Phidias

2014-01-30T15:11:42Z

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{{for|the record label|Rectangle (label)}}
{{Infobox Polygon
| name = Rectangle
| image = Rect Geometry.png
| caption = Rectangle
| type = [[quadrilateral]], [[parallelogram]], [[orthotope]]
| edges = 4
| symmetry = [[Dihedral symmetry|Dih2]], [2], (*22), order 4
| schläfli = { } × { } or { }2
| wythoff =
| coxeter = {{CDD|node_1|2|node_1}}
| area =
| dual = [[rhombus]]
| properties = [[convex polygon|convex]], [[isogonal figure|isogonal]], [[Cyclic polygon|cyclic]] Opposite angles and sides are congruent
}}

In [[Euclidean geometry|Euclidean plane geometry]], a '''rectangle''' is any [[quadrilateral]] with four [[right angle]]s. It can also be defined as an 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. A rectangle with four sides of equal length is a [[square]]. The term '''[[wikt:oblong|oblong]]''' is occasionally used to refer to a non-[[square]] rectangle.<ref>http://www.cimt.plymouth.ac.uk/resources/topics/art002.pdf</ref><ref>[http://www.mathsisfun.com/definitions/oblong.html Definition of Oblong]. Mathsisfun.com. Retrieved 2011-11-13.
</ref><ref>
[http://www.icoachmath.com/SiteMap/Oblong.html Oblong – Geometry – Math Dictionary]. Icoachmath.com. Retrieved 2011-11-13.
</ref>
A rectangle with [[Vertex (geometry)|vertices]] ''ABCD'' would be denoted as {{rectanglenotation|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.<ref>{{Cite journal |doi=10.1098/rsta.1954.0003 |last1=Coxeter |first1=Harold Scott MacDonald |author1-link=Harold Scott MacDonald Coxeter |last2=Longuet-Higgins |first2=M.S. |last3=Miller |first3=J.C.P. |title=Uniform polyhedra |jstor=91532 |mr=0062446 |year=1954 |journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences |issn=0080-4614 |volume=246 |pages=401–450 |issue=916 |publisher=The Royal Society}}
</ref>
It is a special case of an [[antiparallelogram]], and its angles are not right angles. Other geometries, such as [[Spherical geometry|spherical]], [[Elliptic geometry|elliptic]], and [[Hyperbolic geometry|hyperbolic]], have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.

Rectangles are involved in many [[#Tessellations|tiling]] problems, such as tiling the plane by rectangles or tiling a rectangle by [[polygon]]s.

==Characterizations==
A [[Convex and concave polygons|convex]] [[quadrilateral]] is a rectangle iff ([[if and only if]]) it is any one of the following:<ref>Zalman Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, pp. 34–36 ISBN 1-59311-695-0.
</ref><ref>
{{cite book |author1=Owen Byer |author2=Felix Lazebnik |author3=Deirdre L. Smeltzer |title=Methods for Euclidean Geometry |url=http://books.google.com/books?id=W4acIu4qZvoC&pg=PA53 |accessdate=2011-11-13 |date=19 August 2010 |publisher=MAA |isbn=978-0-88385-763-2 |pages=53–}}
</ref>
* a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is <math>\tfrac{1}{4}(a+c)(b+d)</math>.<ref name=Josefsson/>{{rp|fn.1}}
* a convex quadrilateral with successive sides ''a'', ''b'', ''c'', ''d'' whose area is <math>\tfrac{1}{2} \sqrt{(a^2+c^2)(b^2+d^2)}.</math><ref name=Josefsson>Martin Josefsson, [http://forumgeom.fau.edu/FG2013volume13/FG201304.pdf "Five Proofs of an Area Characterization of Rectangles"], ''Forum Geometricorum'' 13 (2013) 17–21.</ref>
* equiangular
* a [[parallelogram]] ''ABCD'' where [[triangles]] ''ABD'' and ''DCA'' are congruent
* a parallelogram with at least one right angle
* a parallelogram with diagonals of equal length
* a quadrilateral with four right angles

==Classification==

===Traditional hierarchy===
A rectangle is a special case of a [[parallelogram]] in which each pair of adjacent [[side (geometry)|side]]s 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 (geometry)|parallel]] and [[equality (mathematics)|equal]] in [[length]].

A trapezium is a [[Convex polygon|convex]] [[quadrilateral]] which has at least one pair of [[parallel (geometry)|parallel]] opposite sides.

A convex quadrilateral is
* '''[[Simple polygon|Simple]]''': The boundary does not cross itself.
* '''[[Star-shaped polygon|Star-shaped]]''': The whole interior is visible from a single point, without crossing any edge.

===Alternative hierarchy===
De Villiers defines a rectangle more generally as any quadrilateral with [[Reflection symmetry|axes of symmetry]] through each pair of opposite sides.<ref>[http://mysite.mweb.co.za/residents/profmd/quadclassify.pdf An Extended Classification of Quadrilaterals] (An excerpt from De Villiers, M. 1996. ''Some Adventures in Euclidean Geometry.'' University of Durban-Westville.)
</ref>
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 of symmetry|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==

===Symmetry===
A rectangle is [[Cyclic polygon|cyclic]]: all [[Corner angle|corner]]s lie on a single [[circle]].

It is [[equiangular polygon|equiangular]]: all its corner [[angle]]s are equal (each of 90 [[Degree (angle)|degrees]]).

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

It has two [[line (geometry)|line]]s of [[reflectional symmetry]] and [[rotational symmetry]] of order 2 (through 180°).

===Rectangle-rhombus duality===
The [[dual polygon]] of a rectangle is a [[rhombus]], as shown in the table below.<ref>de Villiers, Michael, "Generalizing Van Aubel Using Duality", ''Mathematics Magazine'' 73 (4), Oct. 2000, pp. 303-307.</ref>
{|class="wikitable" style="text-align:center"
|-
!Rectangle !! Rhombus
|-
|All ''angles'' are equal.
||All ''sides'' are equal.
|-
|Alternate ''sides'' are equal.
||Alternate ''angles'' are equal.
|-
|Its centre is equidistant from its ''[[Vertex (geometry)|vertices]]'', hence it has a ''[[circumcircle]]''.
||Its centre is equidistant from its ''sides'', hence it has an ''incircle''.
|-
|Its axes of symmetry bisect opposite ''sides''.
||Its axes of symmetry bisect opposite ''angles''.
|-
|Diagonals are equal in ''length''.
||Diagonals intersect at equal ''angles''.
|}
*The figure formed by joining, in order, the midpoints of the sides of a rectangle is a [[rhombus]] and vice-versa.

===Miscellaneous===
The two [[diagonal]]s are equal in length and [[Bisection|bisect]] each other. Every quadrilateral with both these properties is a rectangle.

A rectangle is [[rectilinear polygon|rectilinear]]: its sides meet at right angles.

A rectangle in the plane can be defined by five independent [[Degrees of freedom (mechanics)|degrees of freedom]] consisting, for example, of three for position (comprising two of [[Translation (geometry)|translation]] and one of [[rotation]]), one for shape ([[Aspect_ratio#Rectangles|aspect ratio]]), and one for overall size (area).

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

==Formulae==
[[File:PerimeterRectangle.svg|thumb|150px|The formula for the perimeter of a rectangle.]]
If a rectangle has length <math>\ell</math> and width <math>w</math>
*it has [[area]] <math>A = \ell w\,</math>,
*it has [[perimeter]] <math>P = 2\ell + 2w = 2(\ell + w)\,</math>,
*each diagonal has length <math>d=\sqrt{\ell^2 + w^2}</math>,
*and when <math>\ell = w\,</math>, the rectangle is a [[Square (geometry)|square]].

==Theorems==
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]]<ref>[http://math.kennesaw.edu/~mdevilli/cyclic-incentre-rectangle.html Cyclic Quadrilateral Incentre-Rectangle] with interactive animation illustrating a rectangle that becomes a 'crossed rectangle', making a good case for regarding a 'crossed rectangle' as a type of rectangle.
</ref>
states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.

The [[British flag theorem]] states that with vertices denoted ''A'', ''B'', ''C'', and ''D'', for any point ''P'' on the same plane of a rectangle:<ref>{{cite journal |author=Hall, Leon M., and Robert P. Roe |title=An Unexpected Maximum in a Family of Rectangles |journal=Mathematics Magazine |volume=71 |issue=4 |year=1998 |pages=285–291 |url=http://web.mst.edu/~lmhall/Personal/HallRoe/Hall_Roe.pdf |jstor=2690700}}
</ref>
:<math>\displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.</math>

==Crossed rectangles==
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]] [[Space frame|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 angle]]s (two acute and two [[Reflex angle|reflex]]), as with any crossed quadrilateral, is 720°.<ref>[http://mysite.mweb.co.za/residents/profmd/stars.pdf Stars: A Second Look]. (PDF). Retrieved 2011-11-13.</ref>

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°).

[[File:Crossed rectangles.png|320px]]

==Other rectangles==
[[File:Saddle rectangle example.png|thumb|A '''saddle rectangle''' has 4 nonplanar vertices, [[Alternation (geometry)|alternated]] from vertices of a [[cuboid#Rectangular cuboid|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 polygon|skew]] rectangle''' is a non-planar quadrilateral with opposite sides equal in length and four equal [[acute angle]]s.<ref>[http://mathworld.wolfram.com/SkewQuadrilateral.html Skew Quadrilateral – from Wolfram MathWorld]. Mathworld.wolfram.com. Retrieved 2011-11-13.
</ref>{{citation needed|date=May 2010|reason=source does not mention rectangle – does call it a quadrilateral}}
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.<ref>{{The Geometrical Foundation of Natural Structure (book)}} "Skew Polygons (Saddle Polygons)." §2.2
</ref>
The [[convex hull]] of this skew rectangle is a special [[tetrahedron]] called a [[rhombic disphenoid]]. (The term "skew rectangle" is also used in [[computer graphics|2D graphics]] to refer to a distortion of a rectangle using a "skew" tool. The result can be a parallelogram or a [[trapezoid|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==
The rectangle is used in many periodic [[tessellation]] patterns, in [[brickwork]], for example, these tilings:
{|class=wikitable
|- align=center
|[[File:Stacked bond.png|182px]] Stacked bond
|[[File:Wallpaper group-cmm-1.jpg|150px]] Running bond
|[[File:Wallpaper group-p4g-1.jpg|150px]] Basket weave
|[[File:Basketweave bond.svg|150px]] Basket weave
|[[File:Herringbone bond.svg|150px]] [[Herringbone pattern]]
|}

==Squared, perfect, and other tiled rectangles==
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''<ref name="BSST"/><ref>{{cite journal |author=J.D. Skinner II, C.A.B. Smith and W.T. Tutte |date=November 2000 |title=On the Dissection of Rectangles into Right-Angled Isosceles Triangles |journal=[[Journal of Combinatorial Theory|J. Combinatorial Theory]] Series B |volume=80 |issue=2 |pages=277–319 |doi=10.1006/jctb.2000.1987}}
</ref>
if the tiles are [[Similarity (geometry)|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 triangle]]s.

A rectangle has [[Commensurability (mathematics)|commensurable]] sides if and only if it is tileable by a finite number of unequal squares.<ref name="BSST">{{cite journal |author=R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte |year=1940 |title=The dissection of rectangles into squares |journal=[[Duke Mathematical Journal|Duke Math. J.]] |volume=7 |issue=1 |pages=312–340 |doi=10.1215/S0012-7094-40-00718-9 |url=http://projecteuclid.org/euclid.dmj/1077492259}}
</ref><ref>
{{cite journal |author=R. Sprague |title=Ũber die Zerlegung von Rechtecken in lauter verschiedene Quadrate |journal=[[Crelle's Journal|J. fũr die reine und angewandte Mathematik]] |volume=182 |year=1940 |pages=60–64}}
</ref>
The same is true if the tiles are unequal isosceles [[wikt:right triangle|right triangles]].

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

==See also==
*[[Cuboid]]
*[[Golden rectangle]]
*[[Hyperrectangle]]

==References==
{{reflist|2}}

==External links==
{{Commons category|Rectangles}}
*{{MathWorld |urlname=Rectangle |title=Rectangle}}
*[http://www.mathopenref.com/rectangle.html Definition and properties of a rectangle] with interactive animation.
*[http://www.mathopenref.com/rectanglearea.html Area of a rectangle] with interactive animation.

[[Category:Quadrilaterals]]
[[Category:Elementary shapes]]

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Phidias