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Geometry

From Academic Kids

Geometry (from the Greek words Geo = earth and metro = measure) is the branch of mathematics first popularized in ancient Greek culture by Thales (circa 624-547 BC) dealing with spatial relationships. The earliest beginnings of geometry may be traced to Ancient Egypt (see Egyptian mathematics: Geometry). The Rhind Mathematical Papyrus describes an astoundingly precise means of obtaining an approximation of Pi, accurate to within less than one per cent. The Rhind Mathematical Papyrus also describes one of the earliest attempts at squaring the circle as well as a kind of an analogue of the cotangent.

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 collinearity is.

A discrete form of geometry is treated under Pick's theorem. Pick's theorem used dot paper and a certain formula to find the area of odd shapes.

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Topics in mathematics related to structure

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Abstract algebra | Universal algebra | Graph theory | Category theory | Order theory | Model theory | Structural proof theory
Geometry | Topology | General topology | Algebraic geometry | Algebraic topology | Differential geometry and topology
Analysis | Measure theory | Functional analysis | Harmonic analysis

Topics in mathematics related to space

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Geometry | Trigonometry | Non-Euclidean geometry | Fractal geometry | Algebraic geometry
Topology | Metric geometry | Algebraic topology | Differential geometry and topology
Linear algebra | Functional analysis
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