In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
a polygon (icon) is a flat shape consisting of straight lines that are joined to form a
closed chainA polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of line segments. More formally, a polygonal chain P is a curve specified by a sequence of points \scriptstyle called its vertices so that the curve consists of the line segments connecting the...
or
circuit.
A polygon is traditionally a
planeIn mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
figureThe shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
that is bounded by a closed path, composed of a finite sequence of straight
line segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
s (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 (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the more general
polytopeIn elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
in any number of dimensions.
The word "polygon" derives from the
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
πολύς (polús) "much", "many" and γωνία (gōnía) "corner" or "angle". (The word γόνυ gónu, with a short o, is unrelated and means "knee".) Today a polygon is more usually understood in terms of sides.
The basic geometrical notion has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the closed polygonal chain and with
simple polygonIn geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....
s which do not self-intersect, and may define a polygon accordingly. Geometrically 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 - however mathematically, such corners may sometimes be allowed. In fields relating to computation, the term polygon has taken on a slightly altered meaning derived from the way the shape is stored and manipulated in computer graphics (image generation). Some other generalizations of polygons are described below.
Number of sides
Polygons are primarily classified by the number of sides, see naming polygons below.
Convexity and types of non-convexity
Polygons may be characterised by their convexity or type of non-convexity:
- Convex
In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...
: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. Equivalently, all its interior angles are less than 180°.
- Non-convex: a line may be found which meets its boundary more than twice. In other words, it contains at least one interior angle with a measure larger than 180°.
- Simple
In geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....
: the boundary of the polygon does not cross itself. All convex polygons are simple.
- Concave: Non-convex and simple.
- Star-shaped
A star-shaped polygon is a polygonal region in the plane which is a star domain, i.e., a polygon P is star-shaped, if there exists a point z such that for each point p of P the segment zp lies entirely within P.The set of all points z with the described property is called the kernel of...
: 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
Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
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 polygonA complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one....
as one which exists in the complex HilbertThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
plane consisting of two complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
dimensions.
- Star polygon: a polygon which self-intersects in a regular way.
Symmetry
- Equiangular
In Euclidean geometry, an equiangular polygon is a polygon whose vertex angles are equal. If the lengths of the sides are also equal then it is a regular polygon.The only equiangular triangle is the equilateral triangle...
: all its corner angles are equal.
- Cyclic: all corners lie on a single circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
.
- Isogonal or vertex-transitive
In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...
: all corners lie within the same symmetry orbit. The polygon is also cyclic and equiangular.
- Equilateral
In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths...
: all edges are of the same length. (A polygon with 5 or more sides can be equilateral without being convex.) http://mathworld.wolfram.com/EquilateralPolygon.html
- Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral.
- Regular
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
. 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 rectilinear polygon is a polygon all of whose edges meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons....
: a polygon whose sides meet at right angles, i.e., all its interior angles are 90 or 270 degrees.
- Monotone
In geometry, a polygon P in the plane is called monotone with respect to a straight line L, if every line orthogonal to L intersects P at most twice....
with respect to a given line L, if every line orthogonal to L intersects the polygon not more than twice.
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
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s or 
degreeA degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s. 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 
radians or 
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 – Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way round the polygon makes one full turn
A turn is an angle equal to a 360° or 2 radians or \tau radians. A turn is also referred to as a revolution or complete rotation or full circle or cycle or rev or rot....
, 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 pentagramA pentagram is the shape of a five-pointed star drawn with five straight strokes...
and 0° for an angular "eight", where d is the density or starriness of the polygon. See also orbit (dynamics)In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...
.
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 numberIn mathematics, the winding number 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...
of the orientation of the sides, where at every vertex the contribution is between −½ and ½ winding.)
Area and centroid
The
areaArea is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
of a polygon is the measurement of the 2-dimensional region enclosed by the polygon. For a non-self-intersecting (
simpleIn geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....
) polygon with n vertices, the area and
centroidIn geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
are given by:
To close the polygon, the first and last vertices are the same, i.e.,
. The vertices must be ordered according to positive or negative orientation (counterclockwise or clockwise); if they are ordered negatively, the value given by the area formula will be negative but correct in
absolute valueIn mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
. This is commonly called the Surveyor's Formula.
The area formula is derived by taking each edge AB, and calculating the (signed) area of triangle ABO with a vertex at the origin O, by taking the cross-product (which gives the area of a parallelogram) and dividing by 2. As one wraps around the polygon, these triangles with positive and negative area will overlap, and the areas between the origin and the polygon will be cancelled out and sum to 0, while only the area inside the reference traingle remains. This is why the formula is called the Surveyor's Formula, since the "surveyor" is at the origin; if going counterclockwise, positive area is added when going from left to right and negative area is added when going from right to left, from the perspective of the origin.
The formula was described by Meister in 1769 and by
GaussJohann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
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 theoremIn mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
.
The area A of a
simple polygonIn geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....
can also be computed if the lengths of the sides, a
1,a
2, ..., a
n and the exterior angles,
are known. The formula is
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 theoremGiven a simple polygon constructed on a grid of equal-distanced points such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the...
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.
The area of a regular polygon is also given in terms of its inscribed circle of radius r by
.
The area of a regular n-gon with side s inscribed in a unit circle is
.
The area of a regular n-gon inscribed in a circle of radius R is given by
.
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
.
The sides of a polygon do not in general determine the area. However, if the polygon is cyclic the sides do determine the area. Of all n-gons with given sides, the one with the largest area is cyclic. Of all n-gons with a given perimeter, the one with the largest area is regular (and therefore cyclic).
Self-intersecting polygons
The area of a
self-intersecting polygonThe term complex polygon can mean two different things:*In computer graphics, as a polygon which is neither convex nor concave.*In geometry, as a polygon in the unitary plane, which has two complex dimensions.-Computer graphics:...
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 freedomA degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...
, including 2 for position, 1 for rotational orientation, and 1 for over-all size, so 2n − 4 for
shapeThe shape of an object located in some space is a geometrical description of the part of that space occupied by the object, as determined by its external boundary – abstracting from location and orientation in space, size, and other properties such as colour, content, and material...
. 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 (C
k), there are 2n − 2 degrees of freedom for the shape. With additional mirror-image symmetry (D
k) there are n − 1 degrees of freedom.
Product of diagonals of a regular polygon
For a regular n-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices equals n.
Generalizations of polygons
In a broad sense, a polygon is an unbounded (without ends) sequence or circuit of alternating segments (sides) and angles (corners). An ordinary polygon is unbounded because the sequence closes back in itself in a loop or circuit, while an
apeirogonAn apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...
(infinite polygon) is unbounded because it goes on for ever so you can never reach any bounding end point. The modern mathematical understanding is to describe such a structural sequence in terms of an "
abstractIn mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...
" polygon which is a
partially ordered setIn mathematics, especially order theory, a partially ordered set 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 binary relation that indicates that, for certain pairs of elements in the...
(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. 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
polyhedronIn elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
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—although many authorities do not regard this as a proper polygon.
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
degenerateIn mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
.
The idea of a polygon has been generalized in various ways. Here is a short list of some
degenerateIn mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
cases (or special cases, depending on your point of view):
- Digon
In geometry, a digon is a polygon with two sides and two vertices. It is degenerate in a Euclidean space, but may be non-degenerate in a spherical space.A digon must be regular because its two edges are the same length...
. Interior angle of 0° in the Euclidean plane. See remarks above re. on the sphere.
- Interior angle of 180°: In the plane this gives an apeirogon
An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...
(see below), on the sphere a dihedronA dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat faces can be thought of as a lens, an example of which is...
- A skew polygon
In geometry, a skew polygon is a polygon whose vertices do not lie in a plane. Skew polygons must have at least 4 vertices.A regular skew polygon is a skew polygon with equal edge lengths and which is vertex-transitive....
does not lie in a flat plane, but zigzags in three (or more) dimensions. The Petrie polygonIn geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...
s 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
An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...
is an infinite sequence of sides and angles, which is not closed but it has no ends because it extends infinitely.
- A complex polygon
A complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one....
is a figure analogous to an ordinary polygon, which exists in the complex Hilbert planeThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
.
Naming polygons
The word "polygon" comes from
Late LatinLate Latin is the scholarly name for the written Latin of Late Antiquity. The English dictionary definition of Late Latin dates this period from the 3rd to the 6th centuries AD extending in Spain to the 7th. This somewhat ambiguously defined period fits between Classical Latin and Medieval Latin...
polygōnum (a noun), from
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
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
GreekGreek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
-derived
numerical prefixNumber prefixes are prefixes derived from numbers or numerals. In English and other European languages, they are used to coin numerous series of words, such as unicycle – bicycle – tricycle, dyad – triad – decade, biped – quadruped, September – October – November – December, decimal – hexadecimal,...
with the suffix -gon, e.g.
pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
,
dodecagonIn geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°...
. The
triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
,
quadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
or quadrangle, and nonagon are exceptions. For large numbers,
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
s usually write the
numeralA numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
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
formulaIn mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
.
Some special polygons also have their own names; for example the
regularA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
star pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
is also known as the
pentagramA pentagram is the shape of a five-pointed star drawn with five straight strokes...
.
Polygon names
| Name |
Edges |
Remarks |
| henagon (or monogon) |
1 |
In the Euclidean plane, degenerates to a closed curve with a single vertex point on it. |
digonIn geometry, a digon is a polygon with two sides and two vertices. It is degenerate in a Euclidean space, but may be non-degenerate in a spherical space.A digon must be regular because its two edges are the same length...
|
2 |
In the Euclidean plane, degenerates to a closed curve with two vertex points on it. |
triangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
(or trigon) |
3 |
The simplest polygon which can exist in the Euclidean plane. |
quadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
(or quadrangle or tetragon) |
4 |
The simplest polygon which can cross itself. |
pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
|
5 |
The simplest polygon which can exist as a regular star. A star pentagon is known as a pentagramA pentagram is the shape of a five-pointed star drawn with five straight strokes...
or pentacle. |
| hexagon |
6 |
avoid "sexagon" = Latin [sex-] + Greek |
| heptagon |
7 |
avoid "septagon" = Latin [sept-] + Greek |
| octagon |
8 |
|
enneagonIn geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a prefix hybrid formation, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century...
or nonagon |
9 |
"nonagon" is commonly used but mixes Latin [novem = 9] with Greek. Some modern authors prefer "enneagon". |
decagonIn geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°...
|
10 |
|
| hendecagon In geometry, a hendecagon is an 11-sided polygon....
|
11 |
avoid "undecagon" = Latin [un-] + Greek |
dodecagonIn geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°...
|
12 |
avoid "duodecagon" = Latin [duo-] + Greek |
| tridecagon (or triskaidecagon) |
13 |
|
| tetradecagon (or tetrakaidecagon) |
14 |
|
pentadecagonIn geometry, a pentadecagon is any 15-sided, 15-angled, polygon.- Regular pentadecagon:A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by...
(or quindecagon or pentakaidecagon) |
15 |
|
hexadecagonIn mathematics, a hexadecagon is a polygon with 16 sides and 16 vertices.- Regular hexadecagon :A regular hexadecagon is constructible with a compass and straightedge....
(or hexakaidecagon) |
16 |
|
heptadecagonIn geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
(or heptakaidecagon) |
17 |
|
octadecagonAn octadecagon is a polygon with 18 sides and 18 vertices. Another name for an octadecagon is octakaidecagon.- Construction :A regular octadecagon cannot be constructed using compass and straightedge.- Petrie polygons :...
(or octakaidecagon) |
18 |
|
enneadecagonIn geometry, an enneadecagon is a polygon with 19 sides and angles. It is also known as an enneakaidecagon or a nonadecagon.The radius of the circumcircle of the regular enneadecagon with side length t is...
(or enneakaidecagon or nonadecagon) |
19 |
|
icosagonIn geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.One interior angle in a regular icosagon is 162° meaning that one exterior angle would be 18°...
|
20 |
|
| triacontagon In geometry, an triacontagon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.One interior angle in a regular triacontagon is 168° meaning that one exterior angle would be 12°...
|
30 |
|
| hectogon |
100 |
"hectogon" is the Greek name (see hectometre A hectometre is a somewhat uncommonly used unit of length in the metric system, equal to one hundred metres. It derives from the Greek word "ekato", meaning "hundred". A regulation football or soccer field is approximately 1 hectometre in length.*For area the square hectometre is a common unit...
), "centagon" is a Latin-Greek hybrid; neither is widely attested. |
| chiliagon |
1000 |
The measure of each angleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
in a regular chiliagon is 179.64°. |
| myriagon |
10,000 |
The internal angle of a regular myriagon is 179.964°. |
| megagon |
1,000,000 |
The internal angle of a regular megagon is 179.99964 degrees. |
apeirogonAn apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...
|
 |
A degenerate polygon of infinitely many sides |
Constructing higher names
To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows
| Tens |
and |
Ones |
final suffix |
| -kai- |
1 |
-hena- |
-gon |
| 20 |
icosi- |
2 |
-di- |
| 30 |
triaconta- |
3 |
-tri- |
| 40 |
tetraconta- |
4 |
-tetra- |
| 50 |
pentaconta- |
5 |
-penta- |
| 60 |
hexaconta- |
6 |
-hexa- |
| 70 |
heptaconta- |
7 |
-hepta- |
| 80 |
octaconta- |
8 |
-octa- |
| 90 |
enneaconta- |
9 |
-ennea- |
The "kai" is not always used. Opinions differ on exactly when it should, or need not, be used (see also examples above).
Alternatively, the system used for naming the
higher alkanesHigher alkanes are often defined as alkanes having nine or more carbon atoms. Nonane is the lightest alkane to have a flash point above 25 °C, and so not to be classified as dangerously flammable....
(completely saturated hydrocarbons) can be used:
| Ones |
Tens |
final suffix |
| 1 |
hen- |
10 |
deca- |
-gon |
| 2 |
do- |
20 |
-cosa- |
| 3 |
tri- |
30 |
triaconta- |
| 4 |
tetra- |
40 |
tetraconta- |
| 5 |
penta- |
50 |
pentaconta- |
| 6 |
hexa- |
60 |
hexaconta- |
| 7 |
hepta- |
70 |
heptaconta- |
| 8 |
octa- |
80 |
octaconta- |
| 9 |
ennea- (or nona-) |
90 |
enneaconta- (or nonaconta-) |
This has the advantage of being consistent with the system used for 10- through 19-sided figures.
That is, a 42-sided figure would be named as follows:
| Ones |
Tens |
final suffix |
full polygon name |
| do- |
tetraconta- |
-gon |
dotetracontagon |
and a 50-sided figure
| Tens |
and |
Ones |
final suffix |
full polygon name |
| pentaconta- |
|
-gon |
pentacontagon |
But beyond enneagons and decagons, professional mathematicians generally prefer the aforementioned numeral notation (for example,
MathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
has articles on 17-gons and 257-gons). Exceptions exist for side counts that are more easily expressed in verbal form.
History
Polygons have been known since ancient times. The
regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
s were known to the ancient Greeks, and the
pentagramA pentagram is the shape of a five-pointed star drawn with five straight strokes...
, a non-convex regular polygon (
star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C.. Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bredwardine.
In 1952, Shephard generalised the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create
complex polygonA complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one....
s.
Polygons in nature
Numerous regular polygons may be seen in nature. In the world of
geologyGeology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...
, crystals have flat faces, or facets, which are polygons.
QuasicrystalA quasiperiodic crystal, or, in short, quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry...
s can even have regular pentagons as faces. Another fascinating example of regular polygons occurs when the cooling of
lavaLava refers both to molten rock expelled by a volcano during an eruption and the resulting rock after solidification and cooling. This molten rock is formed in the interior of some planets, including Earth, and some of their satellites. When first erupted from a volcanic vent, lava is a liquid at...
forms areas of tightly packed
hexagonal columns of
basaltBasalt is a common extrusive volcanic rock. It is usually grey to black and fine-grained due to rapid cooling of lava at the surface of a planet. It may be porphyritic containing larger crystals in a fine matrix, or vesicular, or frothy scoria. Unweathered basalt is black or grey...
, which may be seen at the
Giant's CausewayThe Giant's Causeway is an area of about 40,000 interlocking basalt columns, the result of an ancient volcanic eruption. It is located in County Antrim on the northeast coast of Northern Ireland, about three miles northeast of the town of Bushmills...
in
IrelandIreland is an island to the northwest of continental Europe. It is the third-largest island in Europe and the twentieth-largest island on Earth...
, or at the Devil's Postpile in
CaliforniaCalifornia is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...
.
The most famous hexagons in nature are found in the animal kingdom. The wax
honeycombA honeycomb is a mass of hexagonal waxcells 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...
made by
beeBees are flying insects closely related to wasps and ants, and are known for their role in pollination and for producing honey and beeswax. Bees are a monophyletic lineage within the superfamily Apoidea, presently classified by the unranked taxon name Anthophila...
s 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,
sea starStarfish or sea stars are echinoderms belonging to the class Asteroidea. The names "starfish" and "sea star" essentially refer to members of the class Asteroidea...
s display the symmetry of a
pentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
or, less frequently, the
heptagon or other polygons. Other
echinodermEchinoderms are a phylum of marine animals. Echinoderms are found at every ocean depth, from the intertidal zone to the abyssal zone....
s, such as
sea urchinSea urchins or urchins are small, spiny, globular animals which, with their close kin, such as sand dollars, constitute the class Echinoidea of the echinoderm phylum. They inhabit all oceans. Their shell, or "test", is round and spiny, typically from across. Common colors include black and dull...
s, sometimes display similar symmetries. Though echinoderms do not exhibit exact radial symmetry,
jellyfishJellyfish are free-swimming members of the phylum Cnidaria. Medusa is another word for jellyfish, and refers to any free-swimming jellyfish stages in the phylum Cnidaria...
and
comb jelliesThe Ctenophora are a phylum of animals that live in marine waters worldwide. Their most distinctive feature is the "combs", groups of cilia that they use for swimming, and they are the largest animals that swim by means of cilia – adults of various species range from a few millimeters to in size...
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
StarfruitCarambola, also known as starfruit, is the fruit of Averrhoa carambola, a species of tree native to the Philippines, Indonesia, Malaysia, India, Bangladesh and Sri Lanka. The fruit is a popular food throughout Southeast Asia, the South Pacific and parts of East Asia...
, 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'sSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
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 pointThe Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...
s, 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
asteroidsThe Jupiter Trojans, commonly called Trojans or Trojan asteroids, are a large group of objects that share the orbit of the planet Jupiter around the Sun. Relative to Jupiter, each Trojan librates around one of the planet's two Lagrangian points of stability, and , that respectively lie 60° ahead...
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.
In computer graphics
A polygon in a
computer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
(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°.
- 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.
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).
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. When analysing the characteristics of a particular imaging system, the exact definition of polygon count should be obtained as it applies to that system as there is some flexibility in processing which causes comparisons to become non-trivial.
Vertex Count. Although using this metric appears to be closer to reality it still must be taken with some salt. Since each vertex can be augmented with other attributes (such as color or normal) the amount of processing involved cannot be trivially inferred. Furthermore, the applied vertex transform is to be accounted, as well topology information specific to the system being evaluated as post-transform caching can introduce consistent variations in the expected results.
Point in polygon test. In
computer graphicsComputer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer with help from specialized software and hardware....
and
computational geometryComputational 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 are also considered to be part of computational...
, 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. It is known as the
Point in polygonIn computational geometry, the point-in-polygon problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon...
test.
See also
- Boolean operations on polygons
Boolean operations on polygons are a set of Boolean operations operating on one or more sets of polygons in computer graphics...
- 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....
- Cyclic polygon
- Geometric shape
- Golygon
A golygon is any polygon with all right angles, whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A. K. Dewdney in a 1990 Scientific American column...
- Polygon soup
A Polygon soup is a group of unorganized triangles, with generally no relationship whatsoever. Polygon soups are the default output format when geometry is exported from a 3D modeling package, such as Maya or Blender. As soon as any kind of hierarchical sorting or clustering scheme is applied, then...
- Polygon triangulation
In computational geometry, polygon triangulation is the decomposition of a polygonal area P into a set of triangles, i.e., finding the set of triangles with pairwise non-intersecting interiors whose union is P....
- Polyform
In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as...
- Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
- Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
- Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
- Simple polygon
In geometry, a simple polygon is a closed polygonal chain of line segments in the plane which do not have points in common other than the common vertices of pairs of consecutive segments....
- Star polygon
- Synthetic geometry
Synthetic or axiomatic geometry is the branch of geometry which makes use of axioms, theorems and logical arguments to draw conclusions, as opposed to analytic and algebraic geometries which use analysis and algebra to perform geometric computations and solve problems.-Logical synthesis:The process...
- Tiling
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
- 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 . Some tiling puzzles ask you to dissect a given shape first and then rearrange the pieces into another shape...
External links