The Spherics (Greek: τὰ σφαιρικά, tà sphairiká) is a three-volume treatise on spherical geometry written by the Hellenistic mathematician Theodosius of Bithynia in the 2nd or 1st century BC.
Book I and the first half of Book II establish basic geometric constructions needed for spherical geometry using the tools of Euclidean solid geometry, while the second half of Book II and Book III contain propositions relevant to astronomy as modeled by the celestial sphere.
Mīrzā Muhammad Tarāghāy bin Shāhrukh (Chagatay: میرزا محمد تراغای بن شاهرخ; Persian: میرزا محمد طارق بن شاهرخ), better known as Ulugh Beg (Persian: الغبیک; 22 March 1394 – 27 October 1449), was a Timurid sultan, as well as an astronomer and mathematician.
Ulugh Beg was notable for his work in astronomy-related mathematics, such as trigonometry and spherical geometry, as well as his general interests in the arts and intellectual activities. It is thought that he spoke five languages: Arabic, Persian, Chaghatai Turkic, Mongolian, and a small amount of Chinese. During his rule (first as a governor, then outright) the Timurid Empire achieved the cultural peak of the Timurid Renaissance through his attention and patronage. Samarkand was captured and given to Ulugh Beg by his father Shah Rukh.
Theodosius of Bithynia (Ancient Greek: Θεοδόσιος Theodosios; 2nd–1st century BC) was a Hellenistic astronomer and mathematician from Bithynia who wrote the Spherics, a treatise about spherical geometry, as well as several other books on mathematics and astronomy, of which two survive, On Habitations and On Days and Nights.
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries.
Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations, otherwise known as geometric analysis.
In spherical geometry, a spherical circle (often shortened to circle) is the locus of points on a sphere at constant spherical distance (the spherical radius) from a given point on the sphere (the pole or spherical center). It is a curve of constant geodesic curvature relative to the sphere, analogous to a line or circle in the Euclidean plane; the curves analogous to straight lines are called great circles, and the curves analogous to planar circles are called small circles or lesser circles. If the sphere is embedded in three-dimensional Euclidean space, its circles are the intersections of the sphere with planes, and the great circles are intersections with planes passing through the center of the sphere.
In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer .
The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term hypersphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry.
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
The appearance of work on this geometry in the nineteenth century stimulated the development of non-Euclidean geometry generally, including hyperbolic geometry.
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and navigation.
The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in History of trigonometry and Mathematics in medieval Islam. The subject came to fruition in Early Modern times with important developments by John Napier, Delambre and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Isaac Todhunter's textbook Spherical trigonometry for the use of colleges and Schools.Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.
This article lists the regular polytopes in Euclidean, spherical and hyperbolic spaces.
