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| {{for|the record label|Rectangle (label)}}
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| {{Infobox Polygon
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| | name = Rectangle
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| | image = Rect Geometry.png
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| | caption = Rectangle
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| | type = [[quadrilateral]], [[parallelogram]], [[orthotope]]
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| | edges = 4
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| | symmetry = [[Dihedral symmetry|Dih<sub>2</sub>]], [2], (*22), order 4
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| | schläfli = { } × { } or { }<sup>2</sup>
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| | wythoff =
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| | coxeter = {{CDD|node_1|2|node_1}}
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| | area =
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| | dual = [[rhombus]]
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| | properties = [[convex polygon|convex]], [[isogonal figure|isogonal]], [[Cyclic polygon|cyclic]] Opposite angles and sides are congruent
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| }}
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| 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.
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| </ref><ref>
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| [http://www.icoachmath.com/SiteMap/Oblong.html Oblong – Geometry – Math Dictionary]. Icoachmath.com. Retrieved 2011-11-13.
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| </ref>
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| A rectangle with [[Vertex (geometry)|vertices]] ''ABCD'' would be denoted as {{rectanglenotation|ABCD}}.
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|
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| The word rectangle comes from the [[Latin]] ''rectangulus'', which is a combination of ''rectus'' (right) and ''angulus'' ([[angle]]).
| | What will the twelve constellation do following they failure in enjoy? Will they go to buy quite a few lavish products these as Chanel fragrance UGG Naomi boot or LV handbag? <br><br>Perfectly, for the appreciate, especially for the initially like, the previous declaring that a good starting makes a superior ending is not often correct. Usually, the very first appreciate will e+9nd soon and leave us damage and regrets one following a different because of to our childish feelings and impulse. <br><br><br>Some people stated that adore can make us turn into poetry however, are unsuccessful in love will make us additional near to the poets. In any other case, how the romance professionals can say that like is just like the hourglass and our tears and heart-breaking is the sand, even though each individual time we skip our fans would lead to a burst of the hourglass. Now let us see what will the twelve constellation do following they failure in love. <br><br>ARIES Aries will by no means depress their terrible mood even while they were being unsuccessful in adore. They also would not choose to keep at a corner on your own silently to burst out tears nor feel that a human being respect the blue sea and sky by yourself will release their broken heart to some extent. <br><br>So what will they do? Sure our Aries will phone several friends go to the bar to consume and to get the fall short-adore off their upper body. They really don't treatment the rule that drowns sorrow getting benefit of the liquor fearful that worried they just want to consume till their blue. <br><br>TAURUS Confronting with the are unsuccessful adore, Taurus may well be is the most contradictory since they constantly want to relaxed down their damaged coronary heart but are not able to do it. Taurus' modest and rational attitude determine that they will never expose their accurate thoughts quickly. <br><br>However, if they have set real determination into a connection but have not gotten a gorgeous ending before they missing their enjoy, then they will develop into sorrowful and unpleasant. You may possibly be capable to find them out for a couple of times just after the Taurus got a fail-like. <br><br>They have hid by themselves in a corner that just about every a single can not locate them, they will not need to have comfort and pour out their coronary heart, what they will need is just to remain by itself and think by on their own silently. Certainly, Taurus just feels like utilizing these methods to dissolve the painful feelings. <br><br>GEMINI Gemini is not individuals people today who are incredibly straightforward to fall in love with somebody, so if they decide to really like anyone, it indicates that they have set all preventions down and determine to enjoy that individual with coronary heart and soul. So, the Gemini will experience that mountains have fallen down and the earth has spitted when they experiencing failure in appreciate. They get use to using text to convey their thoughts. Even though the words are extremely implicit, it is more painful than tears to convey their disappointment. <br><br>Most cancers For good like is the fit method of Most cancers. In other words, they really don't know how to leave leeway for on their own. For the Most cancers, the lovesick is just like the end of the entire world. If feminine are compared to the flower, our Cancer's heart can be in comparison to the wild sea. If going through the sea there is no blooming flower, their tears will turn out to be the solutions. <br><br>They are very effortless to sink into disappointment and hatred but do not know how to wake up from them, the only way is to wait for the time to lighter the discomfort in their inner heart. In common text, their lovesick are constantly start out and finish with tears. <br><br>LEO Leo likes to present their sweet adore wherever and anytime, on the other hand, at the time the appreciate is failure, they will disguise in an unidentified corner to heal their damaged coronary heart by itself. Just like a hurt little cat and contrary to his appearance that freely to just take something up and easily place them down. <br><br>On the opposite, dealing with the agony of shedding really like, they will sink into the game environment or slumber all the working day in buy to escape the realistic. Till they arrive out of the shade completely they commenced turn out to be self-confident once again. <br><br>if you want to know more,be sure to go to ugginstock (about ugg boots)<br><br>If you have any questions with regards to wherever and how to use [http://tinyurl.com/k7shbtq ugg outlet], you can get hold of us at the webpage. |
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| 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}}
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| </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.
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| Rectangles are involved in many [[#Tessellations|tiling]] problems, such as tiling the plane by rectangles or tiling a rectangle by [[polygon]]s.
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| ==Characterizations==
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| 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.
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| </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–}}
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| </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}}
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| * 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>
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| * equiangular
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| * a [[parallelogram]] ''ABCD'' where [[triangles]] ''ABD'' and ''DCA'' are congruent
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| * a parallelogram with at least one right angle
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| * a parallelogram with diagonals of equal length
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| * a quadrilateral with four right angles
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| ==Classification==
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| ===Traditional hierarchy===
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| A rectangle is a special case of a [[parallelogram]] in which each pair of adjacent [[side (geometry)|side]]s is [[perpendicular]].
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| 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]].
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| A trapezium is a [[Convex polygon|convex]] [[quadrilateral]] which has at least one pair of [[parallel (geometry)|parallel]] opposite sides.
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| A convex quadrilateral is
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| * '''[[Simple polygon|Simple]]''': The boundary does not cross itself.
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| * '''[[Star-shaped polygon|Star-shaped]]''': The whole interior is visible from a single point, without crossing any edge.
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| ===Alternative hierarchy===
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| 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.)
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| </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.
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| 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).
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| ==Properties==
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| ===Symmetry===
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| A rectangle is [[Cyclic polygon|cyclic]]: all [[Corner angle|corner]]s lie on a single [[circle]].
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| It is [[equiangular polygon|equiangular]]: all its corner [[angle]]s are equal (each of 90 [[Degree (angle)|degrees]]).
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| It is isogonal or [[vertex-transitive]]: all corners lie within the same [[symmetry orbit]].
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| It has two [[line (geometry)|line]]s of [[reflectional symmetry]] and [[rotational symmetry]] of order 2 (through 180°).
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| ===Rectangle-rhombus duality===
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| 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>
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| {|class="wikitable" style="text-align:center"
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| |-
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| !Rectangle !! Rhombus
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| |-
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| |All ''angles'' are equal.
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| ||All ''sides'' are equal.
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| |-
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| |Alternate ''sides'' are equal.
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| ||Alternate ''angles'' are equal.
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| |-
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| |Its centre is equidistant from its ''[[Vertex (geometry)|vertices]]'', hence it has a ''[[circumcircle]]''.
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| ||Its centre is equidistant from its ''sides'', hence it has an ''incircle''.
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| |-
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| |Its axes of symmetry bisect opposite ''sides''.
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| ||Its axes of symmetry bisect opposite ''angles''.
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| |-
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| |Diagonals are equal in ''length''.
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| ||Diagonals intersect at equal ''angles''.
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| |}
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| *The figure formed by joining, in order, the midpoints of the sides of a rectangle is a [[rhombus]] and vice-versa.
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| ===Miscellaneous===
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| The two [[diagonal]]s are equal in length and [[Bisection|bisect]] each other. Every quadrilateral with both these properties is a rectangle.
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| A rectangle is [[rectilinear polygon|rectilinear]]: its sides meet at right angles.
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| 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).
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| Two rectangles, neither of which will fit inside the other, are said to be [[Comparability|incomparable]].
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| ==Formulae==
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| [[File:PerimeterRectangle.svg|thumb|150px|The formula for the perimeter of a rectangle.]]
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| If a rectangle has length <math>\ell</math> and width <math>w</math>
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| *it has [[area]] <math>A = \ell w\,</math>,
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| *it has [[perimeter]] <math>P = 2\ell + 2w = 2(\ell + w)\,</math>,
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| *each diagonal has length <math>d=\sqrt{\ell^2 + w^2}</math>,
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| *and when <math>\ell = w\,</math>, the rectangle is a [[Square (geometry)|square]].
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| ==Theorems==
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| The [[isoperimetric theorem]] for rectangles states that among all rectangles of a given [[perimeter]], the square has the largest [[area]].
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| The midpoints of the sides of any [[quadrilateral]] with [[perpendicular]] [[diagonals]] form a rectangle.
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| A [[parallelogram]] with equal [[diagonals]] is a rectangle.
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| 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.
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| </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.
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| 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}}
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| </ref> | |
| :<math>\displaystyle (AP)^2 + (CP)^2 = (BP)^2 + (DP)^2.</math>
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| ==Crossed rectangles==
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| 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.
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| 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".
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| 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.
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| 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>
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| A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:
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| *Opposite sides are equal in length.
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| *The two diagonals are equal in length.
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| *It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).
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| [[File:Crossed rectangles.png|320px]]
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| ==Other rectangles==
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| [[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.]]
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| 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.
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| </ref>{{citation needed|date=May 2010|reason=source does not mention rectangle – does call it a quadrilateral}}
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| 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
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| </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]].)''<!--these 2D figures ARE planar, not non-planar.-->
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| 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. | |
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| 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.
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| 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.
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| ==Tessellations==
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| The rectangle is used in many periodic [[tessellation]] patterns, in [[brickwork]], for example, these tilings:
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| {|class=wikitable
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| |- align=center
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| |[[File:Stacked bond.png|182px]]<br>Stacked bond
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| |[[File:Wallpaper group-cmm-1.jpg|150px]]<br>Running bond
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| |[[File:Wallpaper group-p4g-1.jpg|150px]]<br>Basket weave
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| |[[File:Basketweave bond.svg|150px]]<br>Basket weave
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| |[[File:Herringbone bond.svg|150px]]<br>[[Herringbone pattern]]
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| |}
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| ==Squared, perfect, and other tiled rectangles==
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| 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
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| ''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}}
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| </ref>
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| 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.
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| 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}}
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| </ref> | |
| The same is true if the tiles are unequal isosceles [[wikt:right triangle|right triangles]].
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| 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.
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| ==See also==
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| *[[Cuboid]]
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| *[[Golden rectangle]]
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| *[[Hyperrectangle]]
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| {{Commons category|Rectangles}}
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| *{{MathWorld |urlname=Rectangle |title=Rectangle}}
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| *[http://www.mathopenref.com/rectangle.html Definition and properties of a rectangle] with interactive animation.
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| *[http://www.mathopenref.com/rectanglearea.html Area of a rectangle] with interactive animation.
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| | |
| [[Category:Quadrilaterals]]
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| [[Category:Elementary shapes]]
| |
What will the twelve constellation do following they failure in enjoy? Will they go to buy quite a few lavish products these as Chanel fragrance UGG Naomi boot or LV handbag?
Perfectly, for the appreciate, especially for the initially like, the previous declaring that a good starting makes a superior ending is not often correct. Usually, the very first appreciate will e+9nd soon and leave us damage and regrets one following a different because of to our childish feelings and impulse.
Some people stated that adore can make us turn into poetry however, are unsuccessful in love will make us additional near to the poets. In any other case, how the romance professionals can say that like is just like the hourglass and our tears and heart-breaking is the sand, even though each individual time we skip our fans would lead to a burst of the hourglass. Now let us see what will the twelve constellation do following they failure in love.
ARIES Aries will by no means depress their terrible mood even while they were being unsuccessful in adore. They also would not choose to keep at a corner on your own silently to burst out tears nor feel that a human being respect the blue sea and sky by yourself will release their broken heart to some extent.
So what will they do? Sure our Aries will phone several friends go to the bar to consume and to get the fall short-adore off their upper body. They really don't treatment the rule that drowns sorrow getting benefit of the liquor fearful that worried they just want to consume till their blue.
TAURUS Confronting with the are unsuccessful adore, Taurus may well be is the most contradictory since they constantly want to relaxed down their damaged coronary heart but are not able to do it. Taurus' modest and rational attitude determine that they will never expose their accurate thoughts quickly.
However, if they have set real determination into a connection but have not gotten a gorgeous ending before they missing their enjoy, then they will develop into sorrowful and unpleasant. You may possibly be capable to find them out for a couple of times just after the Taurus got a fail-like.
They have hid by themselves in a corner that just about every a single can not locate them, they will not need to have comfort and pour out their coronary heart, what they will need is just to remain by itself and think by on their own silently. Certainly, Taurus just feels like utilizing these methods to dissolve the painful feelings.
GEMINI Gemini is not individuals people today who are incredibly straightforward to fall in love with somebody, so if they decide to really like anyone, it indicates that they have set all preventions down and determine to enjoy that individual with coronary heart and soul. So, the Gemini will experience that mountains have fallen down and the earth has spitted when they experiencing failure in appreciate. They get use to using text to convey their thoughts. Even though the words are extremely implicit, it is more painful than tears to convey their disappointment.
Most cancers For good like is the fit method of Most cancers. In other words, they really don't know how to leave leeway for on their own. For the Most cancers, the lovesick is just like the end of the entire world. If feminine are compared to the flower, our Cancer's heart can be in comparison to the wild sea. If going through the sea there is no blooming flower, their tears will turn out to be the solutions.
They are very effortless to sink into disappointment and hatred but do not know how to wake up from them, the only way is to wait for the time to lighter the discomfort in their inner heart. In common text, their lovesick are constantly start out and finish with tears.
LEO Leo likes to present their sweet adore wherever and anytime, on the other hand, at the time the appreciate is failure, they will disguise in an unidentified corner to heal their damaged coronary heart by itself. Just like a hurt little cat and contrary to his appearance that freely to just take something up and easily place them down.
On the opposite, dealing with the agony of shedding really like, they will sink into the game environment or slumber all the working day in buy to escape the realistic. Till they arrive out of the shade completely they commenced turn out to be self-confident once again.
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