Linearly ordered group: Difference between revisions

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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])
 
[[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>.]]
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").
 
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.
 
== Non-mathematical uses ==
[[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]]
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.
 
[[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.
 
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.
 
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.
 
[[File:Display size measurements.png|thumb|right|The diagonal is a common measurement of [[Two-dimensional display size|display size]].]]
 
== Polygons ==
{{See also|Quadrilateral#Diagonals}}
 
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.
 
Any ''n''-sided polygon (''n'' ≥ 3), [[Convex polygon|convex]] or [[Concave polygon|concave]], has
:<math>\frac{n^2-3n}{2}\, </math>
or
:<math>\frac{n(n-3)}{2}\, </math>
diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or ''n''&nbsp;−&nbsp;3 diagonals.
 
{|rules="none" border="0" cellspacing="4" cellpadding="0" style="background:transparent;text-align:right"
|-valign="top"
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 3 || 0
|-
| 4 || 2
|-
| 5 || 5
|-
| 6 || 9
|-
| 7 || 14
|-
| 8 || 20
|-
| 9 || 27
|-
| 10 || 35
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 11 || 44
|-
| 12 || 54
|-
| 13 || 65
|-
| 14 || 77
|-
| 15 || 90
|-
| 16 || 104
|-
| 17 || 119
|-
| 18 || 135
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 19 || 152
|-
| 20 || 170
|-
| 21 || 189
|-
| 22 || 209
|-
| 23 || 230
|-
| 24 || 252
|-
| 25 || 275
|-
| 26 || 299
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 27 || 324
|-
| 28 || 350
|-
| 29 || 377
|-
| 30 || 405
|-
| 31 || 434
|-
| 32 || 464
|-
| 33 || 495
|-
| 34 || 527
|}
|
{|class="wikitable"
|-
! Sides !! Diagonals
|-
| 35 || 560
|-
| 36 || 594
|-
| 37 || 629
|-
| 38 || 665
|-
| 39 || 702
|-
| 40 || 740
|-
| 41 || 779
|-
| 42 || 819
|}
|}
 
== Matrices ==
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:
:<math>\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}</math>
 
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.
 
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:
:<math>\begin{pmatrix}
0 & 2 & 0 \\
0 & 0 & 3 \\
0 & 0 & 0
\end{pmatrix}</math>
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>.
 
==Geometry==
By analogy, the [[subset]] of the [[Cartesian product]] ''X''&times;''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.
 
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.
 
==See also==
* [[Jordan normal form]]
* [[Main diagonal]]
* [[Diagonal functor]]
 
==References==
{{reflist}}
 
==External links==
{{Wiktionarypar|diagonal}}
*[http://www.mathopenref.com/polygondiagonal.html Diagonals of a polygon] with interactive animation
*[http://mathworld.wolfram.com/PolygonDiagonal.html Polygon diagonal] from [[MathWorld]].
*[http://mathworld.wolfram.com/Diagonal.html Diagonal] of a matrix from [[MathWorld]].
 
[[Category:Elementary geometry]]

Latest revision as of 04:00, 20 December 2014

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)