Geometry

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'''Geometry''' is the branch of mathematics 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, '''R'''^''n'') 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. See also List of geometry topics, Important publications in geometry. ==External links== {| border="0" width="35%" cellpadding="5" class="noprint floatright" style="border:solid #008 2px; margin:0 0 .5em .5em;" |- | |style="font-weight:bold;"|Our sister project, Wikibooks, provides an electronic book on Geometry. |} * Geometry. From Interactive Mathematics Miscellany and Puzzles {| style="margin:0 auto;" align=center width=80% id=toc |align=center style="background:#ccccff"| '''Topics in mathematics related to spaces''' |align="center" style="background:#ccccff" |Edit |- |align=center|Topology | Geometry | Trigonometry | Algebraic geometry | Differential geometry and topology | Algebraic topology | Linear algebra | Fractal geometry | Compact space |} bg:Геометрия ca:Geometria cs:Geometrie da:Geometri de:Geometrie et:Geomeetria el:Γεωμετρία es:Geometr�a eo:Geometrio fr:G�om�trie hi:रेखागणित it:Geometria hu:Geometria nl:Meetkunde ja:幾何学 pl:Geometria pt:Geometria ru:Геометрия simple:Geometry sl:Geometrija fi:Geometria sv:Geometri vi:Hình học uk:Геометрія zh:几何学