Non-Euclidean geometry is geometry based on alternatives to the parallel postulate of Euclidean geometry . Among
the classical constant-curvature geometries, Euclidean
geometry has zero curvature, hyperbolic geometry
has negative curvature, and elliptic geometry
has positive curvature. Spherical geometry
is a two-dimensional positive-curvature model closely related to elliptic
geometry . These geometries are special cases within the much broader framework
of Riemannian geometry , rather than "Riemannian
geometry" being a synonym for elliptic geometry .
In 1868, Beltrami constructed models of hyperbolic geometry within Euclidean geometry . This
established that the former is logically consistent if the latter is (Greenberg 1994).
After reading János Bolyai's 1832 appendix on non-Euclidean geometry, Gauss wrote to Bolyai's father that "to praise it would be to praise myself," explaining that the work's subject, methods, and results substantially coincided with investigations he had pursued privately for decades (Gauss 1832; Bonola 1955, p. 101).
See also Absolute Geometry ,
Elliptic Geometry ,
Euclid's Postulates ,
Euclidean
Geometry ,
Hyperbolic Geometry ,
Parallel
Postulate ,
Riemannian Geometry ,
Spherical
Geometry
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References Bolyai, J. "Scientiam spatii absolute veritam exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) indepentem:
adjecta ad casum falsitatis, quadratura circuli geometrica." Reprinted as "The
Science of Absolute Space" in Bonola, R. Non-Euclidean
Geometry, and the Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing
the Science of Absolute Space by John Bolyai. Bonola,
R. Non-Euclidean
Geometry, and the Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing
the Science of Absolute Space by John Bolyai. Borsuk, K. Foundations
of Geometry: Euclidean and Bolyai-Lobachevskian Geometry. Projective Geometry. Carslaw, H. S. The
Elements of Non-Euclidean Plane Geometry and Trigonometry. Coxeter, H. S. M. Non-Euclidean
Geometry, 6th ed. Dunham,
W. Journey
through Genius: The Great Theorems of Mathematics. Gauss, C. F. Letter to W. Bolyai, Mar. 6, 1832.
Translated in Bonola (1955), p. 101. Greenberg, M. J. Euclidean
and Non-Euclidean Geometries: Development and History, 3rd ed. Iversen, B. An
Invitation to Hyperbolic Geometry. Iyanaga, S. and Kawada, Y. (Eds.). "Non-Euclidean Geometry."
§283 in Encyclopedic
Dictionary of Mathematics. Lobachevski,
N. Reprinted as "Theory of Parallels" in Bonola, R. Non-Euclidean Geometry,
and The Theory of Parallels by Nikolas Lobachevski, with a Supplement Containing
The Science of Absolute Space by John Bolyai. New York: Dover, 1955. Martin,
G. E. The
Foundations of Geometry and the Non-Euclidean Plane. Pappas, T. "A Non-Euclidean World." The
Joy of Mathematics. Ramsay, A. and Richtmeyer, R. D. Introduction
to Hyperbolic Geometry. Sommerville,
D. Y. The
Elements of Non-Euclidean Geometry. Sommerville,
D. Y. Bibliography
of Non-Euclidean Geometry, 2nd ed. Sved,
M. Journey
into Geometries. Trudeau,
R. J. The
Non-Euclidean Revolution. Veritasium.
"How One Line in the Oldest Math Text Hinted at Hidden Universes." Oct. 21,
2023. https://www.youtube.com/watch?v=lFlu60qs7_4 . Weisstein,
E. W. "Books about Non-Euclidean Geometry." http://www.ericweisstein.com/encyclopedias/books/Non-EuclideanGeometry.html . "Welcome
to the Non-Euclidean Geometry Homepage." http://members.tripod.com/~noneuclidean/ . Woods,
F. S. "Non-Euclidean Geometry." Ch. 3 in Monographs
on Topics of Modern Mathematics Relevant to the Elementary Field Referenced on Wolfram|Alpha Non-Euclidean Geometry
Cite this as:
Weisstein, Eric W. "Non-Euclidean Geometry."
From MathWorld https://mathworld.wolfram.com/Non-EuclideanGeometry.html
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