In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a diagonal is a
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
joining two
vertices of a
polygon or
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
, when those vertices are not on the same
edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the
ancient Greek
Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), 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
Strabo''Strabo'' (meaning "squinty", as in strabismus) was a term employed by the Romans for anyone whose eyes were distorted or deformed. The father of Pompey was called "Gnaeus Pompeius Strabo, Pompeius Strabo". A native of Sicily so clear-si ...
and
Euclid
Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
to refer to a line connecting two vertices of a
rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhom ...
or
cuboid, and later adopted into Latin as ''diagonus'' ("slanting line").
Polygons
As applied to a
polygon, a diagonal is a
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
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
''total'' diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or diagonals, and each diagonal is shared by two vertices.
In general, a regular ''n''-sided polygon has
''distinct'' diagonals in length, which follows the pattern 1,1,2,2,3,3... starting from a square.
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
:
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
.
This holds, for example, for any
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
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:
:
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.
Special cases include:
A
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
has two diagonals of equal length, which intersect at the center of the square. The ratio of a diagonal to a side is
A
regular pentagon has five diagonals all of the same length. The ratio of a diagonal to a side is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
,
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
.
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
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
(a
solid object in
three-dimensional space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values ('' coordinates'') are required to determine the position of a point. Most commonly, it is the three- ...
, 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
Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a ...
. The longest diagonal of an n-cube is
. Additionally, there are
of the ''xth'' shortest diagonal. As an example, a 5-cube would have the diagonals:
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.
Here, v represents the number of vertices and e represents the number of edges.
Geometry
By analogy, the
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
of the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
''X''×''X'' of any set ''X'' with itself, consisting of all pairs , 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
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
''S''
1 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
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
''S''
1×S
1 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
External links
{{Wiktionary, diagonal
Diagonals of a polygonwith interactive animation
from
MathWorld.
Elementary geometry