DIGON

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}.

Contents

In spherical tilings

In polyhedra

See also

References

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.

One antipodal 'digon' on the sphere.

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.

4.4.4

3.8.8

3.4.3.4

4.6.6

3.3.3.3

3.4.4.4

4.6.8

See also

★ Dihedron - a degenerate polyhedron with 2 faces.

★ Hosohedron - a degenerate polyhedron with 2 vertices.

References

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