Abstract
Hyperbolic geometry was created in the nineteenth century to better understand Euclid’s axiomatic basis for geometry. However, hyperbolic geometry is similar to Euclidean geometry in many respects. It has the concepts of distance and angle, and there are many theorems common to both. However, there are also striking differences, e.g., the sum of the angles of a hyperbolic triangle is always less than π.
“One geometry cannot be more true than another; it can only be more convenient.”
Henri Poincaré (1854–1912)
“Mathematics is the art of giving the same name to different things.”
Henri Poincaré (1854–1912)
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References
Lewis, F.P.: Questions and discussions: history of the parallel postulate. Am. Math. Mon. 27(1), 16–23 (1920). https://www.jstor.org/stable/2973238 [Chapter 4]
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Lee, NH. (2020). Hyperbolic Plane. In: Geometry: from Isometries to Special Relativity. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-42101-4_4
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DOI: https://doi.org/10.1007/978-3-030-42101-4_4
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