1 at an angle; especially connecting two nonadjacent corners of a plane figure or any two corners of a solid that are not in the same face; "a diagonal line across the page"
1 (geometry) a straight line connecting any two vertices of a polygon that are not adjacent
2 a line or cut across a fabric that is not at right angles to a side of the fabric [syn: bias]
3 an oblique line of squares of the same color on a checkerboard; "the bishop moves on the diagonals"
4 (mathematics) a set of entries in a square matrix running diagonally either from the upper left to lower right entry or running from the upper right to lower left entry
5 a punctuation mark (/) used to separate related items of information [syn: solidus, slash, virgule, stroke, separatrix]
EtymologyFrom diagonalis, from διαγώνιος, from διά + γωνία.
geometry: joining two nonadjacent vertices
having a slanted or oblique direction
of, or relating the legs of quadruped
- Swedish: diagonal
- (punctuation mark): backslash
line or cut across a fabric
- Dutch: diagonaal
- Swedish: snedstreck
geometry: diagonal line or plane
A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. The word "diagonal" derives from the Greek διαγώνιος (diagonios), used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and is formed from dia- ("through", "across") and gonia ("angle", related to gony "knee."), later adopted into Latin as diagonus ("slanting line").
In mathematics, in addition to its geometric meaning, a diagonal is also used in matrices to refer to a set of entries along a diagonal line.
Non mathematical uses
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 system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch.
PolygonsAs 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 polygons, some diagonals are outside of the polygon.
Any n-sided polygon (n ≥ 3), convex or concave, has
MatricesIn the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left to bottom-right corners. For example, the identity matrix can be defined as having entries of 1 on the main diagonal, and 0s elsewhere. The top-right to bottom-left diagonal is sometimes described as the minor diagonal or antidiagonal. A superdiagonal entry is one that is directly above and to the right of the main diagonal. If otherwise unqualified, it refers to 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.
GeometryBy 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 the identity relation. This 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.
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 fields. For example, the circle S1 has Betti numbers 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 S1xS1 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.
diagonal in Arabic: قطر (رياضيات)
diagonal in Asturian: Diagonal
diagonal in Czech: Úhlopříčka
diagonal in Danish: Diagonal
diagonal in German: Diagonale (Geometrie)
diagonal in Spanish: Diagonal
diagonal in Esperanto: Diagonalo
diagonal in French: Diagonale
diagonal in Korean: 대각선
diagonal in Italian: Diagonale
diagonal in Hebrew: אלכסון
diagonal in Swahili (macrolanguage): Ulalo
diagonal in Hungarian: Átló
diagonal in Dutch: Diagonaal
diagonal in Japanese: 対角線
diagonal in Polish: Przekątna
diagonal in Portuguese: Diagonais de um polígono
diagonal in Quechua: Ch'iqan
diagonal in Russian: Диагональ
diagonal in Simple English: Diagonal
diagonal in Slovak: Uhlopriečka
diagonal in Slovenian: Diagonala
diagonal in Thai: เส้นทแยงมุม
diagonal in Ukrainian: Діагональ
diagonal in Chinese: 對角線
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