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  2. Hyperbolic geometry - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_geometry

    Compared to Euclidean geometry, hyperbolic geometry presents many difficulties for a coordinate system: the angle sum of a quadrilateral is always less than 360°; there are no equidistant lines, so a proper rectangle would need to be enclosed by two lines and two hypercycles; parallel-transporting a line segment around a quadrilateral causes ...

  3. János Bolyai - Wikipedia

    en.wikipedia.org/wiki/János_Bolyai

    János Bolyai (Hungarian: [ˈjaːnoʃ ˈboːjɒi]; 15 December 1802 – 27 January 1860) or Johann Bolyai, [2] was a Hungarian mathematician who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consistent alternative geometry that might correspond to the structure of the ...

  4. Gustav von Escherich - Wikipedia

    en.wikipedia.org/wiki/Gustav_von_Escherich

    Following Eugenio Beltrami's (1868) discussion of hyperbolic geometry, Escherich in 1874 published a paper named "The geometry on surfaces of constant negative curvature". He used coordinates initially introduced by Christoph Gudermann (1830) for spherical geometry, which were adapted by Escherich using hyperbolic functions.

  5. Giovanni Girolamo Saccheri - Wikipedia

    en.wikipedia.org/wiki/Giovanni_Girolamo_Saccheri

    There is some minor argument on whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly-illogical aspects of hyperbolic geometry.

  6. Timeline of geometry - Wikipedia

    en.wikipedia.org/wiki/Timeline_of_geometry

    1829 – Bolyai, Gauss, and Lobachevsky invent hyperbolic non-Euclidean geometry, 1837 – Pierre Wantzel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructibility of regular polygons

  7. Hyperboloid model - Wikipedia

    en.wikipedia.org/wiki/Hyperboloid_model

    In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S + of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m ...

  8. Geometric group theory - Wikipedia

    en.wikipedia.org/wiki/Geometric_group_theory

    The notion of a relatively hyperbolic groups was originally introduced by Gromov in 1987 [8] and refined by Farb [21] and Brian Bowditch, [22] in the 1990s. The study of relatively hyperbolic groups gained prominence in the 2000s. Interactions with mathematical logic and the study of the first-order theory of free groups.

  9. Hyperbolic motion - Wikipedia

    en.wikipedia.org/wiki/Hyperbolic_motion

    Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is the unit circle in the complex plane. Hyperbolic motions can also be described on the hyperboloid model of hyperbolic ...