Home | Site Map | Anoca.org

# Geometry

(geometry)

Geometry is the branch of mathematics dealing withspatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities,which are termed axioms in geometry. Such axioms are insusceptible to proof, but can beused in conjunction with mathematical definitions for points , straight lines , curves , surfaces ,and solids to draw logical conclusions.

Because of its immediate practical applications, geometry was one of the first branches of mathematics to be developed.Likewise, it was the first field to be put on an axiomatic basis, by Euclid . The Greeks were interested in many questions about ruler-and-compass constructions . The nextmost significant development had to wait until a millennium later, and that was analytic geometry , in which coordinatesystems are introduced and points are represented as ordered pairs or triples of numbers. This sort of representation hassince then allowed us to construct new geometries other than the standard Euclidean version.

The central notion in geometry is that of congruence. In Euclidean geometry , two figures are said to be congruent if they are related by a series of reflections , rotations , and translations .

Other geometries can be constructed by choosing a new underlying space to work with (Euclidean geometry uses Euclidean space , Rn) or by choosing anew group of transformations to work with (Euclideangeometry uses the inhomogeneous orthogonal transformations, E(n)). The latter point of view is called the Erlangen program . In general, the more congruences we have, the fewerinvariants there are. As an example, in affine geometry any lineartransformation is allowed, and so the first three figures are all congruent; distances and angles are no longer invariants, butlinearity is.

A discrete form of geometry is treated under Pick's theorem .