Digon

Digon
Digon
Complete graph K2.svg
A degenerate digon with two coinciding edges sharing the same vertices
Edges and vertices 2
Schläfli symbol {2}
Coxeter–Dynkin diagrams CDel node 1.png
CDel node 1.pngCDel 2.pngCDel node.png
Area depends on geometric surface
Internal angle (degrees) depends on geometric surface
On a circle, a nondegenerate antipodal digon is a tessellation composed of two vertices and two 180 degree arcs.

In geometry, a digon is a polygon with two sides (edges) and two vertices. It is degenerate in a Euclidean space, but may be non-degenerate in a spherical space.

A digon must be regular because its two edges are the same length. It has Schläfli symbol {2}.

Some authorities do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case, but most formulae on general polygons do work on the digon. For example, the angle sum of an n-gon, (n − 2)π, would become 0 when n = 2, which is correct for a Euclidean digon.

Contents

In spherical tilings

In Euclidean geometry a digon is always degenerate. However, in spherical geometry a nondegenerate digon (with a nonzero interior area) can exist if the vertices are antipodal. The internal angle of the spherical digon vertex can be any angle between 0 and 180 degrees. Such a spherical polygon can also be called a lune.

In polyhedra

A digon is considered a degenerate face of a polyhedron because it has no geometric area and edges are overlapping. But sometimes it can have a useful topological existence in transforming polyhedra.

Any polyhedron can be topologically modified by replacing an edge with a digon. Such an operation adds one edge and one face to the polyhedron, although the result is geometrically identical. This transformation has no effect on the Euler characteristic (χ=V-E+F).

A digon face can also be created by geometrically collapsing a quadrilateral face by moving pairs of vertices to coincide in space. This digon can then be replaced by a single edge. It loses one face, two vertices, and three edges, again leaving the Euler characteristic unchanged.

Classes of polyhedra can be derived as degenerate forms of a primary polyhedron, with faces sometimes being degenerated into coinciding vertices. For example, this class of 7 uniform polyhedron with octahedral symmetry exist as degenerate forms of the truncated cuboctahedron (4.6.8). This principle is used in the Wythoff construction.

Uniform polyhedron-43-t0.png
4.4.4
Uniform polyhedron-43-t01.png
3.8.8
Uniform polyhedron-43-t1.png
3.4.3.4
Uniform polyhedron-43-t12.png
4.6.6
Uniform polyhedron-43-t2.png
3.3.3.3
Uniform polyhedron-43-t02.png
3.4.4.4
Uniform polyhedron-43-t012.png
4.6.8

See also

References

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • digon — digon …   Dictionnaire des rimes

  • digon — [ digɔ̃ ] n. m. • 1678; de diguer « faire une digue » 1 ♦ Mar. Hampe portant une flamme, un pavillon. 2 ♦ (1769) Fer barbelé ajusté à une perche, servant à harponner les poissons plats à basse mer. ⇒ harpon; angon. digon [digɔ̃] n. m. ÉTYM. 1678; …   Encyclopédie Universelle

  • digon — noun A polygon having two edges and two vertices. Syn: bigon …   Wiktionary

  • digon — (di gon) s. m. 1°   Terme de marine. Pièce de bois posée entre la gorgère et l étrave.    Bâton qui porte une flamme ou un pavillon, et qu on attache au bout d une vergue. 2°   Terme de pêche. Fer barbelé que l on ajuste à une perche pour prendre …   Dictionnaire de la Langue Française d'Émile Littré

  • DIGON — DIGital ONline local network (Berlin) …   Acronyms

  • DIGON — DIGital ONline local network (Berlin) …   Acronyms von A bis Z

  • Hotel Digon — (Ортизеи,Италия) Категория отеля: 3 звездочный отель Адрес: Via Digon 22, 39046 Ортизеи, Италия …   Каталог отелей

  • Matúš Digoň — Personal information Full name Matúš Digoň Date of birth 18 November 1988 (1988 …   Wikipedia

  • Polygon — For other uses, see Polygon (disambiguation). Some polygons of different kinds In geometry a polygon (   …   Wikipedia

  • Liste der Weltmeister im Zweier-Kunstradfahren der Männer — Weltmeister im Zweier Kunstradfahren der Männer Von 1958 ab wurde das Europakriterium ausgefahren, das 1970 durch die Europameisterschaften ersetzt wurde. Ab 1986 gibt es schließlich auch Weltmeisterschaften im Zweier Kunstradfahren der Männer.… …   Deutsch Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”