# Digon

Regular digon | |
---|---|

Type | Regular polygon |

Edges and vertices | 2 |

Schläfli symbol | {2} |

Coxeter diagram | |

Symmetry group |
D_{2}, [2], (*2•) |

Dual polygon | Self-dual |

In geometry, a **digon** is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because the two sides would either coincide or one or both would have to be curved.

A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol {2}. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a lune.

The digon is the simplest abstract polytope of rank 2.

A truncated *digon*, t{2} is a square, {4}. An alternated digon, h{2} is a monogon, {1}.

## In Euclidean geometry

A 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.^{[1]}

Some definitions of a polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case.^{[2]}

## In elementary polyhedra

A digon as a face of a polyhedron 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 on the sphere.^{[3]}

A spherical polyhedron constructed from such digons is called a hosohedron.

- A lune on the sphere.
- Six digon faces on a regular hexagonal hosohedron.

## 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 groups may be obtained as rotation symmetries of polygons: the rotational symmetries of the digon provide the group C_{2}.

## See also

## References

### Citations

- ↑ Eric T. Eekhoff; Constructibility of Regular Polygons, Iowa State University. (retrieved 20 December 2015)
- ↑ Coxeter (1973), Chapter 1,
*Polygons and Polyhedra*, p.4 - ↑ Coxeter (1973), Chapter 1,
*Polygons and Polyhedra*, pages 4 and 12.

### Bibliography

- Herbert Busemann, The geometry of geodesics. New York, Academic Press, 1955
- Coxeter,
*Regular Polytopes*(third edition), Dover Publications Inc, 1973 ISBN 0-486-61480-8 - Weisstein, Eric W. "Digon".
*MathWorld*.

- A.B. Ivanov (2001), "Digon", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4

## External links

Look up in Wiktionary, the free dictionary.digon |

- Media related to Digons at Wikimedia Commons