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.
One antipodal digon on the sphere.
Six antipodal digon faces on a hexagonal hosohedron tiling on the sphere.
In polyhedra Edit
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.
See also Edit
- Dihedron - a degenerate polyhedron with 2 faces.
- Hosohedron - a degenerate polyhedron with 2 vertices.
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