geometri

Geometri is the branch of mathematics that studies shapes, sizes, and the relative positions of objects in space. It concerns properties that remain invariant under transformations such as translation, rotation, and reflection, including distance, angles, area, and volume. The subject typically begins with informal ideas of points, lines, and planes and then develops rigorous theorems from a system of axioms.

Ancient cultures used geometric methods for land measurement and construction. In antiquity, Euclid's Elements (circa 300 BCE) organized geometry as an axiomatic theory and became its foundational text for two thousand years. Later developments introduced coordinates (analytic geometry), allowing algebraic methods to describe geometric relations. The 19th century saw the discovery of non-Euclidean geometries by Bolyai, Lobachevsky, and Riemann, showing that Euclid's parallel postulate is not the only possible framework for geometry.

Geometri encompasses several subfields. Plane geometry studies figures on a flat surface; solid geometry studies three-dimensional figures. Analytic or Cartesian geometry represents geometric objects with coordinates. Differential geometry uses calculus to examine curves and surfaces, while algebraic geometry studies solutions to polynomial equations using geometric methods. Other branches include topology and projective, affine, and computational geometry.

Applications span sciences and engineering. Geometri underpins architectural design, surveying, computer-aided design, and computer graphics. It informs robotics, navigation, physics, and astronomy by providing models of space, motion, and form.