geometria

Geometria, known in English as geometry, is the branch of mathematics that studies the properties and relations of points, lines, surfaces, and solids in space. It addresses questions of shape, size, position, and orientation, and uses logical reasoning to prove general statements about figures and spaces. Throughout history, geometry has combined practical techniques with formal deduction and has served as a bridge between concrete measurement and abstract reasoning.

Geometric ideas originated in ancient civilizations for surveying, construction, and astronomy. The Greeks, especially Euclid, organized geometry into a rigorous axiomatic system, presenting proven results as theorems derived from a small set of postulates in the Elements. In the early modern period, Descartes introduced analytic geometry, linking geometric problems to algebra. The 19th century saw the development of non-Euclidean geometries by Gauss, Bolyai, and Lobachevsky, and later the emergence of differential and algebraic geometry, expanding the subject far beyond its classical confines. Today geometry exists in many subfields, including Euclidean and non-Euclidean geometry, differential geometry, algebraic geometry, topology, and computational geometry. Each area emphasizes different tools, such as coordinates, curvature, invariants, or polynomials.

Applications are widespread: architecture and engineering rely on precise measurements and shapes; computer graphics and vision model scenes and objects; robotics and navigation use geometric reasoning; geospatial information systems map and analyze spatial data. Geometry also serves as a core example of mathematical thinking, illustrating concepts of proof, abstraction, and the relationship between form and space.