In geometry a polygon (IPA: /ˈpɒlɨɡɒn, ˈpɒliɡɒn/) is traditionally a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments (i. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points e. , by a closed polygonal chain). A polygonal chain, polygonal curve, polygonal path, or piecewise linear curve, is a connected series of Line segments More formally a polygonal 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. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations

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. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit

An assortment of polygons

Classification

Number of sides

Polygons are primarily classified by the number of sides, see naming polygons below. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit

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. In Geometry, a simple polygon is a polygon whose sides do not intersect 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. A star-shaped polygon (not to be confused with Star polygon) is a Polygonal region in the plane which is a Star domain, i 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. A complex polytope is one which exists in a complex Hilbert space, where each real dimension is accompanied by an imaginary one This article assumes some familiarity with Analytic geometry and the concept of a limit. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
• Star polygon: a polygon which self-intersects in a regular way. Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one

Symmetry

• Equiangular: all its corner angles are equal. In Euclidean geometry, an equiangular polygon is a Polygon whose vertex angles are equal
• Cyclic: all corners lie on a single circle. In Geometry, the circumscribed circle or circumcircle of a Polygon is a Circle which passes through all the vertices of the polygon Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the
• Isogonal or vertex-transitive: all corners lie within the same symmetry orbit. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. The polygon is also cyclic and equiangular.
• Equilateral: all edges are of the same length. In Geometry, an equilateral polygon is a Polygon which has all sides 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. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. The polygon is also equilateral.
• Regular. General properties These properties apply to both convex and star regular polygons A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon. Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one

Miscellaneous

• Rectilinear: a polygon whose sides meet at right angles, i. 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° 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. In Geometry, a Polygon P in the plane is called monotone with respect to a straight line L, if every line orthogonal to L

Properties

We will assume Euclidean geometry throughout. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

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. Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 This article describes the unit of angle For other meanings see Degree. 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. Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one The Kepler-Poinsot polyhedra is a popular name for the regular star polyhedra.
• Exterior angle - Imagine walking around a simple n-gon marked on the floor. Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle 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. Early history Sumer The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC 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 exterior angle is the supplementary angle to the interior angle. A pair of Angles is supplementary if their measurements add up to 180 degrees If the two supplementary angles are adjacent (i 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. The term winding number may also refer to the Rotation number of an Iterated map. )

Area and centroid

Nomenclature of a 2D polygon.

The area of a polygon is the measurement of the 2-dimensional region enclosed by the polygon. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. For a non-self-intersecting (simple) polygon with n vertices, the area and centroid are given by[1]:

$A = \frac{1}{2} \sum_{i = 0}^{n - 1} x_i y_{i + 1} - x_{i + 1} y_i\,$
$\bar x = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (x_i + x_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i)\,$
$\bar y = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (y_i + y_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i)\,$

To close the polygon, the first and last vertices are the same, ie xn,yn = x0,y0. In Geometry, a simple polygon is a polygon whose sides do not intersect Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Geometry, the centroid or barycenter of an object X in n- Dimensional space is the intersection of all Hyperplanes The vertices must be ordered clockwise or counterclockwise, if they are ordered clockwise the area will be negative but correct in absolute value. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

The formula was described by Meister in 1769 and by Gauss in 1795. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem. In Physics and Mathematics, Green's theorem gives the relationship between a Line integral around a simple closed curve C and a Double integral

The area A of a simple polygon can also be computed if the lengths of the sides, a1,a2, . In Geometry, a simple polygon is a polygon whose sides do not intersect . . , an and the exterior angles, θ12, . Geometry, an interior angle (or internal angle) is an Angle formed by two sides of a Simple polygon that share an endpoint namely the angle . . , θn are known. The formula is

\begin{align}A = \frac12 ( a_1[a_2 sin(\theta_1) + a_3 sin(\theta_1 + \theta_2) + ... + a_{n-1} sin(\theta_1 + \theta_2 + ... + \theta_{n-2}] \\+ a_2[a_3 sin(\theta_2) + a_4 sin(\theta_2 + \theta_3) + ... + a_{n-1} sin(\theta_2 + ... + \theta_{n-2})] \\+ \; ... \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \\+ a_{n-2}[a_{n-1} sin(\theta_{n-2})] ) \end{align}

The formula was described by Lopshits in 1963. [2]

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. Given a Simple polygon constructed on a grid of equal-distanced points (i

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. In Geometry, the Bolyai–Gerwien theorem states that if two simple Polygons of equal Area are given one can cut the first into finitely many polygonal

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

$A = \frac{n}{4} s^2 \cot{\cfrac{\pi}{n}}.$

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. The term complex polygon can mean two different things In Computer graphics, as a polygon which is neither convex nor concave. 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. For information on degrees of freedom in other sciences see Degrees of freedom. The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined In the case of a line of symmetry the latter reduces to n-2. Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect

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. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement 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. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations 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. for the degeneracy of a Graph, see Arboricity#Related_concepts.

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. for the degeneracy of a Graph, see Arboricity#Related_concepts. In Geometry a digon is a degenerate Polygon with two sides (edges and two vertices. 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. As a polyhedron A dihedron can be considered a degenerate prism consisting of two (planar n -sided Polygons connected "back-to-back" In Geometry, a skew polygon is a Polygon whose vertices do not lie in a Plane. 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. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations
• 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. A complex polytope is one which exists in a complex Hilbert space, where each real dimension is accompanied by an imaginary one This article assumes some familiarity with Analytic geometry and the concept of a limit.

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". Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Numerical prefixes are usually derived from the words for numbers in various languages most commonly Greek and Latin, although this is not always the case g. pentagon, dodecagon. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Regular dodecagon It usually refers to a regular dodecagon having all sides of equal length and all angles equal to 150° The triangle, quadrilateral, and nonagon are exceptions. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. Graphs and stars The K9 Complete graph is often drawn as a regular nonagon with all 36 edges connected For large numbers, mathematicians usually write the numeral itself, e. A mathematician is a person whose primary area of study and research is the field of Mathematics. 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. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information

Some special polygons also have their own names; for example, the regular star pentagon is also known as the pentagram. Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Early history Sumer The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC

Polygon names
NameEdges
henagon (or monogon)1
digon2
triangle (or trigon)3
pentagon5
hexagon6
heptagon (avoid "septagon" = Latin [sept-] + Greek)7
octagon8
enneagon (or nonagon)9
decagon10
hendecagon (avoid "undecagon" = Latin [un-] + Greek)11
dodecagon (avoid "duodecagon" = Latin [duo-] + Greek)12
tridecagon (or triskaidecagon)13
icosagon20
No established English name

"hectogon" is the Greek name (see hectometre),
"centagon" is a Latin-Greek hybrid; neither is widely attested. In Geometry a henagon (or monogon) is a Polygon with one edge and one vertex. In Geometry a digon is a degenerate Polygon with two sides (edges and two vertices. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, a quadrilateral is a Polygon with four sides or edges and four vertices or corners. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees Construction A regular heptagon is not constructible with Compass and straightedge but is constructible with a marked Ruler and compass Regular octagons A regular octagon is an octagon whose sides are all the same length and whose internal angles are all the same size Graphs and stars The K9 Complete graph is often drawn as a regular nonagon with all 36 edges connected Construction A regular decagon is Constructible with a Compass and straightedge. Use in coinage The Canadian dollar coin the Loonie, is patterned on a regular Hendecagonal prism, as is the Indian two-rupee coin Regular dodecagon It usually refers to a regular dodecagon having all sides of equal length and all angles equal to 150° In Geometry, a triskaidecagon (or tridecagon is a Polygon with 13 sides and angles In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In Geometry, a pentadecagon is any 15-sided 15-angled Polygon. Heptadecagon construction The regular heptadecagon is a Constructible polygon, as was shown by Carl Friedrich Gauss in 1796 See also The E7 polytope can be drawn with an Orthogonal projection inside a regular octadecagon. A hectometre ( American spelling: hectometer, symbol hm) is a somewhat uncommonly used unit of Length in the Metric system

100
chiliagon1000
myriagon10,000
googolgongoogol

To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows

TensandOnesfinal suffix
-kai-1-hena--gon
20icosi-2-di-
30triaconta-3-tri-
40tetraconta-4-tetra-
50pentaconta-5-penta-
60hexaconta-6-hexa-
70heptaconta-7-hepta-
80octaconta-8-octa-
90enneaconta-9-ennea-

The 'kai' is not always used. A googol is the Large number 10100 that is the digit 1 followed by one hundred zeros (in Decimal representation 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:

TensandOnesfinal suffixfull polygon name
tetraconta--kai--di--gontetracontakaidigon

and a 50-sided figure

TensandOnesfinal suffixfull polygon name
pentaconta- -gonpentacontagon

But beyond enneagons and decagons, professional mathematicians generally prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). MathWorld is an online Mathematics reference work created and largely written by Eric W Exceptions exist for side numbers that are difficult to express in numerical form.

Polygons in nature

The Giant's Causeway, in Ireland

Numerous regular polygons may be seen in nature. The Giant's Causeway (or Clochán na bhFómharach is an area of about 40000 interlocking Basalt columns the result of an ancient volcanic eruption In the world of minerals, crystals often have faces which are triangular, square or hexagonal. Quasicrystals can even have regular pentagons as faces. Quasicrystals are structural forms that are both ordered and nonperiodic 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. Lava is molten rock expelled by a Volcano during an eruption When first expelled from a volcanic vent it is a Liquid at Temperatures Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees Basalt (bəˈsɔːlt ˈbeisɔːlt ˈbæsɔːlt is a common Extrusive Volcanic rock. The Giant's Causeway (or Clochán na bhFómharach is an area of about 40000 interlocking Basalt columns the result of an ancient volcanic eruption Ireland (pronounced /ˈaɾlənd/ Éire) is the third largest island in Europe, and the twentieth-largest island in the world Devils Postpile is a dark cliff of Columnar basalt near Mammoth Mountain in California ( is a US state on the West Coast of the United States, along the Pacific Ocean.

Starfruit, a popular fruit in Southeast Asia

The most famous hexagons in nature are found in the animal kingdom. ||-||-|}The carambola is a species of Tree native to Indonesia, India and Sri Lanka and is popular throughout Southeast Asia, 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. A honeycomb is a mass of Hexagonal Wax cells built by Honey bees in their nests to contain their larvae and stores of Honey and Bees are flying Insects closely related to Wasps and Ants Bees are a Monophyletic lineage within the superfamily Apoidea Regular hexagon The internal Angles of a regular hexagon (one where all sides and all angles are equal are all 120 ° and the hexagon has 720 degrees 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. Starfish (also called sea stars) are any Echinoderms belonging to the class Asteroidea. Regular pentagons The term pentagon is commonly used to mean a regular convex pentagon, where all sides are equal and all interior angles are equal (to Construction A regular heptagon is not constructible with Compass and straightedge but is constructible with a marked Ruler and compass Other echinoderms, such as sea urchins, sometimes display similar symmetries. Echinoderms (Phylum Echinodermata) are a phylum of marine Animals (including Sea stars) Sea urchins are small globular spiny sea cat animals composing most of class Echinoidea. Though echinoderms do not exhibit exact radial symmetry, jellyfish and comb jellies do, usually fourfold or eightfold. "Bilateral symmetry" redirects here For bilateral symmetry in mathematics see Reflection symmetry. Jellyfish are free-swimming members of the phylum Cnidaria. They have several different basic morphologies that represent several different cnidarian classes including the The phylum Ctenophora (tɨˈnɒfərə commonly known as comb jellies, is a phylum that includes the Sea gooseberry ( Pleurobrachia pileus) and

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. ||-||-|}The carambola is a species of Tree native to Indonesia, India and Sri Lanka and is popular throughout Southeast Asia,

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. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements 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. Tangram ( is a Dissection puzzle. It consists of seven pieces called tans, which fit together to form a shape of some sort
• Join many edge-to-edge as a tiling or tessellation. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps
• Join several edge-to-edge and fold them all up so there are no gaps, to make a three-dimensional polyhedron. What is a polyhedron? We can at least say that a polyhedron is built up from different kinds of element or entity each associated with a different number of dimensions
• Join many edge-to-edge, folding them into a crinkly thing called an infinite polyhedron. In Geometry, infinite skew polyhedra are an extended definition of polyhedra, created by Regular polygon faces and nonplanar Vertex figures
• 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. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data 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. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. It is known as the Point in polygon test. In Computational geometry, the point-in-polygon ( PIP) problem asks whether a given point in the plane lies inside outside or on the boundary of a Polygon