Diagonal

Informally, any sloping line is called diagonal.

The word diagonal derives from the ancient Greek διαγώνιος diagonios,[1] "from corner to corner" (from διά- dia-, "through", "across" and γωνία gonia, "corner", related to gony "knee"); it was used by both Strabo[2] and Euclid[3] to refer to a line connecting two vertices of a rhombus or cuboid,[4] and later adopted into Latin as diagonus ("slanting line").

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.

distinct diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.

In a convex polygon, if no three diagonals are concurrent at a single point in the interior, the number of regions that the diagonals divide the interior into is given by[5] For n-gons with n=3, 4, ... the number of regions is This is OEIS sequence A006522.

[7][8] This holds, for example, for any regular polygon with an odd number of sides.

Although the number of distinct diagonals in a polygon increases as its number of sides increases, the length of any diagonal can be calculated.

In a regular n-gon with side length a, the length of the xth shortest distinct diagonal is: This formula shows that as the number of sides approaches infinity, the xth shortest diagonal approaches the length ⁠

Additionally, the formula for the shortest diagonal simplifies in the case of x = 1: If the number of sides is even, the longest diagonal will be equivalent to the diameter of the polygon's circumcircle because the long diagonals all intersect each other at the polygon's center.

A regular hexagon has nine diagonals: the six shorter ones are equal to each other in length; the three longer ones are equal to each other in length and intersect each other at the center of the hexagon.

The lengths of an n-dimensional hypercube's diagonals can be calculated by mathematical induction.

which describes the total number of face and space diagonals in convex polytopes.

By analogy, the subset of the Cartesian product X×X of any set X with itself, consisting of all pairs ⁠

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.

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.