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In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement.
Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.
Within contemporary geometry there are many kinds of geometry that are quite different from Euclidean geometry, first encountered in the forms of elementary geometry, plane geometry of triangles and circles, and solid geometry. The conventional meaning of Non-Euclidean geometry is the one set in the nineteenth century: the fields of elliptic ...
Gauss called it "non-Euclidean geometry" [13] causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self-consistent, but still believed in the special role of Euclidean geometry.
Non-Euclidean geometry; Elliptic geometry. Spherical geometry; Sphere-world; Angle excess; hyperbolic geometry. hyperbolic space; hyperboloid model; Poincaré disc model; Poincaré half-plane model; Poincaré metric; Angle of parallelism
Geometry of Complex Numbers is an undergraduate textbook on geometry, whose topics include circles, the complex plane, inversive geometry, and non-Euclidean geometry. It was written by Hans Schwerdtfeger , and originally published in 1962 as Volume 13 of the Mathematical Expositions series of the University of Toronto Press .
Journey into Geometries is a book on non-Euclidean geometry. It was written by Hungarian-Australian mathematician Márta Svéd and published in 1991 by the Mathematical Association of America in their MAA Spectrum book series.
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three points called angles or vertices. Just as in the Euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. Hence planar hyperbolic triangles also ...
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