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non-Euclidean geometry
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example of straight line not being shortest distance between two points
Riemann sphere
hyperbolic plane
models of hyperbolic geometry
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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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