geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a digon is a

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

with two sides (

edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...

s) and two vertices. Its construction is degenerate in a

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space. A regular digon has both angles equal and both sides equal and is represented by

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...

. It may be constructed on a

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

as a pair of 180 degree arcs connecting

antipodal point In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...

s, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated ''digon'', t is a

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, . An alternated digon, h is a monogon, .

In Euclidean geometry

The digon can have one of two visual representations if placed in Euclidean space. One representation is degenerate, and visually appears as a double-covering of a

line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...

. Appearing when the minimum distance between the two edges is 0, this form arises in several situations. This double-covering form is sometimes used for defining degenerate cases of some other polytopes; for example, a

regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...

can be seen as an

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

formed of such a digon. It can be derived from the alternation of a square (h), as it requires two opposing vertices of said square to be connected. When higher-dimensional polytopes involving squares or other tetragonal figures are alternated, these digons are usually discarded and considered single edges. A second visual representation, infinite in size, is as two parallel lines stretching to (and projectively meeting at; i.e. having vertices at) infinity, arising when the shortest distance between the two edges is greater than zero. This form arises in the representation of some degenerate polytopes, a notable example being the

apeirogonal hosohedron In geometry, an apeirogonal hosohedron or infinite hosohedronConway (2008), p. 263 is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol ...

, the limit of a general spherical

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...

at infinity, composed of an infinite number of digons meeting at two antipodal points at infinity. However, as the vertices of these digons are at infinity and hence are not bound by closed line segments, this tessellation is usually not considered to be an additional regular tessellation of the Euclidean plane, even when its dual order-2 apeirogonal tiling (infinite dihedron) is. Regular star figure 2(2,1).svg, A compound of two "line segment" digons, as the two possible alternations of a square (note the

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

). Apeirogonal hosohedron.svg, The apeirogonal

, containing infinitely narrow digons. Any straight-sided ''digon'' is regular even though it is degenerate, because its two edges are the same length and its two angles are equal (both being zero degrees). As such, the regular digon is a constructible polygon.Eric T. Eekhoff
Constructibility of Regular Polygons
, Iowa State University. (retrieved 20 December 2015) Some definitions of a polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case.

In elementary polyhedra

A digon as a

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

of a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

is degenerate because it is a degenerate polygon. But sometimes it can have a useful topological existence in transforming polyhedra.

As a spherical lune

A spherical lune is a digon whose two vertices are

antipodal points In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...

on the sphere.Coxeter (1973), Chapter 1, ''Polygons and Polyhedra'', pages 4 and 12. A spherical polyhedron constructed from such digons is called a

. Regular digon in spherical geometry-2.svg, A lune on the sphere. Hexagonal Hosohedron.svg, Six digon faces on a regular hexagonal

Theoretical significance

The digon is an important construct in the topological theory of networks such as graphs and polyhedral surfaces. Topological equivalences may be established using a process of reduction to a minimal set of polygons, without affecting the global topological characteristics such as the Euler value. The digon represents a stage in the simplification where it can be simply removed and substituted by a line segment, without affecting the overall characteristics. The

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

s may be obtained as rotation symmetries of polygons: the rotational symmetries of the digon provide the group C₂.

References

Citations

Bibliography

Herbert Busemann Herbert Busemann (12 May 1905 – 3 February 1994) was a German-American mathematician specializing in convex and differential geometry. He is the author of Busemann's theorem in Euclidean geometry and geometric tomography. He was a member ...

, The geometry of geodesics. New York, Academic Press, 1955 * Coxeter, ''Regular Polytopes'' (third edition), Dover Publications Inc, 1973 * *

External links

* {{polygons Polygons by the number of sides 2 (number)