The diagonals of acube with side length 1. AC' (shown in blue) is aspace diagonal with length, while AC (shown in red) is aface diagonal and has length.
Ingeometry, adiagonal is aline segment joining twovertices of apolygon orpolyhedron, when those vertices are not on the sameedge. Informally, any sloping line is called diagonal. The worddiagonal derives from theancient Greek διαγώνιοςdiagonios,[1] "from corner to corner" (from διά-dia-, "through", "across" and γωνίαgonia, "corner", related togony "knee"); it was used by bothStrabo[2] andEuclid[3] to refer to a line connecting two vertices of arhombus orcuboid,[4] and later adopted into Latin asdiagonus ("slanting line").
As applied to apolygon, a diagonal is aline segment joining any two non-consecutive vertices. Therefore, aquadrilateral has two diagonals, joining opposite pairs of vertices. For anyconvex polygon, all the diagonals are inside the polygon, but forre-entrant polygons, some diagonals are outside of the polygon.
Anyn-sided polygon (n ≥ 3),convex orconcave, hastotal diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, orn − 3 diagonals, and each diagonal is shared by two vertices.
In general, a regularn-sided polygon hasdistinct diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.
In aconvex polygon, if no three diagonals areconcurrent at a single point in the interior, the number of regions that the diagonals divide the interior into is given by[5]
Forn-gons withn=3, 4, ... the number of regions is
If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by.[7][8] This holds, for example, for anyregular polygon with an odd number of sides. The formula follows from the fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of then vertices four at a time.
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 lengtha, the length of thexth shortest distinct diagonal is:
This formula shows that as the number of sides approaches infinity, thexth 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.
Special cases include:
Asquare has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is
Aregular pentagon has five diagonals all of the same length. The ratio of a diagonal to a side is thegolden ratio,
A regularhexagon 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 ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is.
A regularheptagon has 14 diagonals. The seven shorter ones equal each other, and the seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a short and a long diagonal.
Apolyhedron (asolid object inthree-dimensional space, bounded bytwo-dimensionalfaces) may have two different types of diagonals: face diagonals on the various faces, connecting non-adjacent vertices on the same face; and space diagonals, entirely in the interior of the polyhedron (except for the endpoints on the vertices).
The lengths of an n-dimensionalhypercube's diagonals can be calculated bymathematical induction. The longest diagonal of an n-cube is. Additionally, there are of thexth shortest diagonal. As an example, a 5-cube would have the diagonals:
Diagonal length
Number of diagonals
√2
160
√3
160
2
80
√5
16
Its total number of diagonals is 416. In general, an n-cube has a total of diagonals. This follows from the more general form of which describes the total number of face and space diagonals in convex polytopes.[9] Here, v represents the number of vertices and e represents the number of edges.
By analogy, thesubset of theCartesian productX×X of any setX with itself, consisting of all pairs, is called the diagonal, and is thegraph of theequalityrelation onX or equivalently thegraph of theidentity function fromX toX. This plays an important part in geometry; for example, thefixed points of amappingF fromX to itself may be obtained by intersecting the graph ofF with the diagonal.
In geometric studies, the idea of intersecting the diagonalwith itself is common, not directly, but by perturbing it within anequivalence class. This is related at a deep level with theEuler characteristic and the zeros ofvector fields. For example, thecircleS1 hasBetti 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-torusS1×S1 and observe that it can moveoff 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 theLefschetz fixed-point theorem; the self-intersection of the diagonal is the special case of the identity function.
^Poonen, Bjorn; Rubinstein, Michael. "The number of intersection points made by the diagonals of a regular polygon".SIAM J. Discrete Math. 11 (1998), no. 1, 135–156;link to a version on Poonen's website
