# Dictionary Definition

diagonal adj

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"

### Noun

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]

# User Contributed Dictionary

## English

### Etymology

From diagonalis, from διαγώνιος, from διά + γωνία.### Pronunciation

/daɪˈægənəl/### Adjective

#### Synonyms

#### Derived terms

#### Translations

geometry: joining two nonadjacent vertices

having a slanted or oblique direction

of, or relating the legs of quadruped

- Swedish: diagonal

### Noun

- something arranged diagonally or obliquely
- a line or cut across a fabric that is not at right angles to a side of the fabric
- a punctuation mark used to separate related items of information
- a diagonal line or plane

#### Synonyms

#### Antonyms

- (punctuation mark): backslash

#### Translations

line or cut across a fabric

- Dutch: diagonaal

punctuation mark

- Swedish: snedstreck

geometry: diagonal line or plane

## Swedish

### Adjective

#### Derived terms

### Noun

diagonal#### Derived terms

# Extensive Definition

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.

## 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 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

- \frac\,

## Matrices

In 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.## 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, 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.

## External links

- Diagonals of a polygon with interactive animation
- Polygon diagonal from MathWorld.
- Diagonal of a matrix from MathWorld.
- Diagonal media from Diagonal.

## References

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

# Synonyms, Antonyms and Related Words

across,
air line, athwart,
axis, band, bar, beeline, bend, bendwise, bevel, beveled, bias, biased, biaswise, catercorner, catercornered, chord, cross-hatching, crossways, crosswise, dash, delineation, diameter, direct line, directrix, dotted line,
edge, great-circle course,
hachure, hairline, hatching, kittycorner, line, lineation, normal, oblique, oblique angle, oblique
figure, oblique line, perpendicular, radius, radius vector, rhomboid, right line, score, scratch comma, secant, segment, separatrix, shortcut, side, slant, slanted, slanting, slash, solidus, straight, straight course,
straight line, straight stretch, straightaway, streak, streaking, streamline, striation, strip, stripe, striping, stroke, sublineation, tangent, thwart, transversal, transverse, underline, underlining, underscore, underscoring, vector, virgule