Diagonal

For other uses, seeDiagonal (disambiguation).

The diagonals of acubewith side length 1. AC' (shown in blue) is aspace diagonalwith length${\sqrt {3}}$,while AC (shown in red) is aface diagonaland has length${\sqrt {2}}$.

Ingeometry,adiagonalis aline segmentjoining twoverticesof apolygonorpolyhedron,when those vertices are not on the sameedge.Informally, any sloping line is called diagonal. The worddiagonalderives from theancient Greekδιαγώνιοςdiagonios,"from corner to corner" (from διά-dia-,"through", "across" and γωνίαgonia,"corner", related togony"knee" ); it was used by bothStraboandEuclidto refer to a line connecting two vertices of arhombusorcuboid,and later adopted into Latin asdiagonus( "slanting line" ).

Polygons

As applied to apolygon,a diagonal is aline segmentjoining any two non-consecutive vertices. Therefore, aquadrilateralhas 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),convexorconcave,has${\tfrac {n(n-3)}{2}}$totaldiagonals, 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 has$\lfloor {\frac {n-2}{2}}\rfloor$distinctdiagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.

Sides Diagonals 3 0 4 2 5 5 6 9 7 14 8 20 9 27 10 35

Sides Diagonals 11 44 12 54 13 65 14 77 15 90 16 104 17 119 18 135

Sides Diagonals 19 152 20 170 21 189 22 209 23 230 24 252 25 275 26 299

Sides Diagonals 27 324 28 350 29 377 30 405 31 434 32 464 33 495 34 527

Sides Diagonals 35 560 36 594 37 629 38 665 39 702 40 740 41 779 42 819

Regions formed by diagonals

In aconvex polygon,if no three diagonals areconcurrentat a single point in the interior, the number of regions that the diagonals divide the interior into is given by

${\binom {n}{4}}+{\binom {n-1}{2}}={\frac {(n-1)(n-2)(n^{2}-3n+12)}{24}}.$

Forn-gons withn=3, 4,... the number of regions is

1, 4, 11, 25, 50, 91, 154, 246...

This isOEISsequence A006522.

Intersections of diagonals

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$\textstyle {\binom {n}{4}}$.This holds, for example, for anyregular polygonwith 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 thenvertices four at a time.

Regular polygons

See also:Quadrilateral § Diagonals,Hexagon § Convex equilateral hexagon,andHeptagon § Diagonals and heptagonal triangle

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 thexthshortest distinct diagonal is:

$\sin({\frac {\pi (x+1)}{n}})\csc({\frac {\pi }{n}})*a$

This formula shows that as the number of sides approaches infinity, thexthshortest diagonal approaches the length(x+1)a.Additionally, the formula for the shortest diagonal simplifies in the case of x = 1:

$\sin({\frac {2\pi }{n}})\csc({\frac {\pi }{n}})*a=2\cos({\frac {\pi }{n}})*a$

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:

Asquarehas two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is${\sqrt {2}}\approx 1.414.$

Aregular pentagonhas five diagonals all of the same length. The ratio of a diagonal to a side is thegolden ratio,${\frac {1+{\sqrt {5}}}{2}}\approx 1.618.$

A regularhexagonhas 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${\sqrt {3}}$.

A regularheptagonhas 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.

Polyhedrons

See also:Face diagonalandSpace diagonal

Apolyhedron(asolid objectinthree-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).

Higher dimensions

N-Cube

The lengths of an n-dimensionalhypercube's diagonals can be calculated bymathematical induction.The longest diagonal of an n-cube is${\sqrt {n}}$.Additionally, there are$2^{n-1}{\binom {n}{x+1}}$of thexthshortest diagonal. As an example, a 5-cube would have the diagonals:

Diagonal length Number of diagonals ${\sqrt {2}}$ 160 ${\sqrt {3}}$ 160 2 80 ${\sqrt {5}}$ 16

Its total number of diagonals is 416. In general, an n-cube has a total of$2^{n-1}(2^{n}-n-1)$diagonals. This follows from the more general form of${\frac {v(v-1)}{2}}-e$which describes the total number of face and space diagonals in convex polytopes.Here, v represents the number of vertices and e represents the number of edges.

Geometry

By analogy, thesubsetof theCartesian productX×Xof any setXwith itself, consisting of all pairs (x,x), is called the diagonal, and is thegraphof theequalityrelationonXor equivalently thegraphof theidentity functionfromXtoX.This plays an important part in geometry; for example, thefixed pointsof amappingFfromXto itself may be obtained by intersecting the graph ofFwith the diagonal.

In geometric studies, the idea of intersecting the diagonalwith itselfis common, not directly, but by perturbing it within anequivalence class.This is related at a deep level with theEuler characteristicand the zeros ofvector fields.For example, thecircleS¹hasBetti numbers1, 1, 0, 0, 0, and therefore Euler characteristic 0. A geometric way of expressing this is to look at the diagonal on the two-torusS¹xS¹and observe that it can moveoff itselfby 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.

External links

• Diagonals of a polygonwith interactive animation
• Polygon diagonalfromMathWorld.