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Nikolai Ivanovich Lobachevsky (Russian: Никола́й Ива́нович Лобаче́вский, IPA: [nʲɪkɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕefskʲɪj] ⓘ; 1 December [O.S. 20 November] 1792 – 24 February [O.S. 12 February] 1856) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for ...
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 ...
Nikolai Ivanovich Lobachevsky (1792–1856) – hyperbolic geometry, a non-Euclidean geometry Michel Chasles (1793–1880) – projective geometry Germinal Dandelin (1794–1847) – Dandelin spheres in conic sections
The hyperbolic plane (and more generally any Hadamard manifolds of sectional curvature) is -hyperbolic. If we scale the Riemannian metric by a factor λ > 0 {\displaystyle \lambda >0} then the distances are multiplied by λ {\displaystyle \lambda } and thus we get a space that is λ ⋅ δ {\displaystyle \lambda \cdot \delta } -hyperbolic.
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
The area of a triangle in hyperbolic geometry is bounded while triangles exist with arbitrarily large areas in Euclidean geometry. The set of points on the same side and equally far from a given straight line themselves form a line in Euclidean geometry, but don't in hyperbolic geometry (they form a hypercycle.)
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 ...
Geometry Dash is a side-scrolling music platforming game series developed by Robert Topala. It was released on 13 August 2013 for iOS and Android , with versions for Windows and macOS following on 22 December 2014.