Geometry

Geometry is the study of figures in a Space of a given number of dimensions and of a given type. The most common types of geometry are Plane Geometry (dealing with objects like the Line, Circle, Triangle, and Polygon), Solid Geometry (dealing with objects like the Line, Sphere, and Polyhedron), and Spherical Geometry (dealing with objects like the Spherical Triangle and Spherical Polygon).

Historically, the study of geometry proceeds from a small number of accepted truths (Axioms or Postulates), then builds up true statements using a systematic and rigorous step-by-step Proof. However, there is much more to geometry than this relatively dry textbook approach, as evidenced by some of the beautiful and unexpected results of Projective Geometry (not to mention Schubert's powerful but questionable Enumerative Geometry).

Formally, a geometry is defined as a complete locally homogeneous Riemannian Metric. In $\Bbb{R}^2$, the possible geometries are Euclidean planar, hyperbolic planar, and elliptic planar. In $\Bbb{R}^3$, the possible geometries include Euclidean, hyperbolic, and elliptic, but also include five other types.

See also Absolute Geometry, Affine Geometry, Coordinate Geometry, Differential Geometry, Enumerative Geometry, Finsler Geometry, Inversive Geometry, Minkowski Geometry, Nil Geometry, Non-Euclidean Geometry, Ordered Geometry, Plane Geometry, Projective Geometry, Sol Geometry, Solid Geometry, Spherical Geometry, Thurston's Geometrization Conjecture

References

Geometry

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Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, 1994.

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Ghyka, M. C. The Geometry of Art and Life, 2nd ed. New York: Dover, 1977.

Hilbert, D. The Foundations of Geometry, 2nd ed. Chicago, IL: The Open Court Publishing Co., 1921.

Johnson, R. A. Advanced Euclidean Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. New York: Dover, 1960.

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Klein, F. Famous Problems of Elementary Geometry and Other Monographs. New York: Dover, 1956.

Melzak, Z. A. Invitation to Geometry. New York: Wiley, 1983.

Moise, E. E. Elementary Geometry from an Advanced Standpoint, 3rd ed. Reading, MA: Addison-Wesley, 1990.

Ogilvy, C. S. ``Some Unsolved Problems of Modern Geometry.'' Ch. 11 in Excursions in Geometry. New York: Dover, pp. 143-153, 1990.

Simon, M. Über die Entwicklung der Elementargeometrie im XIX Jahrhundert. Berlin, pp. 97-105, 1906.

Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, 1961.