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Encyclopedia > Geometric

Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial 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 be used 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 next most significant development had to wait until a millennium later, and that was analytic geometry, in which coordinate systems are introduced and points are represented as ordered pairs or triples of numbers. This sort of representation has since 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 a new group of transformations to work with (Euclidean geometry 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 fewer invariants there are. As an example, in affine geometry any linear transformation is allowed, and so the first three figures are all congruent; distances and angles are no longer invariants, but linearity is.

A discrete form of geometry is treated under Pick's theorem. Results from FactBites:

 Geometric progression - Wikipedia, the free encyclopedia (511 words) In mathematics, a geometric progression (also inaccurately known as a geometric series, see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. A non-zero geometric progression shows exponential growth or exponential decay. An infinite geometric series is an infinite series whose successive terms have a common ratio.
 Geometric Sequences (234 words) A geometric sequence is a sequence in which each successive term is formed by multiplying the same number. In a geometric sequence, the ratio between consecutive terms is constant. The sequence is geometric with a first term of 2 and a common ratio of -4/2 = -2.
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