Diagonal

In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word diagonal derives from the ancient Greek διαγώνιος diagonios, "from corner to corner" (from διά- dia-, "through", "across" and γωνία gonia, "corner", related to gony "knee"); it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as diagonus ("slanting line").

The diagonals of a cube with side length 1. AC' (shown in blue) is a space diagonal with length ${\displaystyle {\sqrt {3}}}$, while AC (shown in red) is a face diagonal and has length ${\displaystyle {\sqrt {2}}}$.

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 ${\displaystyle {\tfrac {n(n-3)}{2}}}$ total diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or n − 3 diagonals, and each diagonal is shared by two vertices.

In general, a regular n-sided polygon has ${\displaystyle \lfloor {\frac {n-2}{2}}\rfloor }$ distinct diagonals 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 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

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

For n-gons with n=3, 4, ... the number of regions is

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

This is OEIS sequence 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 ${\displaystyle \textstyle {\binom {n}{4}}}$. This holds, for example, for any regular 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 the n vertices four at a time.

Regular polygons

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:

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

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

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

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:

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

A regular pentagon has five diagonals all of the same length. The ratio of a diagonal to a side is the golden ratio, ${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.618.}$

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 ratio of a long diagonal to a side is 2, and the ratio of a short diagonal to a side is ${\displaystyle {\sqrt {3}}}$.

A regular heptagon 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.

Polyhedrons

A polyhedron (a solid object in three-dimensional space, bounded by two-dimensional faces) 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-dimensional hypercube's diagonals can be calculated by mathematical induction. The longest diagonal of an n-cube is ${\displaystyle {\sqrt {n}}}$. Additionally, there are ${\displaystyle 2^{n-1}{\binom {n}{x+1}}}$ of the xth shortest diagonal. As an example, a 5-cube would have the diagonals:

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

Its total number of diagonals is 416. In general, an n-cube has a total of ${\displaystyle 2^{n-1}(2^{n}-n-1)}$ diagonals. This follows from the more general form of ${\displaystyle {\frac {v(v-1)}{2}}-e}$ 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, 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 equality relation on X or equivalently the graph of the identity function from X to X. 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.

Notes

1. Harper, Douglas R. (2018). "diagonal (adj.)". Online Etymology Dictionary.
2. Strabo, Geography 2.1.36–37
3. Euclid, Elements book 11, proposition 28
4. Euclid, Elements book 11, proposition 38
5. Honsberger (1973). "A Problem in Combinatorics". Mathematical Gems. Mathematical Association of America. Ch. 9, pp. 99–107. ISBN 0-88385-301-9.
   Freeman, J. W. (1976). "The Number of Regions Determined by a Convex Polygon". Mathematics Magazine. 49 (1): 23–25. doi:10.2307/2689875. JSTOR 2689875.
6. Sloane, N. J. A. (ed.). "Sequence A006522". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
7. 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
8. 3Blue1Brown (2015-05-23). Circle Division Solution (old version). Retrieved 2024-09-01 – via YouTube.{{cite AV media}}: CS1 maint: numeric names: authors list (link)
9. "Counting Diagonals of a Polyhedron – the Math Doctors".

External links

Look up diagonal in Wiktionary, the free dictionary.

• Diagonals of a polygon with interactive animation
• Polygon diagonal from MathWorld.