geometrie

Geometrie, commonly known in English as geometry, is the branch of mathematics dedicated to the study of shapes, sizes, relative positions of figures, and the properties of space. It considers points, lines, planes, and solids, as well as more abstract spaces that arise in higher mathematics. Core concerns include measuring length, area, and volume, understanding similarity and congruence, and analyzing symmetry and spatial relations. Geometric reasoning often proceeds through proofs based on axioms, definitions, and previously established theorems.

The field has ancient roots in practical problems of land surveying, construction, and astronomy. The term derives from Greek words meaning “earth measure.” In classical antiquity, Euclid’s Elements presented a comprehensive, axiomatic treatment of geometry that shaped the subject for centuries. Through the Middle Ages and early modern period, geometry incorporated contributions from diverse cultures, expanding from the plane to three dimensions and beyond.

Modern geometry encompasses many branches. Euclidean geometry describes flat, two- and three-dimensional spaces. Non‑Euclidean geometries, such as hyperbolic and elliptic geometry, explore curved spaces. Analytic or coordinate geometry links geometry with algebra by using coordinates to describe geometric figures. Differential geometry studies curved spaces with calculus, while algebraic geometry investigates shapes defined by polynomial equations. Topology focuses on properties preserved under continuous deformations, and computational geometry addresses algorithmic aspects of geometric problems. Together, these branches form a framework for both theory and application.

Geometric methods underpin science, engineering, computer graphics, architecture, and navigation. Key concepts include distance, angle, area, curvature, and symmetry, as well as transformations such as rotations, translations, and reflections. Geometry integrates with many areas of mathematics, informing intuition about space and guiding practical problem-solving.