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A degenerate digon with two coinciding edges sharing the same vertices

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 degenerate polygon with two sides (edges) and two vertices.

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

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.

Regular digon in spherical geometry-2.svg
One antipodal digon on the sphere.
Hexagonal hosohedron.png
Six antipodal digon faces on a hexagonal hosohedron tiling on the sphere.

In polyhedra

A digon is considered degenerate face of a polyhedron because it has no geometric area and overlapping edges, but it can sometimes 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 great rhombicuboctahedron (4.6.8). This principle is used in the Wythoff construction.

Uniform polyhedron-43-t0.png
Uniform polyhedron-43-t01.png
Uniform polyhedron-43-t1.png
Uniform polyhedron-43-t12.png
Uniform polyhedron-43-t2.png
Uniform polyhedron-43-t02.png
Uniform polyhedron-43-t012.png

See also

  • Dihedron - a degenerate polyhedron with 2 faces.
  • Hosohedron - a degenerate polyhedron with 2 vertices.
  • Demihypercube


External links


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