Geometrien

Geometrien, in mathematics, are theories that describe properties and relations of points, lines, surfaces, and solids in space, typically through axioms and logical proofs. The classical foundation is Euclidean geometry, which analyzes flat space and the behavior of angles, distances, and parallel lines. In the 19th century, the parallel postulate was shown not to be a logical necessity, leading to non-Euclidean geometries such as hyperbolic and elliptic geometry that describe curved spaces. This transformation broadened geometry beyond flat space and gave rise to differential and Riemannian geometry, where curvature plays a central role and which underpins general relativity.

Geometries can be characterized by the sets of axioms they adopt and the properties they preserve under transformations. Projective geometry focuses on incidence structures and invariants under projection, while affine geometry preserves parallelism and ratios along lines. Metric geometry studies distance and lengths, and analytic geometry uses coordinates to translate geometric problems into algebra.

Modern developments span several branches. Algebraic geometry investigates solution sets of polynomial equations and their geometric properties; differential geometry studies smooth shapes and their curvature; symplectic geometry concerns structures governing phase space in physics; complex, tropical, and geometric topology explore more specialized aspects. These disciplines increasingly interact with physics, computer graphics and vision, robotics, and geographic information systems.

Etymology-wise, the term geometry derives from Greek ge, earth, and metron, measure, reflecting historical origins in measuring the earth and its features.