Some polygons of different kinds

In geometry a polygon ( /ˈpɒlɪɡɒn/) is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.

A polygon is traditionally a plane figure that is bounded by a closed path, 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 (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 polytope in any number of dimensions.

The word "polygon" derives from the Greek 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 polygons 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.

Classification[link]

Number of sides[link]

Polygons are primarily classified by the number of sides. See table below.

Convexity and types of non-convexity[link]

Polygons may be characterized by their convexity or type of non-convexity:

• Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. 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: the boundary of the polygon does not cross itself. All convex polygons are simple.
• Concave: Non-convex and simple.
• Star-shaped: the whole interior is visible from a single point, without crossing any edge. The polygon must be simple, and may be convex or concave.
• Self-intersecting: the boundary of the polygon crosses itself. Branko Grünbaum calls these coptic, though this term does not seem to be widely used. The term complex is sometimes used in contrast to simple, but this risks confusion with the idea of a complex polygon as one which exists in the complex Hilbert plane consisting of two complex dimensions.
• Star polygon: a polygon which self-intersects in a regular way.

Symmetry[link]

• 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.) [1]
• Isotoxal or edge-transitive: all sides lie within the same symmetry orbit. The polygon is also equilateral.
• Tangential: all sides are tangent to an inscribed circle.
• Regular: A polygon is regular if it is both cyclic and equilateral. A non-convex regular polygon is called a regular star polygon.

Miscellaneous[link]

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

Properties[link]

Euclidean geometry is assumed throughout.

Angles[link]

Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides. Each corner has several angles. The two most important ones are:

• Interior angle – The sum of the interior angles of a simple n-gon is (n − 2)π radians or (n − 2)180 degrees. This is because any simple n-gon can be considered to be made up of (n − 2) triangles, each of which has an angle sum of π radians or 180 degrees. The measure of any interior angle of a convex regular n-gon is Failed to parse (Missing texvc executable; please see math/README to configure.): \left(1-\tfrac{2}{n}\right)\pi

radians or Failed to parse (Missing texvc executable; please see math/README to configure.): 180-\tfrac{360}{n}
degrees. The interior angles of regular star polygons were first studied by Poinsot, in the same paper in which he describes the four regular star polyhedra.

• Exterior angle – Tracing around a convex n-gon, the angle "turned" at a corner is the exterior or external angle. Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360°. This argument can be generalized to concave simple polygons, if external angles that turn in the opposite direction are subtracted from the total turned. Tracing around an n-gon in general, the sum of the exterior angles (the total amount one rotates at the vertices) can be any integer multiple d of 360°, e.g. 720° for a pentagram and 0° for an angular "eight", where d is the density or starriness of the polygon. See also orbit (dynamics).

The exterior angle is the supplementary angle to the interior angle. From this the sum of the interior angles can be easily confirmed, even if some interior angles are more than 180°: going clockwise around, it means that one sometime turns left instead of right, which is counted as turning a negative amount. (Thus we consider something like the winding number of the orientation of the sides, where at every vertex the contribution is between −¹⁄₂ and ¹⁄₂ winding.)

Area and centroid[link]

Nomenclature of a 2D polygon.

The area of a polygon is the measurement of the 2-dimensional region enclosed by the polygon. For a non-self-intersecting (simple) polygon with n vertices, the area and centroid are given by:^[1]

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{1}{2} \sum_{i = 0}^{n - 1}( x_i y_{i + 1} - x_{i + 1} y_i)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): C_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)\,

Failed to parse (Missing texvc executable; please see math/README to configure.): C_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, i.e., x_n, y_n = x₀, y₀. The vertices must be ordered according to positive or negative orientation (counterclockwise or clockwise, respectively); if they are ordered negatively, the value given by the area formula will be negative but correct in absolute value. This is commonly called the Surveyor's Formula.^[2]

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 triangle 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^{[citation needed]} in 1769 and by Gauss in 1795. It can be verified by dividing the polygon into triangles, but it can also be seen as a special case of Green's theorem.

The area A of a simple polygon can also be computed if the lengths of the sides, a₁, a₂, ..., a_n and the exterior angles, θ₁, θ₂, ..., θ_n are known. The formula is

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align}A = \frac12 ( a_1[a_2 \sin(\theta_1) + a_3 \sin(\theta_1 + \theta_2) + \cdots + a_{n-1} \sin(\theta_1 + \theta_2 + \cdots + \theta_{n-2})] \\ {} + a_2[a_3 \sin(\theta_2) + a_4 \sin(\theta_2 + \theta_3) + \cdots + a_{n-1} \sin(\theta_2 + \cdots + \theta_{n-2})] \\ {} + \cdots + a_{n-2}[a_{n-1} \sin(\theta_{n-2})] ) \end{align}

The formula was described by Lopshits in 1963.^[3]

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.

In every polygon with perimeter p and area A , the isoperimetric inequality Failed to parse (Missing texvc executable; please see math/README to configure.): p^2 > 4\pi A

holds.[4]

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 the radius r of its inscribed circle and its perimeter p by

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \tfrac{1}{2} \cdot p \cdot r

.

This radius is also termed its apothem and is often represented as a.

The area of a regular n-gon with side s inscribed in a unit circle is

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{ns}{4} \sqrt{4-s^{2}}

.

The area of a regular n-gon in terms of the radius r of its circumscribed circle and its perimeter p is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac {r}{2} \cdot p \cdot \sqrt{1- \tfrac{p^{2}}{4n^{2}r^{2}}}

.

The area of a regular n-gon, inscribed in a unit-radius circle, with side s and interior angle θ can also be expressed trigonometrically as

Failed to parse (Missing texvc executable; please see math/README to configure.): A = \frac{ns^{2}}{4}\cot \frac{\pi}{n} = \frac{ns^{2}}{4}\cot\frac{\theta}{n-2}=n \cdot \sin \frac{\pi}{n} \cdot \cos \frac{\pi}{n} = n \cdot \sin \frac{\theta}{n-2} \cdot \cos \frac{\theta}{n-2}

.

The sides of a polygon do not in general determine the area.^[5] 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).^[6]

Self-intersecting polygons[link]

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 center of a pentagram has density 2. The two triangular regions of a cross-quadrilateral (like a figure 8) have opposite-signed densities, and adding their areas together can give a total area of zero for the whole figure.
• Considering the enclosed regions as point sets, we can find the area of the enclosed point set. This corresponds to the area of the plane covered by the polygon, or to the area of 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[link]

An n-gon has 2n degrees of freedom, including 2 for position, 1 for rotational orientation, and 1 for overall size, so 2n − 4 for shape. In the case of a line of symmetry the latter reduces to n − 2.

Let k ≥ 2. For an nk-gon with k-fold rotational symmetry (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 distances from a vertex to other vertices of a regular polygon[link]

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[link]

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 apeirogon (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 "abstract" polygon which is a partially ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.

A geometric polygon is understood to be a "realization" of the associated abstract polygon; this involves some "mapping" of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. 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 around, 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 degenerate.

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

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

Naming polygons[link]

The word "polygon" comes from Late Latin polygōnum (a noun), from Greek πολύγωνον (polygōnon/polugōnon), noun use of neuter of πολύγωνος (polygōnos/polugōnos, the masculine adjective), meaning "many-angled". Individual polygons are named (and sometimes classified) according to the number of sides, combining a Greek-derived numerical prefix with the suffix -gon, e.g. pentagon, dodecagon. The triangle, quadrilateral or quadrangle, and nonagon are exceptions. For large numbers, mathematicians usually write the numeral itself, e.g. 17-gon. A variable can even be used, usually n-gon. This is useful if the number of sides is used in a formula.

Some special polygons also have their own names; for example the regular star pentagon is also known as the pentagram.

Constructing higher names[link]

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

The "kai" is not always used. Opinions differ on exactly when it should, or need not, be used (see also examples above).

Alternatively, the system used for naming the higher alkanes (completely saturated hydrocarbons) can be used:

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:

and a 50-sided figure

But beyond enneagons and decagons, professional mathematicians generally prefer the aforementioned numeral notation (for example, MathWorld has articles on 17-gons and 257-gons). Exceptions exist for side counts that are more easily expressed in verbal form.

History[link]

historical image of polygons (1699)

Polygons have been known since ancient times. The regular polygons were known to the ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on the vase of Aristophonus, Caere, dated to the 7th century B.C.^{[citation needed]} Non-convex polygons in general were not systematically studied until the 14th century by Thomas Bradwardine.^[21]

In 1952, Geoffrey Colin Shephard generalized the idea of polygons to the complex plane, where each real dimension is accompanied by an imaginary one, to create complex polygons.^[22]

Polygons in nature[link]

The Giant's Causeway, in Northern Ireland

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

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, sea stars display the symmetry of a pentagon or, less frequently, the heptagon or other polygons. Other echinoderms, such as sea urchins, sometimes display similar symmetries. Though echinoderms do not exhibit exact radial symmetry, jellyfish and comb jellies do, usually fourfold or eightfold.

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

Moving off the earth into space, early mathematicians doing calculations using Newton's law of gravitation discovered that if two bodies (such as the sun and the earth) are orbiting one another, there exist certain points in space, called Lagrangian points, where a smaller body (such as an asteroid or a space station) will remain in a stable orbit. The sun-earth system has five Lagrangian points. The two most stable are exactly 60 degrees ahead and behind the earth in its orbit; that is, joining the center 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.

Polygons in computer graphics[link]

This unreferenced section requires citations to ensure verifiability.

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 colored, shaded and textured, and its position in the database is defined by the coordinates 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 / 2(n²) 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 analyzing 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 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.

See also[link]

References[link]

Notes[link]

Bibliography[link]

• Coxeter, H.S.M.; Regular Polytopes, (Methuen and Co., 1948).
• Cromwell, P.; Polyhedra, CUP hbk (1997), pbk. (1999).
• Grünbaum, B.; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461–488. (pdf)

External links[link]

Look up polygon in Wiktionary, the free dictionary.

Wikimedia Commons has media related to: Polygons

• Polygon name generator, enter the number of sides to see the polygon's name
• Weisstein, Eric W., "Polygon" from MathWorld.
• What Are Polyhedra?, with Greek Numerical Prefixes
• Polygons, types of polygons, and polygon properties, with interactive animation
• How to draw monochrome orthogonal polygons on screens, by Herbert Glarner
• comp.graphics.algorithms Frequently Asked Questions, solutions to mathematical problems computing 2D and 3D polygons
• Comparison of the different algorithms for Polygon Boolean operations, compares capabilities, speed and numerical robustness
• Interior angle sum of polygons: a general formula, Provides an interactive Java investigation that extends the interior angle sum formula for simple closed polygons to include crossed (complex) polygons

Sam Reynolds

Personal information
Full name	Sam Reynolds
Date of birth	(1981-12-13) December 13, 1981 (age 30)
Place of birth	Fortuna, California, United States
Height	6 ft 1 in (1.85 m)
Playing position	Goalkeeper
Youth career
2001–2004	Fullerton Titans
Senior career*
Years	Team	Apps†	(Gls)†
2005	Portland Timbers	0	(0)
2006	Chivas USA	0	(0)
2007	Toronto FC	2	(0)
2008	Miami FC	3	(0)
2009	Austin Aztex	12	(0)
* Senior club appearances and goals counted for the domestic league only and correct as of 6 November 2009. † Appearances (Goals).

Sam Reynolds (born December 13, 1981 in Fortuna, California) is an American professional soccer player, now an assistant goalkeeper coach at Cal State Fullerton.

Career[link]

Youth and High School[link]

Reynolds attended Davis Senior High School, where he played for the Ashley Yudin coached Blue Devils who were undefeated for two seasons and won two CIF Section championships. He graduated in 2000. Reynolds played his youth soccer for the Davis Devils and in 1999-2000 for the River City Rush, a CYSL team coached by Marcos Mercado that won the California State Cup Championship (U-18) in 2000.

College and Amateur[link]

Reynolds attended Cal State Fullerton where he played four seasons of NCAA Division I soccer under head coach Al Mistri. He red-shirted his freshman year, then played from 2001 to 2004 ^[1] a total of 59 games, starting 57, accumulating 12 shutouts and finishing with a 1.51. goals against average. Reynolds also played for the United States Adult Soccer Association California-South Region IV team that won the 2004 George F. Donnelly Cup, the US national amateur championship. Reynolds was named MVP of the tournament.

Professional[link]

On April 2, 2005 Reynolds signed with the Portland Timbers of the USL First Division, but saw no regular season game time.^[2] In April 2006 Reynolds signed with Major League Soccer as a “pool” goalkeeper for the 2006 season. He spent most of the season as a reserve keeper with Chivas USA, under head coach Bob Bradley and goalkeeper coach Zak Abdel, and played in MLS reserve matches, but never made a first team start.

He re-signed as a pool goalkeeper with MLS in 2007, training with Chivas USA, and gained his MLS debut with Toronto FC on July 1, 2007 in a 1-1 draw with the Kansas City Wizards.^[3] Three days later he started for Toronto again and earned a 2-1 win over Real Salt Lake, but was injured in the 85th minute.^[4] In April 2008, Reynolds signed a contract to be represented by FIFA licensed agent Carlos De Sousa ^[5] and was subsequently signed to a contract with Miami FC of USL First Division.^[6] He made three starting appearances in 2008, earning a 1-1 draw on the road against the Vancouver Whitecaps^[7] and two shutout victories against the Rochester Rhinos^[8] and Atlanta Silverbacks^[9] to close out the Miami season with a .33 goals against average.

He joined Austin Aztex for the 2009 season. He made 12 appearances, but was released at the end of the year.^[10]

In 2010 he returned to Cal State Fullerton to coach goalkeepers for both the men's and women's teams.^[11]

International[link]

Reynolds has played in several international exhibitions including in July 2005 with Portland Timbers in a friendly against Sunderland of the English Premier League; friendly matches with Chivas USA against Mexican Primera A sides Necaxa in October 2006 in Aguascalientes, Mexico, and Chivas Guadalajara at Estadio Jalisco in November 2007; and on August 2, 2008 for Miami FC in a friendly match against the Haiti national football team in Miami.^[12]

In September 2008 Reynolds was invited to the United States men's national soccer team training camp in Miami for three days of training as the team prepared for their World Cup qualifier match against Cuba.

References[link]

External links[link]

Wikimedia Commons has media related to: Sam Reynolds

• Austin Aztex profile
• Miami FC profile
• MLS player profile

Johnny Polygon
Johnny Polygon in Silverlake, California
Background information
Birth name	John Armour
Born	(1984-05-15) May 15, 1984 (age 28) Ohio, United States
Origin	Tulsa, Oklahoma, United States
Genres	Hip hop, alternative hip hop
Occupations	Rapper, singer-songwriter, record producer
Years active	2001-present
Labels	Future Green Entertainment (2008-2010) Bananabeat Records (2012-present)
Associated acts	Gabe Real, Amanda Seales, Cisco Adler, Shwayze, Kid Cudi, Mod Sun, Dead Prez, Nas
Website	www.johnnypolygononline.com

John Armour (born May 15, 1984), better known by his stage name Johnny Polygon is an American independent hip hop recording artist. He first gained attention with his music video for a song titled "Bag", which ultimately led to him being signed to DJ Green Lantern’s record label Future Green Entertainment.^[1] In 2009, he released his first EP titled Group Hug. On February 14, 2010, Polygon released an official mixtape, Rebel Without Applause, which included his promotional single, "The Riot Song". On April 1, 2010, Polygon announced he was no longer signed decided to Future Green Entertainment and will continue to be an independent artist. On February 2, 2012, via his Twitter feed, Polygon announced he signed to Bananabeat Records, where he'll release his full-length project Pussy Gun.^[2]

Polygon participated in local theater groups and rap battles helping him develop a sense of stage presence and projection. Surviving on the underground Midwest scene for several years amassing self-released albums, Polygon headed out to Los Angelos, doubling his back catalog of recorded music. Polygon has also worked with Nas, Kid Cudi, Amanda Diva, and Gabe Real.

Biography[link]

1984-2008: Early life and career beginnings[link]

Tulsa, Oklahoma where Polygon grew up.

Johnny Polygon was born John Armour on May 15, 1984, in Ohio and grew up in Tulsa, Oklahoma.^[3]

Johnny Polygon first gained notice on a local battle scene. Looking for something more, he moved to Hollywood to pursue his music career. It was in Los Angeles where Johnny met Nathan Morse, who approached him about making a music video for his song “Bag”. The video was eventually entered into an online competition and gained notice from DJ Green Lantern.^[3] Liking what he heard and saw, in 2008 Green Lantern signed Johnny to his label Future Green Entertainment.^[4]

Once Johnny signed with DJ Green Lantern he moved to New York where they recorded a song together called, “Price on Your Head” which is featured in 2008’s Grand Theft Auto 4 and its soundtrack.^[4] That same year Johnny was featured on Nas’ 2008 untitled album, singing the chorus for the song “Black President”.

[edit] 2009–2010: Mixtapes, Group Hug and Wolf in Cheap Clothing

In 2009, Polygon made his television debut on BET’s Inaugural Address, performing “Black President” with Nas.^[5] That same year he released his first EP Group Hug on Invasion Records. On July 17, 2009, Polygon was featured in Vapors Magazine^[3] and on September 14, he was again featured in YRB Magazine.^[6] Johnny also made a cameo appearance in Kid Cudi’s video for “Pursuit of Happiness”.^[7]

Polygon’s “Riot Song” was also included on HBO’s 2010 series How to Make it in America and its mixtape, How to Make it in America Mixtape presented by Kid Cudi, DJ Green Lantern, and Broke Mogul.^[8] During the intro of the track Cudi states that Johnny is “…the illest up and coming artists”. Kid Cudi is featured on Rebel Without Applause, performing a remix with Johnny to “The Riot Song”.^[9]

Presented by DJ Green Lantern and Karmaloop.com, Johnny released an official mixtape on February 14, 2010, titled Rebel Without Applause.^[10] The album features guest appearances by Kid Cudi, Amanda Diva, and Gabe Real.^[10] Following his mixtape release, a video for “The Riot Song” debuted on February 23.^[11] The video is a Wile E. Coyote and Roadrunner inspired live action cartoon, using over 3,000 real paper cut outs to tell the story, with no repeating images.

On March 31, 2010, Polygon went on Los Angeles’ KPWR's POWER 106 FM THE TAKEOVER with the LA Leakers and DJ Reflex.^[12] During the show he performed a freestyle over The Black Eyed Peas' "Imma Be". In the July 2010 issue of Oklahoma Magazine Polygon had a featured spread.^[13] Polygon also made an appearance on the Dead Prez song, "NYDP", which was in the 2010 movie Brooklyn’s Finest.

During a show in Los Angeles on April 1, 2010, Polygon announced to his fans that he was moving forward as an independent artist.

Johnny's video for "The Riot Song" premiered on MTV2’s show subterranean on July 28, 2010 ^[14] Also in late July, “The Riot Song” music video was added to VH1 Soul and mtvU^[15] Following the release of his video of “The Riot Song” on MTV eventually breaking MTV's Top 100 peaking at #8.

On November 16, 2010, Polygon released an official 30-track mixtape presented by Orisue, titled Catch Up. It is composed of tracks from his career up to this point as well as remixes, collaborations with other artists, and unreleased tracks. He followed this up with an EP, on December 15, 2010, titled Wolf in Cheap Clothing that contains seven new tracks.^[16]

[edit] 2011-present: Pussy Gun

In 2011, Polygon returned to his Oklahoma roots when he was featured on the back of This Land Press's January print issue as a part of the paper's "True Tulsa" portrait series.^[17] In an interview published in April 2011, Polygon revealed that when a label based out of San Francisco offered him a record deal, on the condition that he transplant himself to the Bay Area and represent it as his hometown, Polygon declined out of loyalty to his hometown of Tulsa, Oklahoma.^[18]

In 2011, Polygon stopped by Los Angeles’ Knocksteady Live, and performed "LIMOSEXSUPERSTAR", a new song from his upcoming project, Pussy Gun, for their podcast.^[19] Polygon's debut studio album, Pussy Gun, has been announced for a 2012 release under Bananabeat Records.^[20] On March 7, 2012, Polygon released a trailer for a song off the album titled "My Shit >" and released the song for free download on his Facebook page.^[21] Almost eight months after performing the song for Knocksteady TV, Polygon released the official audio for Pussy Gun's lead single, "LIMOSEXSUPERSTAR".

The Johnny Polygon Show[link]

Johnny also has his own online show on karmaloop.tv.^[22] The premise of the show revolves around Johnny and his life as an artist. Now on its second season, the show has featured stories about him, Nas, and Cudi and what its like to be in the shoes of Johnny Polygon.

Discography[link]

Studio albums

• Journey to I (2001)
• Soldier and the Dancer (2002)
• Plan B (2003)
• Leggo My Ego (2004)
• Meaty-Ogre (2006)
• Pussy Gun (2012)

EPs

• Group Hug (2009)
• Wolf in Cheap Clothing (2010)

Mixtapes

• Rebel Without Applause (2010)
• Catch-Up (2010)

Guest appearances[link]

References[link]

External links[link]

Wikimedia Commons has media related to: Johnny Polygon

• Official website
• Johnny Polygon at Allmusic
• Johnny Polygon on Myspace
• Johnny Polygon on Twitter
• Johnny Polygon on Tumblr
• Johnny Polygon's channel on YouTube

Aphex Twin
In Turin, Italy in March 2007.
Background information
Birth name	Richard David James
Born	(1971-08-18) 18 August 1971 (age 40) Limerick, Ireland
Origin	Lanner, Cornwall, United Kingdom
Genres	Electronic, ambient, IDM, acid techno
Occupations	Musician, composer, remixer
Instruments	Synthesizers, piano, electronics, harpsichord, organ, percussion, laptop, softsynth, turntables, drum machine, guitar, bass guitar, computer, sequencer, sampler
Years active	1985–present
Labels	Warp Records Sire Records Rephlex Records
Associated acts	Universal Indicator, Mike & Rich

Richard David James (born 18 August 1971), best known under the pseudonym Aphex Twin, is a British electronic musician and composer. He founded the record label Rephlex Records in 1991 with Grant Wilson-Claridge. He was described by The Guardian as "the most inventive and influential figure in contemporary electronic music."^[1]

Aphex Twin has also recorded music under the aliases AFX, Blue Calx, Bradley Strider, Caustic Window, DJ Smojphace, GAK, Martin Tressider, Polygon Window, Power-Pill, Prichard D. Jams, Q-Chastic, Tahnaiya Russell, The Dice Man, Soit-P.P., and speculatively The Tuss.

Aphex Twin has released recordings on Rephlex, Warp, R&S, Sire, Mighty Force, Rabbit City, and Men Records.