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Encyclopedia > Polygon
Look up polygon in
Wiktionary, the free dictionary.

In geometry a polygon (IPA: [ˈpɒlɪˌɡɒn ~ ˈpɒliˌɡɒn]) is 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 called its body. A polygon is a 2-dimensional example of the more general polytope in any number of dimensions. Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Articles with similar titles include the NATO phonetic alphabet, which has also informally been called the “International Phonetic Alphabet”. For information on how to read IPA transcriptions of English words, see IPA chart for English. ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... Figure can refer to any of the following: A persons figure. ... For other uses, see Curve (disambiguation). ... The geometric definition of a line segment In 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. ... A polygonal chain, polygonal curve, polygonal path, piecewise linear curve, a connected series of line segments. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ...


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. Look up polygon in Wiktionary, the free dictionary. ...

An assortment of polygons

Contents

Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ...

Classification

Number of sides

Polygons are primarily classified by the number of sides, see naming polygons below. Look up polygon in Wiktionary, the free dictionary. ...


Convexity

  • Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets it exactly twice.
  • Non-convex: a line may be found which meets it more than twice.
  • Star polygon: a polygon which self-intersects in a regular way.
  • Star-shaped polygon, a polygon whose whole interior is visible from a single point.

A convex pentagon In geometry, a convex polygon is a simple polygon whose interior is a convex set. ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... A star-shaped polygon (top). ...

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 polygon: 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.

Look up Polygon on Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ... In mathematics, a symmetry group describes all symmetries of objects. ... In geometry, an equilateral polygon has all sides of the same length. ... In geometry, a form is isotoxal or edge-transitive if its symmetries act transitively on its edges. ... In mathematics, a symmetry group describes all symmetries of objects. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ...

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.

An rectilinear polygon is a polygon all of whose edges meet at right angles. ... 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. ...

Properties

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


Angles

  • Any polygon, regular or irregular, complex 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. In topology and analysis,
    • 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°. The exterior angle is the supplementary angle to the interior angle, and from this the sum of the interior angles can be easily confirmed.

The reasoning also applies 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.) In geometry, an internal angle is an angle that 2 sides of a polygon form by touching. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... Some common angles, measured in radians. ... This article describes the unit of angle. ... In geometry, an internal angle is an angle that 2 sides of a polygon form by touching. ... Look up supplementary in Wiktionary, the free dictionary. ... A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ...


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. In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... A single face is colored yellow and outlined in red to help identify the faces. ...


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 times 360°, e.g. 720° for a pentagram and 0° for an angular "eight". See also orbit (dynamics). A pentagram A pentagram (sometimes known as pentalpha or pentangle) is the shape of a five-pointed star drawn with five straight strokes. ... In the study of dynamical systems, an orbit is the sequence generated by iterating a map. ...


Area

The area of a polygon is the measurement of the 2-dimensional region enclosed by the polygon.


Simple polygons

The area A of a simple polygon can be computed if the cartesian coordinates (x1, y1), (x2, y2), ..., (xn, yn) of its vertices, listed in order as the area is circulated in counter-clockwise fashion, are known. The formula is Area is a quantity expressing the size of a figure in the Euclidean plane or on a 2-dimensional surface. ... Fig. ...

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. Johann Carl Friedrich Gauss or Gauß ( ; Latin: ) (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... In physics and mathematics, Greens 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. Greens theorem was named after British scientist George Green and is a special two-dimensional case of...


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 pieces and rearrange the pieces to obtain the second polygon. ...


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

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. Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... Shape (OE. sceap Eng. ... Figures with the axes of symmetry drawn in. ...


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.


Generally, a geometric polygon is a 'realization' of this 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. Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... An apeirogon is a degenerate polygon with an infinite number of sides. ...


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

  • 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), 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.

In geometry a digon is a polygon with two sides and two vertices. ... An apeirogon is a degenerate polygon with an infinite number of sides. ... A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. ... In geometry, a skew polygon is a polygon whose vertices do no lie in a plane. ... Spherical triangle Spherical trigonometry is a part of spherical geometry that deals with polygons (especially triangles) on the sphere and explains how to find relations between the involved angles. ... An apeirogon is a degenerate polygon with an infinite number of sides. ... A complex polytope is one which exists in a unitary space, where each real dimension is accompanied by an imaginary one. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. ...

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. Latin was the language originally spoken in the region around Rome called Latium. ... A numerical prefix is a prefix that denotes a number, which is usually a multiplier for the thing being prefixed. ... Look up pentagon in Wiktionary, the free dictionary. ... In geometry, a dodecagon is a polygon with exactly twelve sides. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... In geometry, a quadrilateral is a polygon with four sides or edges and four vertices or corners. ... In geometry, an enneagon or nonagon is a nine-sided polygon. ... Leonhard Euler, one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... A numeral is a symbol or group of symbols that represents a number. ... In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...


Some special polygons also have their own names; for example, the regular star pentagon is also known as the pentagram. In ordinary English, regular is an adjective or noun used to mean in accordance with the usual customs, conventions, or rules, or frequent, periodic, or symmetric. ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... Look up pentagon in Wiktionary, the free dictionary. ... A pentagram A pentagram (sometimes known as pentalpha or pentangle) is the shape of a five-pointed star drawn with five straight strokes. ...

Polygon names
Name Edges
henagon (or monogon) 1
digon 2
triangle (or trigon) 3
quadrilateral (or tetragon) 4
pentagon 5
hexagon 6
heptagon (avoid "septagon" = Latin [sept-] + Greek) 7
octagon 8
enneagon (or nonagon) 9
decagon 10
hendecagon (avoid "undecagon" = Latin [un-] + Greek) 11
dodecagon (avoid "duodecagon" = Latin [duo-] + Greek) 12
tridecagon (or triskaidecagon) 13
tetradecagon (or tetrakaidecagon) 14
pentadecagon (or quindecagon or pentakaidecagon) 15
hexadecagon (or hexakaidecagon) 16
heptadecagon (or heptakaidecagon) 17
octadecagon (or octakaidecagon) 18
enneadecagon (or enneakaidecagon or nonadecagon) 19
icosagon 20
No established English name ("hectagon" is bad Greek,

"centagon" is a Latin-Greek hybrid; neither is widely attested. In geometry a henagon (or monogon) is a polygon with one side and one vertex. ... In geometry a digon is a polygon with two sides and two vertices. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... In geometry, a quadrilateral is a polygon with four sides or edges and four vertices or corners. ... Look up pentagon in Wiktionary, the free dictionary. ... A regular hexagon. ... In geometry, a heptagon is a polygon with seven sides and seven angles. ... For other uses, see Octagon (disambiguation). ... A regular enneagon. ... a regular decagon In 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 all angles equal to 144°, therefore making each angle of a regular decagon be 144°. Its Schläfli symbol is... Categories: Math stubs | Polygons ... In geometry, a dodecagon is a polygon with exactly twelve sides. ... A regular triskaidecagon. ... Look up Polygon in Wiktionary, the free dictionary. ... In geometry, a pentadecagon is any 15-sided, 15-angled, polygon. ... HELLLOOOOO THERE ... Erchingers heptadecagon In geometry, a heptadecagon (or 17-gon) is a seventeen-sided polygon. ... An octadecagon is a polygon with 18 sides and 18 vertexes. ... An enneadecagon. ... A regular icosagon. ...

100
chiliagon 1000
myriagon 10,000
googolgon 10100

To construct the name of a polygon with more than 20 and less than 100 edges, combine the prefixes as follows In geometry, a chiliagon (pronounced /ˈkɪli. ... This article or section does not cite its references or sources. ...

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).


That is, a 42-sided figure would be named as follows:

Tens and Ones final suffix full polygon name
tetraconta- -kai- -di- -gon tetracontakaidigon

and a 50-sided figure

Tens and Ones final suffix full polygon name
pentaconta-   -gon pentacontagon

But beyond enneagons and decagons, professional mathematicians prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...


Polygons in nature

The Giant's Causeway, in Ireland

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 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. Close up of Giants Causeway. ... This article or section includes a list of works cited or a list of external links, but its sources remain unclear because it lacks in-text citations. ... Quasicrystals are aperiodic structures which produce diffraction. ... Look up lava, Aa, pahoehoe in Wiktionary, the free dictionary. ... A regular hexagon. ... Basalt Basalt (IPA: ) is a common gray to black extrusive volcanic rock. ... This article or section includes a list of works cited or a list of external links, but its sources remain unclear because it lacks in-text citations. ... The longer fragments of basalt at the base of the cliff can be larger than a person. ... Official language(s) English Capital Sacramento Largest city Los Angeles Area  Ranked 3rd  - Total 158,302 sq mi (410,000 km²)  - Width 250 miles (400 km)  - Length 770 miles (1,240 km)  - % water 4. ...

Starfruit, a popular fruit in Southeast Asia

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. Download high resolution version (640x797, 110 KB)Carambolas, Arkin variety. ... Download high resolution version (640x797, 110 KB)Carambolas, Arkin variety. ... Binomial name Averrhoa carambola L. Carambolas still on the tree The carambola is a species of tree native to Sri Lanka, India and Indonesia and is popular throughout Southeast Asia Malaysia and parts of East Asia. ... Location of Southeast Asia Southeast Asia is a subregion of Asia. ... Honeycomb Honeycombs on a Sacred fig tree 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 pollen. ... This article or section does not cite any references or sources. ... A regular hexagon. ... Orders Brisingida (100 species[1]) Forcipulatida (300 species[2]) Paxillosida (255 species[3]) Notomyotida (75 species[4]) Spinulosida (120 species[5]) Valvatida (695 species[6]) Velatida (200 species[7]) For other uses, see Starfish (disambiguation). ... Look up pentagon in Wiktionary, the free dictionary. ... In geometry, a heptagon is a polygon with seven sides and seven angles. ... Classes Subphylum Homalozoa Gill & Caster, 1960 Class Homostelea Class Homoiostelea Class Stylophora Gill & Caster, 1960 Class Ctenocystoidea Robison & Sprinkle, 1969 Subphylum Crinozoa Class Eocrinoidea Jaekel, 1899 Class Paracrinoidea Regnéll, 1945 Class Cystoidea von Buch, 1846 Class Blastoidea Class Crinoidea Subphylum Asterozoa Class Ophiuroidea Class Asteroidea Subphylum Echinozoa Helicoplacoidea †  ?Arkarua... Subclasses Euechinoidea Superorder Atelostomata Order Cassiduloida Order Spatangoida (heart urchins) Superorder Diadematacea Order Diadematoida Order Echinothurioida Order Pedinoida Superorder Echinacea Order Arbacioida Order Echinoida Order Phymosomatoida Order Salenioida Order Temnopleuroida Superorder Gnathostomata Order Clypeasteroida (sand dollars) Order Holectypoida Perischoechinoidea Order Cidaroida (pencil urchins) Sea urchins are small spiny sea creatures... The elaborate patterns on the wings of butterflies are one example of biological symmetry. ... Orders Stauromedusae Coronatae Semaeostomeae Rhizostomae Jellyfish are marine invertebrates belonging to the Scyphozoan class. ... This article or section is in need of attention from an expert on the subject. ...


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. Binomial name Averrhoa carambola L. Carambolas still on the tree The carambola is a species of tree native to Sri Lanka, India and Indonesia and is popular throughout Southeast Asia Malaysia and parts of East 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. 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. Sir Isaac Newton (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1726][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... A contour plot of the effective potential (the Hills Surfaces) of a two-body system (the Sun and Earth here), showing the five Lagrange points. ... Image of the Trojan asteroids in front of and behind Jupiter along its orbital path. ...


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..

A typical tangram construction Tangram (Chinese: ; Pinyin: ; literally seven boards of cunning) is a Dissection puzzle. ... A tessellated plane seen in street pavement. ... A tessellated plane seen in street pavement. ... A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... A simple (prismatic) semiregular skew polyhedron with vertex configuration 4. ...

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). Computer graphics is a sub-field of computer science and is concerned with digitally synthesizing and manipulating visual content. ...


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. Computer graphics is a sub-field of computer science and is concerned with digitally synthesizing and manipulating visual content. ... In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry. ... In computational geometry, the point in polygon (also point-in-polygon or PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a simple polygon. ...


External links

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Polygons

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In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ... In geometry, a circumcircle of a given two-dimensional geometric shape is the smallest circle which contains the shape completely within it. ... In geometry, two sets of points are of the same shape precisely if one can be transformed to another by dilating (i. ... Polygon triangulation is a topic in computational geometry. ... In recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. ... A polyhedron (plural polyhedra or polyhedrons) is a geometric object with flat faces and straight edges. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ... A simple concave hexagon In geometry, two edges of a polygon may cross or even overlap in general. ... In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. ... Synthetic geometry is a descriptive term that identifies a methodology of geometry which makes use of theorems and synthetic observations to create theorems or solve problems, as opposed to analytic geometry which uses algebra, numbers, computations to draw theorems or solve problems. ... A tessellated plane seen in street pavement. ... This page is a candidate for speedy deletion. ... An example of a simple 8-sided golygon A golygon (technically referred to as a serial isogon of 90 degrees) is any polygon with all right angles, whose sides are consecutive integer lengths. ...

Polygons
TriangleQuadrilateralPentagon •Hexagon • Heptagon • OctagonEnneagon (Nonagon)Decagon • Hendecagon • DodecagonTriskaidecagonPentadecagonHexadecagon • Heptadecagon • EnneadecagonIcosagonChiliagonMyriagon

  Results from FactBites:
 
SparkNotes: Polygons: Defining a Polygon (479 words)
A polygon is one type of simple closed curve.
A polygon is the union of three or more line segments whose endpoints meet.
The segments are called the sides of the polygon.
Polygon - LoveToKnow 1911 (1465 words)
The term regular polygon is usually restricted to "convex" polygons; a special class of polygons (regular in the wider sense) has been named "star polygons" on account of their resemblance to star-rays; these are, however, concave.
The Arabian geometers of the 9th century showed that the heptagon required the solution of a cubic equation, thus resembling the Pythagorean problems of "duplicating the cube" and "trisecting an angle." Edmund Halley gave solutions for the heptagon and nonagon by means of the parabola and circle, and by a parabola and hyperbola respectively.
In general, the number of star polygons which can be drawn with the vertices of an n-point regular polygon is the number of numbers which are not factors of n and are less than In.
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