Digon

From Wikipedia, the free encyclopedia
Jump to: navigation, search
On a circle, a nondegenerate antipodal digon is a tessellation composed of two vertices and two 180-degree arcs.

In geometry, a digon or 2-gon 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 as a pair of 180 degree arcs connecting antipodal points.

In Euclidean space a digon is regular, because its two edges are the same length and its two angles are equal (both being zero degrees). 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.[citation needed]

In spherical tilings[edit]

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 spherical lune.

In polyhedra[edit]

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, by bringing pairs or groups of vertices into coincidence. For example, the following uniform polyhedra with octahedral symmetry exist as degenerate forms of the truncated cuboctahedron (4.6.8).

Polyhedron Cube Truncated cube Truncated octahedron Octahedron cuboctahedron rhombicuboctahedron Truncated cuboctahedron
Image Near uniform polyhedron-43-t0.png Near uniform polyhedron-43-t01.png Near uniform polyhedron-43-t12.png Near uniform polyhedron-43-t2.png Near uniform polyhedron-43-t1.png Near uniform polyhedron-43-t02.png Near uniform polyhedron-43-t012.png
Vertex figure (2.4)3 3.8.2.8 2.6.4.6 (2.3)4 (3.4)2 3.4.4.4 4.6.8

In these pictures, the edges between red faces in the first two polyhedra and yellow faces the third and fourth can be seen as blue degenerate digonal faces {2}. In the cube, the yellow faces degenerate into points, in the octahedron, the red faces degenerate into points, and in the cuboctahedron the blue faces degenerate to points. This principle is used in the Wythoff construction.

See also[edit]

References[edit]

External links[edit]