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Hyperbolic geometry

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In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.

A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring

The "hyperbolic soccerball" is a paper model which approximates (part of) the hyperbolic plane as a truncated icosahedron approximates the sphere.

János Bolyai

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János Bolyai or Johann Bolyai, 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 universe helped to free mathematicians to study abstract concepts irrespective of any possible connection with the physical world.

Portrait of J. Bolyai by Ferenc Márkos (2012)

János Bolyai; artwork by Attila Zsigmond

Memorial plaque of János Bolyai in Olomouc, Czech Republic

The house in Cluj-Napoca where János Bolyai was born