geometry

Geometry is a branch of mathematics concerned with questions of shape, size, relative position, and properties of space. It studies points, lines, curves, surfaces, and solids and how they interact under a system of measurement and axioms. The field originated in ancient civilizations, with the practical needs of land surveying and architecture, and later matured into an axiomatic discipline through Euclid’s Elements, which organized geometric knowledge using defined objects, postulates, and logical proofs.

Today geometry encompasses several subfields. Plane geometry studies figures in a flat two-dimensional plane, while solid geometry concerns three-dimensional figures. Analytic or coordinate geometry introduces algebraic methods by representing geometric objects with coordinates. Other branches include differential geometry, which uses calculus to study curved spaces; algebraic geometry, which focuses on solution sets of polynomial equations; and computational or discrete geometry, which addresses algorithmic questions and applications in computer science. In the 19th century non-Euclidean geometries expanded notions of space, and Riemannian geometry later underpinned the mathematics of general relativity.

Fundamental concepts include points, lines, angles, shapes, distance, area, and volume, along with ideas of congruence, similarity, and symmetry. Transformations such as translations, rotations, reflections, and scalings preserve or modify geometric properties. The axiomatic method in Euclidean geometry derives theorems such as the Pythagorean theorem and triangle similarity from a small set of postulates; non-Euclidean geometries replace the parallel postulate and reveal different notions of space.

Geometry informs many disciplines and technologies, including architecture, engineering, computer graphics, geographic information systems, and the physical sciences. It provides the language to model space, optimize structures, and simulate real-world phenomena, making it a foundational tool in mathematics and applied sciences.