producing honey. Bees are a monophyly, monophyletic lineage within the ...
s is an array of hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
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.
Any surface is modelled as a tessellation called polygon mesh. 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
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...
, it is often necessary to determine whether a given point lies inside a simple polygon given by a sequence of line segments. This is called the point in polygon test.
See also
* Boolean operations on polygons Boolean operations on polygons are a set of Boolean operations (AND, OR, NOT, XOR, ...) operating on one or more sets of polygons in computer graphics. These sets of operations are widely used in computer graphics, CAD, and in EDA (in integrated ci ...
* Complete graph
* Constructible polygon
* Cyclic polygon
* Geometric shape
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type.
A plane shape or plane figure is constrained to lie on ...
* Golygon
A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a 19 ...
* List of polygons
* Polyform
* Polygon soup
A polygon soup is a set of unorganized polygons, typically triangles, before the application of any structuring operation, such as e.g. octree grouping.
The term must not to be confused with the "PolySoup" operation available in the 3D package Ho ...
* Polygon triangulation
In computational geometry, polygon triangulation is the partition of a polygonal area (simple polygon) into a set of triangles, i.e., finding a set of triangles with pairwise non-intersecting interiors whose union is .
Triangulations may be v ...
* Precision polygon
Precision polygons are basic standards for angle measurement, which are used and Calibration, calibrated by the national standards Laboratory, laboratories in particular. Multifaced polygons are used for calibration of rotary tables and dividing he ...
* Spirolateral
In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...
* Synthetic geometry
* Tiling
* Tiling puzzle
Tiling puzzles are puzzles involving two-dimensional packing problems in which a number of flat shapes have to be assembled into a larger given shape without overlaps (and often without gaps). Some tiling puzzles ask you to dissect a given ...
References
Bibliography
* Coxeter, H.S.M.; '' Regular Polytopes'', 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.''
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Notes
External links
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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
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Euclidean plane geometry