# User Contributed Dictionary

### Noun

polygons- Plural of polygon

# Extensive Definition

In geometry a polygon () is
traditionally a plane
figure that is bounded by
a closed path
or circuit, composed of a finite sequence of straight line
segments (i.e., by a closed
polygonal chain). These segments are called its edges or sides,
and the points where two edges meet are the polygon's vertices or
corners. The interior of the polygon is sometimes called its body.
A polygon is a 2-dimensional example of the more general polytope in any number of
dimensions.

Usually two edges meeting at a corner are
required to form an angle that is not straight (180°); otherwise,
the line segments will be considered parts of a single edge.

The basic geometrical notion has been adapted in
various ways to suit particular purposes. For example in the
computer graphics (image generation) field, the term polygon
has taken on a slightly altered meaning, more related to the way
the shape is stored and manipulated within the computer.

## Classification

### Number of sides

Polygons are primarily classified by the number of sides, see naming polygons below.### Convexity

Polygons may be characterised by their degree of convexity:- Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice.
- Non-convex: a line may be found which meets its boundary more than twice.
- Simple: the boundary of the polygon does not cross itself. All convex polygons are simple.
- Concave: Non-convex and simple.
- Star-shaped: the whole interior is visible from a single point, without crossing any edge. The polygon must be simple, and may be convex or concave.
- Self-intersecting: the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used. The term complex is sometimes used in contrast to simple, but this 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: a polygon which self-intersects in a regular way.

### Symmetry

- Equiangular: all its corner angles are equal.
- Cyclic: all corners lie on a single circle.
- Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.
- Equilateral: all edges are of the same length. (A polygon with 5 or more sides can be equilateral without being convex.) (Williams 1979, pp. 31-32)
- Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral.
- Regular. A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.

### Miscellaneous

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

## Properties

We will assume Euclidean geometry throughout.### Angles

Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides. Each corner has several angles. The two most important ones are:- Interior angle - The sum of the interior angles of a simple n-gon is (n−2)π radians or (n−2)180 degrees. This is because any simple n-gon can be considered to be made up of (n−2) 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 (n−2)π/n radians or (n−2)180/n 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.

- Exterior angle - Imagine walking around a simple n-gon marked on the floor. The amount you "turn" at a corner is the exterior or external angle. Walking all the way round the polygon, you make one full turn, so the sum of the exterior angles must be 360°. Moving around an n-gon in general, the sum of the exterior angles (the total amount one "turns" at the vertices) can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight", where d is the density or starriness of the polygon. See also orbit (dynamics).

The exterior angle is the supplementary
angle to the interior angle. From this the sum of the interior
angles can be easily confirmed, even if some interior angles are
more than 180°: going clockwise around, it means that one sometime
turns left instead of right, which is counted as turning a negative
amount. (Thus we consider something like the winding
number of the orientation of the sides, where at every vertex
the contribution is between -½ and ½ winding.)

### Area and centroid

The area
of a polygon is the measurement of the 2-dimensional region
enclosed by the polygon. For a non-self-intersecting (simple)
polygon with n vertices, the area and centroid are given by:

- A = \frac \sum_^ x_i y_ - x_ y_i\,

- \bar x = \frac \sum_^ (x_i + x_) (x_i y_ - x_ y_i)\,

- \bar y = \frac \sum_^ (y_i + y_) (x_i y_ - x_ y_i)\,

To close the polygon, the first and last vertices
are the same, ie x_n, y_n = x_0, y_0. The vertices must be ordered
clockwise or counterclockwise, if they are ordered clockwise the
area will be negative but correct in absolute
value.

The formula was described by Meister in 1769 and
by Gauss
in 1795. It can be verified by dividing the polygon into triangles,
but it can also be seen as a special case of Green's
theorem.

The area A of
a simple
polygon can also be computed if the lengths of the sides,
a1,a2, ..., an and the exterior
angles, \theta_1, \theta_2, ..., \theta_n are known. The
formula is

- \beginA = \frac12 ( a_1[a_2 sin(\theta_1) + a_3 sin(\theta_1 + \theta_2) + ... + a_ sin(\theta_1 + \theta_2 + ... + \theta_] \\

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.

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.

For a regular polygon with n sides of length s,
the area is given by:

- A = \frac s^2 \cot.

#### Self-intersecting polygons

The area of a self-intersecting polygon can be defined in two different ways, each of which gives a different answer:- Using the above methods for simple polygons, we discover 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 centre 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 a simple polygon having the same outline as the self-intersecting one (or, in the case of the cross-quadrilateral, the two simple triangles).

### Degrees of freedom

An n-gon has 2n degrees of freedom, including 2 for position and 1 for rotational orientation, and 1 for over-all size, so 2n-4 for shape. In the case of a line of symmetry the latter reduces to n-2.Let k≥2. For an nk-gon with k-fold rotational
symmetry (Ck), there are 2n-2 degrees of freedom for the shape.
With additional mirror-image symmetry (Dk) there are n-1 degrees of
freedom.

## Generalizations of polygons

In a broad sense, a polygon is an unbounded sequence or circuit of alternating segments (sides) and angles (corners). The modern mathematical understanding is to describe this structural sequence in terms of an 'abstract' polygon which is a partially-ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.A geometric polygon is understood to be a
'realization' of the associated abstract polygon; this involves
some 'mapping' of elements from the abstract to the geometric. Such
a polygon does not have to lie in a plane, or have straight sides,
or enclose an area, and individual elements can overlap or even
coincide. For example a spherical
polygon is drawn on the surface of a sphere, and its sides are
arcs of great circles. As another example, most polygons are
unbounded because they close back on themselves, while apeirogons (infinite polygons)
are unbounded because they go on for ever so you can never reach
any bounding end point. So when we talk about "polygons" we must be
careful to explain what kind we are talking about.

A digon is a closed polygon having two sides and
two corners. On the sphere, we can mark two opposing points (like
the North and South poles) and join them by half a great circle.
Add another arc of a different great circle and you have a digon.
Tile the sphere with digons and you have a polyhedron called a
hosohedron. Take just one great circle instead, run it all the way
round, and add just one "corner" point, and you have a monogon or
henagon.

Other realizations of these polygons are possible
on other surfaces - but in the Euclidean (flat) plane, their bodies
cannot be sensibly realized and we think of them as degenerate.

The idea of a polygon has been generalised in
various ways. Here is a short list of some degenerate
cases (or special cases, depending on your point of view):

- Digon. Angle of 0° in the Euclidean plane. See remarks above re. on the sphere.
- Angle of 180°: In the plane this gives an apeirogon(see below), on the sphere a dihedron
- A skew polygon does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygons of the regular polyhedra are classic examples.
- A spherical polygon is a circuit of sides and corners on the surface of a sphere.
- An apeirogon is an infinite sequence of sides and angles, which is not closed but it has no ends because it extends infinitely.
- A complex polygon is a figure analogous to an ordinary polygon, which exists in the unitary plane.

## Naming polygons

The word 'polygon' comes from Late Latin polygōnum (a noun), from Greek polygōnon/polugōnon πολύγωνον, noun use of neuter of polygōnos/polugōnos πολύγωνος (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. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.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 less than 100 edges, combine the prefixes as follows The
'kai' is not always used. Opinions differ on exactly when it
should, or need not, be used (see also examples above).

That is, a 42-sided figure would be named as
follows: and a 50-sided figure But beyond enneagons and decagons,
professional mathematicians generally prefer the aforementioned
numeral notation (for example, MathWorld has
articles on 17-gons and 257-gons). Exceptions exist for side
numbers that are difficult to express in numerical form.

## Polygons in nature

Numerous regular polygons may be seen in nature.
In the world of minerals, crystals often have faces which are
triangular, square or hexagonal. Quasicrystals
can even have regular pentagons as faces. Another fascinating
example of regular polygons occurs when the cooling of lava forms areas of tightly packed
hexagonal columns of
basalt, which may be seen
at the Giant's
Causeway in Ireland, or at the
Devil's
Postpile in California.

The most famous hexagons in nature are found in
the animal kingdom. The wax honeycomb made by bees is an array of hexagons used to store honey and
pollen, and as a secure place for the larvae to grow. There also
exist animals who themselves take the approximate form of regular
polygons, or at least have the same symmetry. For example, starfish display the symmetry
of a pentagon or, less
frequently, the heptagon or other polygons.
Other echinoderms,
such as sea urchins,
sometimes display similar symmetries. Though echinoderms do not
exhibit exact
radial symmetry, jellyfish and comb jellies do,
usually fourfold or eightfold.

Radial symmetry (and other symmetry) is also
widely observed in the plant kingdom, particularly amongst flowers,
and (to a lesser extent) seeds and fruit, the most common form of
such symmetry being pentagonal. A particularly striking example is
the Starfruit, a
slightly tangy fruit popular in Southeast Asia, whose cross-section
is shaped like a pentagonal star.

Moving off the earth into space, early
mathematicians doing calculations using Newton's law
of gravitation discovered that if two bodies (such as the sun and
the earth) are orbiting one another, there exist certain points in
space, called Lagrangian
points, where a smaller body (such as an asteroid or a space
station) will remain in a stable orbit. The sun-earth system has
five Lagrangian points. The two most stable are exactly 60 degrees
ahead and behind the earth in its orbit; that is, joining the
centre of the sun and the earth and one of these stable Lagrangian
points forms an equilateral triangle. Astronomers have already
found asteroids
at these points. It is still debated whether it is practical to
keep a space station at the Lagrangian point — although
it would never need course corrections, it would have to frequently
dodge the asteroids that are already present there. There are
already satellites and space observatories at the less stable
Lagrangian points.

## Things to do with polygons

- Cut up a piece of paper into polygons, and put them back together as a tangram.
- Join many edge-to-edge as a tiling or tessellation.
- Join several edge-to-edge and fold them all up so there are no gaps, to make a three-dimensional polyhedron.
- Join many edge-to-edge, folding them into a crinkly thing called an infinite polyhedron.
- Use computer-generated polygons to build up a three-dimensional world full of monsters, theme parks, aeroplanes or anything - see Polygons in computer graphics below..

## Polygons in computer graphics

A polygon in a computer graphics (image generation) system is a two-dimensional shape that is modelled and stored within its database. A polygon can be coloured, shaded and textured, and its position in the database is defined by the co-ordinates of its vertices (corners).Naming conventions differ from those of
mathematicians:

- A simple polygon does not cross itself.
- a concave polygon is a simple polygon having at least one interior angle greater than 180 deg.
- A complex polygon does cross itself.

Use of Polygons in Real-time imagery. 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 so that as the viewing point moves through the scene,
it is perceived in 3D.

Morphing. To avoid artificial effects at polygon
boundaries where the planes of contiguous polygons are at different
angle, so called 'Morphing Algorithms' are used. These blend,
soften or smooth the polygon edges so that the scene looks less
artificial and more like the real world.

Polygon Count. Since a polygon can have many
sides and need many points to define it, in order to compare one
imaging system with another, "polygon count" is generally taken as
a triangle. A triangle is processed as three points in the x,y, and
z axes, needing nine geometrical descriptors. In addition, coding
is applied to each polygon for colour, brightness, shading,
texture, NVG (intensifier or night vision), Infra-Red
characteristics and so on. When analysing the characteristics of a
particular imaging system, the exact definition of polygon count
should be obtained as it applies to that system.

Meshed Polygons. The number of meshed polygons
(`meshed' is like a fish net) can be up to twice that of
free-standing unmeshed polygons, particularly if the polygons are
contiguous. 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) 2/2n2
vertices per triangle. Where n is large, this approaches one half.
Or, each vertex inside the square mesh connects four edges
(lines).

Vertex Count. Because of effects such as the
above, a count of Vertices may be more reliable than Polygon count
as an indicator of the capability of an imaging system.

Point in polygon test. 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. It is known as the Point in
polygon test.

## External links

## See also

## References

- Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948).
- 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)

polygons in Aragonese: Poligono

polygons in Asturian: Polígonu

polygons in Bosnian: Poligon

polygons in Bulgarian: Многоъгълник

polygons in Catalan: Polígon

polygons in Czech: Mnohoúhelník

polygons in Danish: Polygon

polygons in German: Polygon

polygons in Estonian: Hulknurk

polygons in Modern Greek (1453-): Πολύγωνο

polygons in Spanish: Polígono

polygons in Esperanto: Plurlatero

polygons in Basque: Poligono

polygons in French: Polygone

polygons in Scottish Gaelic: Poileagan

polygons in Galician: Polígono

polygons in Korean: 다각형

polygons in Croatian: Mnogokut

polygons in Ido: Poligono

polygons in Italian: Poligono

polygons in Hebrew: מצולע

polygons in Georgian: პენტაკონტაგონი

polygons in Haitian: Poligòn

polygons in Hungarian: Sokszög

polygons in Malayalam: ബഹുഭുജം

polygons in Dutch: Veelhoek

polygons in Japanese: 多角形

polygons in Norwegian: Polygon

polygons in Low German: Veeleck

polygons in Polish: Wielokąt

polygons in Portuguese: Polígono

polygons in Romanian: Poligon

polygons in Russian: Многоугольник

polygons in Simple English: Polygon

polygons in Slovak: Mnohouholník

polygons in Slovenian: Mnogokotnik

polygons in Serbian: Многоугао

polygons in Finnish: Monikulmio

polygons in Swedish: Polygon

polygons in Tamil: பல்கோணம்

polygons in Thai: รูปหลายเหลี่ยม

polygons in Vietnamese: Đa giác

polygons in Turkish: Çokgen

polygons in Ukrainian: Багатокутник

polygons in Yiddish: פילעק

polygons in Contenese: 多邊形

polygons in Chinese: 多边形