Article
Article
Hyperbolic geometry
Article By:
Marden, Albert School of Mathematics, University of Minnesota, Minneapolis, Minnesota.
Last reviewed:August 2020
DOI:https://doi.org/10.1036/1097-8542.901200
A geometry that obeys a consistent set of axioms differing from Euclid's in implying that the angle sum of a triangle is less than 180°. In the early nineteenth century, K. F. Gauss proved the existence of such a geometry. He questioned which geometry—the new geometry, which is now called hyperbolic geometry, or Euclid's—was the “real geometry of the natural world.” Gauss made numerous measurements before confirming that the “real” geometry was Euclidean. Instead of listing axioms, we will approach hyperbolic geometry by introducing a model of hyperbolic space with its distance formula, lines and planes, and distance-preserving motions. See also: Euclidean geometry; Non-Euclidean geometry
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