Non-Euclidean Geometry
In 3 dimensions, there are three classes of constant curvature Geometries. All are based on the firstfour of Euclid's Postulates, but each uses its own version of the Parallel Postulate. The ``flat'' geometryof everyday intuition is called Euclidean Geometry (or Parabolic Geometry), and the non-Euclidean geometriesare called Hyperbolic Geometry (or Lobachevsky-Bolyai-Gauss Geometry) and Elliptic Geometry (orRiemannian Geometry). It was not until 1868 that Beltrami proved thatnon-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
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.
Non-Euclidean Geometry
- Vertical Tangent
- Vesica Piscis
- Vibration Problem
- Vickery Auction
- Viergruppe
- Vieta's Substitution
- Vigesimal
- Vigintillion
- Villarceau Circles
- Vinculum
- Vinogradov's Theorem
- Virtual Group
- Visibility
- Visible Point
- Visible Point Vector Identity
- Vitali's Convergence Theorem
- Viviani's Curve
- Viviani's Theorem
- Vojta's Conjecture
- Volterra Integral Equation of the First Kind
- Volterra Integral Equation of the Second Kind
- Volume
- Volume Element
- Volume Integral
- von Aubel's Theorem