Geometry
simple:GeometryGeometry isbranchmathematics dealingspatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, whichtermed axiomsgeometry. Such axiomsinsusceptibleproof, but can be usedconjunctionmathematical definitionspoints, straight lines, curves, surfaces,solidsdraw logical conclusions.
Becauseits immediate practical applications, geometry was one offirst branchesmathematicsbe developed. Likewise,wasfirst fieldbe put on an axiomatic basis, by Euclid. The Greeks were interestedmany questions about ruler-and-compass constructions. The next most significant development hadwait untilmillennium later,that was analytic geometry,which coordinate systemsintroducedpointsrepresented as ordered pairs or triplesnumbers. This sortrepresentation has since then allowed usconstruct new geometries other thanstandard Euclidean version.
The central notiongeometrythatcongruence. In Euclidean geometry, two figuressaidbe congruent if theyrelated byseriesreflections, rotations,translationss.
Other geometries can be constructed by choosingnew underlying spacework(Euclidean geometry uses Euclidean space, Rn) or by choosingnew group transformationswork(Euclidean geometry usesinhomogeneous orthogonal transformations, E(n)). The latter pointviewcalledErlanger program. In general,more congruences we have,fewer invariants there are. As an example,affine geometry any linear transformationallowed,sofirst three figuresall congruent; distancesanglesno longer invariants, but linearity is.
A discrete formgeometrytreated under Pick's theorem.
See Listgeometry topics.
