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| {{for|the avenue in [[Barcelona]]|Avinguda Diagonal}}
| | I am 22 years old and my name is Arturo Colleano. I life in Rotterdam (Netherlands).<br><br>Feel free to surf to my web blog :: Fifa 15 Coin Generator ([http://www.ahmhzh.com/plus/guestbook.php/ ahmhzh.com]) |
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| [[Image:Cube diagonals.svg|thumb|right|The diagonals of a [[cube]] with side length 1. AC' (shown in blue) is a [[space diagonal]] with length <math>\sqrt 3</math>, while AC (shown in red) is a [[face diagonal]] and has length <math>\sqrt 2</math>.]]
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| A '''diagonal''' is a line joining two nonconsecutive vertices of a [[polygon]] or [[polyhedron]]. Informally, any sloping line is called diagonal. The word "diagonal" derives from the [[ancient Greek]] διαγώνιος ''diagonios'',<ref>[http://www.etymonline.com/index.php?search=diagonal&searchmode=none Online Etymology Dictionary]</ref> "from angle to angle" (from διά- ''dia-'', "through", "across" and γωνία ''gonia'', "angle", related to ''gony'' "knee"); it was used by both Strabo<ref>Strabo, Geography 2.1.36–37</ref> and Euclid<ref>Euclid, Elements book 11, proposition 28</ref> to refer to a line connecting two vertices of a [[rhombus]] or [[cuboid]],<ref>Euclid, Elements book 11, proposition 38</ref> and later adopted into Latin as ''diagonus'' ("slanting line").
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| In [[mathematics]], in addition to its geometric meaning, a diagonal is also used in [[matrix (math)|matrices]] to refer to a set of entries along a diagonal line.
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| == Non-mathematical uses ==
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| [[File:2512-échafaudage-Réunion.jpg|250px|thumb|right|A stand of basic scaffolding on a house construction site, with diagonal braces to maintain its structure]]
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| In [[engineering]], a diagonal brace is a beam used to brace a rectangular structure (such as [[scaffolding]]) to withstand strong forces pushing into it; although called a diagonal, due to practical considerations diagonal braces are often not connected to the corners of the rectangle.
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| [[Diagonal pliers]] are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or "on a diagonal", hence the name.
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| A [[diagonal lashing]] is a type of lashing used to bind spars or poles together applied so that the lashings cross over the poles at an angle.
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| In [[association football]], the [[diagonal (football)|diagonal]] system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch.
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| [[File:Display size measurements.png|thumb|right|The diagonal is a common measurement of [[Two-dimensional display size|display size]].]]
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| == Polygons ==
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| {{See also|Quadrilateral#Diagonals}}
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| As applied to a [[polygon]], a diagonal is a [[line segment]] joining any two non-consecutive vertices. Therefore, a [[quadrilateral]] has two diagonals, joining opposite pairs of vertices. For any [[convex polygon]], all the diagonals are inside the polygon, but for [[re-entrant polygon]]s, some diagonals are outside of the polygon.
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| Any ''n''-sided polygon (''n'' ≥ 3), [[Convex polygon|convex]] or [[Concave polygon|concave]], has
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| :<math>\frac{n^2-3n}{2}\, </math>
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| or
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| :<math>\frac{n(n-3)}{2}\, </math>
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| diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or ''n'' − 3 diagonals.
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| {|rules="none" border="0" cellspacing="4" cellpadding="0" style="background:transparent;text-align:right"
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| |-valign="top"
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| {|class="wikitable"
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| ! Sides !! Diagonals
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| | 3 || 0
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| |-
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| | 4 || 2
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| |-
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| | 5 || 5
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| |-
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| | 6 || 9
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| |-
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| | 7 || 14
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| |-
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| | 8 || 20
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| |-
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| | 9 || 27
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| |-
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| | 10 || 35
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| |}
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| {|class="wikitable"
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| |-
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| ! Sides !! Diagonals
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| |-
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| | 11 || 44
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| |-
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| | 12 || 54
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| |-
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| | 13 || 65
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| |-
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| | 14 || 77
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| |-
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| | 15 || 90
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| |-
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| | 16 || 104
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| |-
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| | 17 || 119
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| |-
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| | 18 || 135
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| |}
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| {|class="wikitable"
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| |-
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| ! Sides !! Diagonals
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| | 19 || 152
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| |-
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| | 20 || 170
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| | 21 || 189
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| | 22 || 209
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| | 23 || 230
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| |-
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| | 24 || 252
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| | 25 || 275
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| |-
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| | 26 || 299
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| |}
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| {|class="wikitable"
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| |-
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| ! Sides !! Diagonals
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| | 27 || 324
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| | 28 || 350
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| | 29 || 377
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| | 30 || 405
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| | 31 || 434
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| | 32 || 464
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| | 33 || 495
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| | 34 || 527
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| |}
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| {|class="wikitable"
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| ! Sides !! Diagonals
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| | 35 || 560
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| | 36 || 594
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| | 37 || 629
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| | 38 || 665
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| | 39 || 702
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| | 40 || 740
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| | 41 || 779
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| | 42 || 819
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| |}
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| |}
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| == Matrices ==
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| In the case of a [[square matrix]], the ''main'' or ''principal diagonal'' is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix <math> A </math> with row index specified by <math>i</math> and column index specified by <math>j</math>, these would be entries <math>A_{ij}</math> with <math>i = j</math>. For example, the [[identity matrix]] can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:
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| :<math>\begin{pmatrix}
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| 1 & 0 & 0 \\
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| 0 & 1 & 0 \\
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| 0 & 0 & 1
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| \end{pmatrix}</math>
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| The top-right to bottom-left diagonal is sometimes described as the ''minor'' diagonal or ''antidiagonal''. The ''off-diagonal'' entries are those not on the main diagonal. A ''[[diagonal matrix]]'' is one whose off-diagonal entries are all zero.
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| A ''superdiagonal'' entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those <math>A_{ij}</math> with <math>j=i</math>, the superdiagonal entries are those with <math>j = i+1</math>. For example, the non-zero entries of the following matrix all lie in the superdiagonal:
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| :<math>\begin{pmatrix} | |
| 0 & 2 & 0 \\
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| 0 & 0 & 3 \\
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| 0 & 0 & 0
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| \end{pmatrix}</math>
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| Likewise, a ''subdiagonal'' entry is one that is directly below and to the left of the main diagonal, that is, an entry <math>A_{ij}</math> with <math>j = i - 1</math>. General matrix diagonals can be specified by an index <math>k</math> measured relative to the main diagonal: the main diagonal has <math>k = 0</math>; the superdiagonal has <math>k = 1</math>; the subdiagonal has <math>k = -1</math>; and in general, the <math>k</math>-diagonal consists of the entries <math>A_{ij}</math> with <math>j = i+k</math>.
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| ==Geometry==
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| By analogy, the [[subset]] of the [[Cartesian product]] ''X''×''X'' of any set ''X'' with itself, consisting of all pairs (x,x), is called the diagonal, and is the [[Graph of a relation|graph]] of the [[Equality (mathematics)|equality]] [[Relation (mathematics)|relation]] on ''X'' or equivalently the [[Graph of a function|graph]] of the [[identity function]] from ''X'' to ''x''. This plays an important part in geometry; for example, the [[fixed point (mathematics)|fixed point]]s of a [[function (mathematics)|mapping]] ''F'' from ''X'' to itself may be obtained by intersecting the graph of ''F'' with the diagonal.
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| In geometric studies, the idea of intersecting the diagonal ''with itself'' is common, not directly, but by perturbing it within an [[equivalence class]]. This is related at a deep level with the [[Euler characteristic]] and the zeros of [[vector field]]s. For example, the [[circle]] ''S''<sup>1</sup> has [[Betti number]]s 1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-[[torus]] ''S''<sup>1</sup>xS<sup>1</sup> and observe that it can move ''off itself'' by the small motion (θ, θ) to (θ, θ + ε). In general, the intersection number of the graph of a function with the diagonal may be computed using homology via the [[Lefschetz fixed point theorem]]; the self-intersection of the diagonal is the special case of the identity function.
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| ==See also==
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| * [[Jordan normal form]]
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| * [[Main diagonal]]
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| * [[Diagonal functor]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| {{Wiktionarypar|diagonal}}
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| *[http://www.mathopenref.com/polygondiagonal.html Diagonals of a polygon] with interactive animation
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| *[http://mathworld.wolfram.com/PolygonDiagonal.html Polygon diagonal] from [[MathWorld]].
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| *[http://mathworld.wolfram.com/Diagonal.html Diagonal] of a matrix from [[MathWorld]].
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| [[Category:Elementary geometry]]
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I am 22 years old and my name is Arturo Colleano. I life in Rotterdam (Netherlands).
Feel free to surf to my web blog :: Fifa 15 Coin Generator (ahmhzh.com)