page_title: Spherical lune
section_title: n-sphere lunes
hierarchical_section_title: Spherical lune / n-sphere lunes
caption_attribution_description: Hypersphere parallel meridian and hypermeridian
context_page_description: In spherical geometry, a spherical lune is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an example of a digon, {2}θ, with dihedral angle θ. The word "lune" derives from luna, the Latin word for Moon.
context_section_description: Lunes can be defined on higher dimensional spheres as well.
  In 4-dimensions a 3-sphere is a generalized sphere. It can contain regular digon lunes as {2}θ,φ, where θ and φ are two dihedral angles.
  For example, a regular hosotope {2,p,q} has digon faces, {2}2π/p,2π/q, where its vertex figure is a spherical platonic solid, {p,q}. Each vertex of {p,q} defines an edge in the hosotope and adjacent pairs of those edges define lune faces. Or more specifically, the regular hosotope {2,4,3}, has 2 vertices, 8 180° arc edges in a cube, {4,3}, vertex figure between the two vertices, 12 lune faces, {2}π/4,π/3, between pairs of adjacent edges, and 6 hosohedral cells, {2,p}π/3.
caption: Stereographic projection of the 3-sphere's parallels (red), meridians (blue) and hypermeridians (green). Lunes exist between pairs of blue meridian arcs.
exif: {}