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File:Complete graph K2.svg
File:Digon.svg

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 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 lune.

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One antipodal digon on the sphere.
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Six antipodal digon faces on a hexagonal hosohedron tiling on the sphere.

In polyhedra Edit

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.

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4.4.4
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3.8.8
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3.4.3.4
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4.6.6
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3.3.3.3
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3.4.4.4
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4.6.8

See also Edit

ReferencesEdit

External linksEdit

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