Diagonal

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A diagonal is: an upward or downward sloping line, or a line across a shape joining two nonadjacent corners. Originally from Greek: διαγωνιος (diagonios) used by both Strabo^[1] and Euclid^[2] for a line across a rhombus but also from corner to corner through the middle of a cuboid.^[3] Formed from dia- ("through", "across") and gonia ("angle", related to gony "knee.") later taken into latin as: diagonus ("slanting line").

In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology.

1 Non mathematical uses
2 Polygons
3 Matrices
4 Geometry
5 See also
6 External links
7 References

[edit] Non mathematical uses

In engineering a diagonal brace is a beam used to cross brace a rectangular structure such as scaffolding to withstand sheer forces pusing into a rhombus; although called diagonal, due to practical considerations in practice diagonal braces are often not connected to the corners of the rectangle. In draughts all pieces move diagonally whilst in chess the Queen, Pawn, King and Bishop are said to move diagonally across the board.[1]. 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. The Turner Diagonal is a short freeway in Kansas City, named because it runs diagonally. In association football In association football, the diagonal system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch. It is also used to refer to the opposite pair of legs of animals. Many flags with a cross from corner to corner such as the Saint Andrew's Flag are referred to as diagonal. It refers to various angled objects such as telescope mirrors and it is used for any sloping pattern such as the "Blue tie with diagonal orange stripes" of The King's School, Worcester. Diagonal is also a city in Ringgold County, Iowa, United States of 312 people in 2000.

In literature Harry Potter in the Sorceror's stone, mistakenly pronounces Diagon Alley as "Diagonally" with the hilarious result of being flung at an angle across the room rather than going to his intended destination.

[edit] Polygons

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 a convex polygon the diagonals run inside the polygon. This is not so for re-entrant polygons. In fact a polygon is convex if and only if the diagonals are internal.

When n is the number of vertices in a polygon and d is the number of possible different diagonals, each vertex has possible diagonals to all other vertices save for itself and the two adjacent vertices, or n-3 diagonals; this multiplied by the number of vertices is

(n − 3) × n,

which counts each diagonal twice (once for each vertex) — therefore,

$d= \frac{n^2-3n}{2}.\,$

[edit] Matrices

In the case of a square matrix, the main or principal diagonal is the diagonal line of entries running north-west to south-east. For example the identity matrix can be described as having entries 1 on main diagonal, and 0 elsewhere. The north-east to south-west diagonal is sometimes described as the minor diagonal or antidiagonal. A superdiagonal entry is one that is above, and to the right of, the main diagonal. If otherwise unqualified, it means the one adjacent to the main diagonal. Likewise, a subdiagonal entry is one that is directly below, and to the left of, the main diagonal. A diagonal matrix is one whose off-diagonal entries are all zero.

[edit] Geometry

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. It is the graph of the identity relation. It plays an important part in geometry: for example the fixed points of a mapping F from X to itself may be obtained by intersecting the graph of F with the diagonal.

Quite a major role is played in geometric studies by the idea of intersecting the diagonal with itself: not directly, but by perturbing it within an equivalence class. This is related at quite a deep level with the Euler characteristic and the zeroes of vector fields. For example the circle S¹ has Betti numbers 1, 1, 0, 0, 0, ... and so Euler characteristic 0. A geometric way of saying that is to look at the diagonal on the two-torus S¹xS¹; and to 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.