Rectangle
Type	quadrilateral, parallelogram, orthotope
Edges and vertices	4
Schläfli symbol	{}×{}
Coxeter–Dynkin diagram
Symmetry group	D2, [2], (*22)
Dual polygon	rhombus
Properties	convex, isogonal, cyclic

In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. Another name is equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. The term oblong is occasionally used to refer to a non-square rectangle.^[1][2] A rectangle with vertices ABCD would be denoted as  ABCD.

The word rectangle comes from the Latin rectangulus, which is a combination of rectus (right) and angulus (angle).

A so-called crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals.^[3] It is a special case of an antiparallelogram, and its angles are not right angles. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.

Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.

Characterizations[link]

A convex quadrilateral is a rectangle if and only if it is any one of the following:^[4][5]

• a parallelogram with at least one right angle
• an equiangular parallelogram
• a parallelogram with diagonals of equal length
• a parallelogram ABCD where triangles ABD and DCA are congruent
• a quadrilateral which has four right angles
• an equiangular quadrilateral

Classification[link]

Traditional hierarchy[link]

A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular.

A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length.

A trapezium is a convex quadrilateral which has at least one pair of parallel opposite sides.

A convex quadrilateral is

• Star-shaped: The whole interior is visible from a single point, without crossing any edge.
• Simple: The boundary does not cross itself.

Alternative hierarchy[link]

De Villiers defines a rectangle more generally as any quadrilateral with axes of symmetry through each pair of opposite sides.^[6] This definition includes both right-angled rectangles and crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, and another which is the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects.

Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides. These quadrilaterals comprise isosceles trapezia and crossed isosceles trapezia (crossed quadrilaterals with the same vertex arrangement as isosceles trapezia).

Properties[link]

Symmetry[link]

A rectangle is cyclic: all corners lie on a single circle.

It is equiangular: all its corner angles are equal (each of 90 degrees).

It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit.

It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

Rectangle-rhombus duality[link]

The dual polygon of a rectangle is a rhombus, as shown in the table below.^[7]

Miscellaneous[link]

The two diagonals are equal in length and bisect each other. Every quadrilateral with both these properties is a rectangle.

A rectangle is rectilinear: its sides meet at right angles.

A non-square rectangle has 5 degrees of freedom, comprising 2 for position, 1 for rotational orientation, 1 for overall size, and 1 for shape.

Two rectangles, neither of which will fit inside the other, are said to be incomparable.

With vertices denoted A, B, C, and D, for any point P in the interior of a rectangle:^[8]

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.

Formulas[link]

The formula for the perimeter of a rectangle.

If a rectangle has length Failed to parse (Missing texvc executable; please see math/README to configure.): \ell

and width w

• it has area Failed to parse (Missing texvc executable; please see math/README to configure.): K = \ell w\,

,

• it has perimeter Failed to parse (Missing texvc executable; please see math/README to configure.): P = 2\ell + 2w = 2(\ell + w)\,

,

• each diagonal has length Failed to parse (Missing texvc executable; please see math/README to configure.): d=\sqrt{\ell^2 + w^2}

,

• and when Failed to parse (Missing texvc executable; please see math/README to configure.): \ell = w\,

, the rectangle is a square.

Theorems[link]

The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area.

The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle.

A parallelogram with equal diagonals is a rectangle.

The Japanese theorem for cyclic quadrilaterals^[9] states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.

Crossed rectangles[link]

A crossed (self-intersecting) quadrilateral consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.

A crossed quadrilateral is sometimes likened to a bow tie or butterfly. A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie. A crossed rectangle is sometimes called an "angular eight".

The interior of a crossed rectangle can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.

A crossed rectangle is not equiangular. The sum of its interior angles (two acute and two reflex), as with any crossed quadrilateral, is 720°.^[10]

A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:

• Opposite sides are equal in length.
• The two diagonals are equal in length.
• It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

Other rectangles[link]

A saddle rectangle has 4 nonplanar vertices, alternated from vertices of a cuboid, with a unique minimal surface interior defined as a linear combination of the four vertices, creating a saddle surface. This example shows 4 blue edges of the rectangle, and two green diagonals, all being diagonal of the cuboid rectangular faces.

In solid geometry, a figure is non-planar if it is not contained in a (flat) plane. A skew rectangle is a non-planar quadrilateral with opposite sides equal in length and four equal acute angles.^{[11][citation needed]} A saddle rectangle is a skew rectangle with vertices that alternate an equal distance above and below a plane passing through its centre, named for its minimal surface interior seen with saddle point at its centre.^[12] The convex hull of this skew rectangle is a special tetrahedron called a rhombic disphenoid. (The term "skew rectangle" is also used in 2D graphics to refer to a distortion of a rectangle using a "skew" tool. The result can be a parallelogram or a trapezoid/trapezium.)

In spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry.

In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.

In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.

Tessellations[link]

The rectangle is used in many periodic tessellation patterns, in brickwork, for example, these tilings:

Stacked bond

Running bond

Basket weave

Basket weave

Herringbone pattern

Squared, perfect, and other tiled rectangles[link]

A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is perfect^[13][14] if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is imperfect. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles.

A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares.^[13][15] The same is true if the tiles are unequal isosceles right triangles.

The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.

See also[link]

• British flag theorem
• Cuboid
• Golden rectangle
• Hyperrectangle

References[link]

External links[link]

Wikimedia Commons has media related to: Rectangles

• Weisstein, Eric W., "Rectangle" from MathWorld.
• Definition and properties of a rectangle with interactive animation.
• Area of a rectangle with interactive animation.

Gigi D'Agostino
Background information
Birth name	Luigino Celestino Di Agostino
Also known as	Dottor Dag, Il Folklorista, Lento Violento Man
Born	(1967-12-17) December 17, 1967 (age 44)
Origin	Turin, Italy
Genres	Italo Dance Eurodance Lento Violento House
Years active	1994–present
Labels	Arista Records, ZYX Music
Associated acts	Luca Noise, DJ Pandolfi
Website	http://www.gigidagostino.com/

Luigino Celestino Di Agostino (born 17 December 1967), better known by his stage name Gigi D'Agostino, is an Italian DJ, remixer and record producer. In 1986, he started his career as a DJ-spinning Italo Disco, and releasing his first mix, titled "Psychodelic".^[1]

Biography[link]

Early life[link]

Born in Turin, D'Agostino spent his childhood between his native city of Turin and Brescia, where the Media Records studios are located. As a child, D'Agostino wanted to be someone in the world of disco music. Starting out working as a stonemason and a fitter, he began his musical career as a DJ by organizing parties in clubs. His debut was in a club near Turin called "Woodstock". He was a resident DJ of an Italian disco named Ultimo Impero from 1993 to 1998.

1990s[link]

D'Agostino's first release was Noise Makers Theme, a double-A sided record with a track by Daniele Gas on the other side, which launched the Noise Maker label, under the direction of Italian house producer Gianfranco Bortolotti. D'Agostino would continue working with Gas, as well as Mauro Picotto's production team in the following years. His 1999 single "Bla Bla Bla" became a major hit in Europe. He described the single as "a piece I wrote thinking of all the people who talk and talk without saying anything!"^[2] As a DJ, D'Agostino is known as one of the "pioneers of Mediterranean Progressive Dance,"^[3] consisting of minimalistic sounds and Latin and Mediterranean melodies. As a record producer, D'Agostino's uses the pseudonym "Gigi Dag"; he transforms a piece, originally destined for the discos, into a success for the mainstream public.

After successes like the track "Sweetly", he joined the team of Bortolotti, the general manager of Media Records and suddenly reached the top of the hit parades. His single "FLY", published in 1996 with BXR Noise Maker, the label created by Media Records for the Mediterranea progressive house, reached the top of the sales in Italy. This was followed by his single "Gigi's Violin".

Subsequently, D'Agostino released the song "Angel's Symphony" with R.A.F. by Mauro Picotto, a friend from Media Records. His biggest project was the self-titled album Gigi D'Agostino, consisting of 19 tracks, which sold over 60,000 copies.

After his rise to success, D'Agostino's musical style changed, with his sound becoming more melodic, at midway between house and progressive, with more energetic and melodious sonorities and less obsessive rhythms, also known as Italo dance.

In 1997, he released the single "Gin Lemon", followed by "'Elisir (Your Love)" (1998), "Cuba Libre" (1999), and "Bla Bla Bla" (1999). He later released the album Eurodance Compilation, which contained five unpublished tracks. With this compilation, he earned a platinum disc and was awarded with the "Best Producer in 1999" award at the Italian Dance Awards. In October, D'Agostino released another hit single, entitled "Another Way".

In 1999, his album L'Amour Toujours (released in 2001 in the United States) included 23 tracks on two CDs, with the hit song "L'Amour Toujours" bringing him to the 10th position in the sales parade in Italy; as a result, he earned another Platinum Disc.

2000s[link]

In 2000, his remake of the Nik Kershaw song "The Riddle" sold 1,000,000 copies in Germany and 200,000 copies in France.^{[citation needed]} He also released the albums Tecno Fes (August 2000) and Tecno Fes - EP, Vol. 2 (December 2000).

In 2001, he released L'Amour Toujours EP, consisting of three songs: two new versions of "L'amour Toujours" and "Un Giorno Credi", a single made in collaboration with Edoardo Bennato. "Un Giorno Credi" reached the top of the parades in Italy and became one of the most played songs in the national and international music networks. He was also awarded with "Best DJ-Producer of the Year" at the "Red Bull Awards" in Italy.

In 2001, he also released the hit "Super", as a result of a collaboration with Albertino (an Italian music artist), which won him the awards "Best Dance Producer" at the PIM (Italian Music Award) and "Public Award" at the Danish DJs Award in Copenhagen, Denmark. In December of that year, he released Il Grande Viaggio, a compilation of his favourite songs, including old electronic pieces and new rhythms and melodies. As a result, he was awarded with "Best Dance Producer" at the Italian Dance Awards. D'Agostino stated during an interview with MTV that he enjoys cycling with his pigs in Rome.

In 2003, he released the song "Ripassa" in the EP Underconstruction 1: Silence, a hit which is very important for his next productions. This song helped influence Lento Violento, a subgenre that had yet to be distinguished.

On 15 July 2004, his compilation "Euro Dance" won Platinum certification with 120,000 copies sold in just a few weeks.^{[citation needed]}

D'Agostino has played in some well known clubs throughout Europe, including in Italy, Spain, France, the Netherlands, the Czech Republic at the Karlovy Lázně, Germany, Switzerland, and England (in 1996, he played at the Ministry of Sound in London with Mauro Picotto).

In 2006, he created his own new record label called "Noisemaker Hard" to differentiate the style of music released under his older record label, "Noisemaker".

In 2007, D'Agostino released Lento Violento e Altre Storie, a compilation which contains two CDs and his new 35 Lento Violento songs. The most important hit is "Ginnastica Mentale"^{[citation needed]}

After some months, he released the compilation La musica che pesta under the name "Lento Violento Man", which contains two CDs for a total of 38 unmixed tracks.

From 2005 until January 2010, he had radio programs, on the Italian Radio m2o Musica allo Stato Puro (in English: "Music at the Pure State"), called "Il Cammino di Gigi D'Agostino" and "Quello che mi piace ". They consist of DJ sets played by Gigi, and featured new remixes of his songs and songs by Italian and international artists.

In August 2011, he released a digital single Stay With Me. This was released a few months after he postponed the release of his upcoming album Mondo Reale.

Discography[link]

Studio albums[link]

• A Journey into Space (1994)
• Gigi D'Agostino (1996)
• L'Amour Toujours (1999)
• Tecno Fes Vol. 2 (2000)
• All In One Night (2001)
• L'Amour Toujours II (2004)
• Disco Tanz (2005)
• Per Sempre: The Hits (2005)
• The Greatest Hits (2006)
• Some Experiments (2006)
• Lento Violento ...e altre storie (2007)
• La Musica Che Pesta (as Lento Violento Man) (2007)
• Suono Libero (2008)
• The Essential Gigi D'Agostino (2009 - Austria and Germany only)
• Ieri & Oggi Mix Vol. 1 (2010)
• Ieri & Oggi Mix Vol. 2 (1 December 2010)

EPs[link]

• Le Voyage Estate (1996)
• Gin Lemon (1997)
• Tanzen (1999)
• Tecno Fes EP (2000)
• Il Grande Viaggio EP (2001)
• Underconstruction 1: Silence E.P. (2003)
• Underconstruction 2: Silence Remix (2004)
• Underconstruction 3: Remix (2004)

Compilations[link]

• The Greatest Hits (1996)
• Il Grande Viaggio Di Gigi D'Agostino Vol. 1 (2001)
• Il programmino di Gigi D'agostino (2003)
• Benessere (2004)
• Laboratorio 1 (2004)
• Laboratorio 2 (2005)
• Laboratorio 3 (2005)

Singles[link]

References[link]

External links[link]

• Official website

DJ Rectangle is an American DJ/turntablist and Grammy Award-nominated hip hop/R&B record producer.

In 1993, DJ Rectangle won the US DMC Championship and became the official DJ for West Coast rapper, Warren G. He then toured with Warren which included appearances on various television shows including Late Night with Conan O’Brien, The Jon Stewart Show, Late Night with David Letterman, MTV Spring Break, and London’s Top of the Pops. Rectangle has also toured with other established artists including R&B singer, R Kelly, and hip hop artist and producer, Jermaine Dupri.

Rectangle was among the first DJs to integrate freestyles from major recording artists into his own mix tapes and has released an extensive set of battle records used by many successive DJs to win DMC World Championships.

DJ Rectangle has contributed his production skills to numerous projects including releases by Eminem, Dirt Nasty, Funkdoobiest, The Luniz, The Whoridas, and Xzibit. Additionally, he has produced tracks used on television shows such as E! True Hollywood Story, Saturday Night Live, and Super Bowl XLIII as well as in movies including 1996's Set It Off starring Jada Pinkett and Queen Latifah, 1997's Fakin' da Funk starring Pam Grier and Dante Basco, and 2005's Get Rich or Die Tryin' starring 50 Cent.

In 2008, DJ Rectangle received a Grammy Award nomination for his production work on the 2007 album, Malamarismo, by Latin music hip hop singer, Mala Rodríguez.

Discography[link]

Television[link]

• 2010 Saturday Night Live
• 2010 MTV's My Super Sweet 16
• 2010 World Poker Tour
• 2009 Super Bowl XLIII
• 2001 E! True Hollywood Story

Movies[link]

• 2005 Get Rich or Die Tryin'
• 1998 The Big Hit
• 1997 Fakin' Da Funk
• 1996 Set It Off

Mixtapes[link]

• 2011 TBA
• 2008 Wax Assassin 2: Training Day, Bad Table Manners
• 2006 Casino Royale Vol. 2: For The Gangsters, Casino Royale Vol. 1 : For The Hustlers, Kill Steelz Vol. 2, Kill Steelz Vol. 1
• 2005 Fear And Loathing, Rectangle's Big Adventure, 2005 Pyro Technics
• 2004 Who Framed DJ Rectangle?, "Resident Evil" Turn Table Apocalypse, Six Million Dollar Hand, Wax On Wax Off
• 2003 1200's Never Die
• 2002 Faders Of Fury, Vinyl Combat 2: Attack Of The Clones, Turntable Torture, Deadly Needles
• 2000 Enter The Diamond Needle Dynasty
• 1999 The Tone Arm Terrorist, Behind Bars, Wax Assassin
• 1998 The Lightning Fist (Millennium Collection Vol. 7)
• 1997 The Tables Have Turned
• 1995 Vinyl Combat, Ill Rated
• 1994 Rollin' Deep, O.G Style, Talkin Shit, Let The Bass Kick, Funky Demolition
• 1992 DJ Rectangle EP / Z-Rock Records

Battle records[link]

• 2009 Scratch Masters Revenge Volume 2
• 2007 Ultimate, Ultimate Battle weapon Vol. 8, Ultimate, Ultimate Battle Weapon Vol. 7
• 2006 Ultimate, Ultimate Battle Weapon Vol. 6
• 2005 Dope Trax Vol. 2
• 2003 Battle Wax Vol. 2
• 2002 The Vinyl Avengers Vol. 2
• 2001 The Best of DJ Rectangle, Ultimate, Ultimate Battle Weapon, Vol. 5, Ultimate, Ultimate Battle Weapon, Vol. 4 (Battle record)
• 2000 Scratch Masters Revenge, Vol. 1, Mixes Beats & More, Vol. 2, Mixes Beats & More, Vol. 1
• 1999 Ultimate, Ultimate Battle Weapon, Vol. 3, The Vinyl Avengers, Vol. 1, Ultimate, Ultimate Battle Weapon, Vol. 2
• 1998 Ultimate, Ultimate Battle Weapon, Vol. 1, Frosted Breaks Vol. 2
• 1997 Deadly Needles, Gorilla Breaks, Frosted Breaks, Battle Wax Vol. 1
• 1996 Ultimate Battle Weapon Vol. 5, Ultimate Battle Weapon Vol. 4, Original Battle Weapon
• 1995 Ultimate Battle Weapon Vol. 3, Ultimate Battle Weapon Vol. 2, Ultimate, Battle Weapon Vol. 1
• 1994 Let the Bass Kick, Funky Demolition
• 1991 Dope Trax Vol. 1

Production[link]

2007[link]

[edit] Dirt Nasty - Dirt Nasty

• "Animal Lover" (Label: Shoot To Kill)

2005[link]

[edit] DJ Rectangle - Get Rich or Die Tryin'

• "Forgot The Name!!" (Label: Aftermath Entertainment)

2003[link]

[edit] DJ Rectangle Feat. Eminem and Dree - 1200's Never Die

• "You Must Be Crazy" (Label: Music 4 Da People)

2001[link]

[edit] Funkdoobiest - Real Urban Latin 1

• "4-1 Nite" (Label: RCA)

DJ Rectangle - Hip Hop + Rap[link]

• "Ya Ya Door" (Label: De Wolfe)
• "What What" (Label: De Wolfe)
• "First Love" (Label: De Wolfe)
• "How Many Times" (Label: De Wolfe)
• "Best Watch Out" (Label: De Wolfe)
• "Rock The Spot" (Label: De Wolfe)
• "Fake a Smile" (Label: De Wolfe)

DJ Rectangle Feat. The Whoridas - The Ultimate[link]

• "The Way We Do" (Label: South Paw)

[edit] DJ Rectangle Feat. Funkdoobiest - The Ultimate

• "Wild Out" (Label: South Paw)

2000[link]

[edit] A Lighter Shade of Brown - Greatest Hits

• "Street Life" (Label: Universal Music Group)

[edit] Xzibit Feat. Montage - Larger Than Life

• "Larger Than Life" (Label: TVT)

1999[link]

DJ Rectangle - Pumping Energy 3[link]

• Various (Label: De Wolfe)

DJ Rectangle - Hard Beats[link]

• Various (Label: De Wolfe)

1998[link]

[edit] Funkdoobiest - The Big Hit (soundtrack)

• "Act On It" (Label: Universal Music Group)

The Whoridas - Cheap Shots, Low Blows and Sucker Punches[link]

• "Get Lifted" (Label: South Paw)

Son Doobie - Cheap Shots, Low Blows and Sucker Punches[link]

• "Cali Life" (Label: South Paw)

1997[link]

DJ Rectangle - Ghetto Politix[link]

• "Turntable Terror" (Label: Universal Music Group/Thump)

[edit] Funkdoobiest Feat. Daz Dillinger - Troubleshooters

• "Papi Chulo" (Label: RCA)

1996[link]

[edit] Da 5 Footaz - Set It Off (soundtrack)

• "The Heist" (Label: Def Jam Recordings)

[edit] The Dove Shack - Fakin' da Funk (soundtrack)

• "Low Low" (Label: Street Solid)

External links[link]

• DJ Rectangle on Billboard
• DJ Rectangle on Allmusic
• DJ Rectangle on Discogs
• DJ Rectangle on Msn Entertainment
• DJ Rectangle on Yahoo! Music
• Soundtrack for Get Rich or Die Tryin'
• Soundtrack for Set It Off
• Soundtrack for Fakin' da Funk
• Soundtrack for The Big Hit