Non-Euclidean Geometry

In 3 dimensions, there are three classes of constant curvature Geometries. All are based on the first four of Euclid's Postulates, but each uses its own version of the Parallel Postulate. The ``flat'' geometry of everyday intuition is called Euclidean Geometry (or Parabolic Geometry), and the non-Euclidean geometries are called Hyperbolic Geometry (or Lobachevsky-Bolyai-Gauss Geometry) and Elliptic Geometry (or Riemannian Geometry). It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean Geometry.

See also Absolute Geometry, Elliptic Geometry, Euclid's Postulates, Euclidean Geometry, Hyperbolic Geometry, Parallel Postulate

References

Non-Euclidean Geometry

Borsuk, K. Foundations of Geometry: Euclidean and Bolyai-Lobachevskian Geometry. Projective Geometry. Amsterdam, Netherlands: North-Holland, 1960.

Carslaw, H. S. The Elements of Non-Euclidean Plane Geometry and Trigonometry. London: Longmans, 1916.

Coxeter, H. S. M. Non-Euclidean Geometry, 6th ed. Washington, DC: Math. Assoc. Amer., 1988.

Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 53-60, 1990.

Iversen, B. An Invitation to Hyperbolic Geometry. Cambridge, England: Cambridge University Press, 1993.

Iyanaga, S. and Kawada, Y. (Eds.). ``Non-Euclidean Geometry.'' §283 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 893-896, 1980.

Martin, G. E. The Foundations of Geometry and the Non-Euclidean Plane. New York: Springer-Verlag, 1975.

Pappas, T. ``A Non-Euclidean World.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 90-92, 1989.

Ramsay, A. and Richtmeyer, R. D. Introduction to Hyperbolic Geometry. New York: Springer-Verlag, 1995.

Sommerville, D. Y. The Elements of Non-Euclidean Geometry. London: Bell, 1914.

Sommerville, D. Y. Bibliography of Non-Euclidean Geometry, 2nd ed. New York: Chelsea, 1960.

Sved, M. Journey into Geometries. Washington, DC: Math. Assoc. Amer., 1991.

Trudeau, R. J. The Non-Euclidean Revolution. Boston, MA: Birkhäuser, 1987.