Spherical lune
In spherical geometry, a spherical lune is an area on a sphere bounded by two half great circles which meet at antipodal points, and is an example of a digon, {2}θ, with dihedral angle θ. The word "lune" derives from luna, the Latin word for Moon.
A spherical wedge is the volume of space bounded by two planes passing through a sphere center and the surface of the sphere.
Great circles are the largest possible circles (circumferences) of a sphere; each one divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points.
Common examples of great circles are lines of longitude (meridians) on a sphere, which meet at the north and south poles.
A spherical lune has two planes of symmetry. It can be bisected into two lunes of half the angle, or it can be bisected by an equatorial line into two right spherical triangles.
The surface area of a spherical lune is 2θ R2, where R is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles.
When this angle equals 2π radians (360°) — i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere as a spherical monogon — the area formula for the spherical lune gives 4πR2, the surface area of the sphere.
A hosohedron is a tessellation of the sphere by lunes. A n-gonal regular hosohedron, {2,n} has n equal lunes of π/n radians. An n-hosohedron has dihedral symmetry Dnh, [n,2], (*22n) of order 4n. Each lune individually has cyclic symmetry C2v, [2], (*22) of order 4.
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