non-Euclidean geometry

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example of straight line not being shortest distance between two points

hyperbolic plane

models of hyperbolic geometry

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Riemannian geometry hyperbolic geometry parallel postulate geometry topology Euclidean geometry analytic geometry differential geometry projective geometry algebraic geometry catastrophe theory Pappus’s theorem Ceva’s theorem length, area, and volume

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Key People

Helmholtz.

Hermann von Helmholtz

German scientist and philosopher

Carl Friedrich Gauss

Carl Friedrich Gauss

German mathematician

Bernhard Riemann, lithograph after a portrait, artist unknown, 1863.

Bernhard Riemann

German mathematician

Nikolay Ivanovich Lobachevsky, detail of a portrait by an unknown artist.

Nikolay Ivanovich Lobachevsky

Russian mathematician

William Clifford, portrait replica by John Collier; in the National Portrait Gallery, London.

William Kingdon Clifford

British mathematician and philosopher

János Bolyai

Hungarian mathematician

Eugenio Beltrami

Italian mathematician

H.S.M. Coxeter

British mathematician

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