In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. [1] It is an example of a digon, {2}θ, with dihedral angle θ. [2] The word "lune" derives from luna , the Latin word for Moon.
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.
Each hosohedra can be divided by an equatorial bisector into two equal spherical triangles.
| n | 2 | 3 | 4 | 5 | 6 | ... |
|---|---|---|---|---|---|---|
| Hosohedra |  |  |  |  |  | ... |
| Bipyramidal tiling |  |  |  |  |  | ... |
The visibly lighted portion of the Moon visible from the Earth is a spherical lune. The first of the two intersecting great circles is the terminator between the sunlit half of the Moon and the dark half. The second great circle is a terrestrial terminator that separates the half visible from the Earth from the unseen half. The spherical lune is a lighted crescent shape seen from Earth.
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.
In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays. Angles are also formed by the intersection of two planes; these are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.
In geometry, a polygon is a plane figure made up of line segments connected to form a closed polygonal chain.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass.
In geometry, a solid angle is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the apex of the solid angle, and the object is said to subtend its solid angle at that point.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance.
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four sides of equal length and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.
In geometry, the (angular) defect means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess.
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
A circular sector, also known as circle sector or disk sector or simply a sector, is the portion of a disk enclosed by two radii and an arc, with the smaller area being known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle, the radius of the circle, and is the arc length of the minor sector.
In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.
In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in Schwarz (1873).
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle. Each triangle group is the symmetry group of a tiling of the Euclidean plane, the sphere, or the hyperbolic plane by congruent triangles called Möbius triangles, each one a fundamental domain for the action.
In spherical geometry, an n-gonalhosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
In geometry, a bigon, digon, or a 2-gon, is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space. It may also be viewed as a representation of a graph with two vertices, see "Generalized polygon".
The goat grazing problem is either of two related problems in recreational mathematics involving a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing an exterior of a circular area. For the exterior problem, the constraint that the rope can not enter the circular area dictates that the grazing area forms an involute. If the goat were instead tethered to a post on the edge of a circular path of pavement that did not obstruct the goat, the interior and exterior problem would be complements of a simple circular area.
In geometry, many uniform tilings on sphere, euclidean plane, and hyperbolic plane can be made by Wythoff construction within a fundamental triangle,, defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles. Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror. Up to 3 face types exist centered on the fundamental triangle corners. Right triangle domains can have as few as 1 face type, making regular forms, while general triangles have at least 2 triangle types, leading at best to a quasiregular tiling.