geometries

Geometries refer to mathematical theories about space and shape. The classical subject, Euclidean geometry, is grounded in Euclid's postulates and describes flat spaces of everyday experience. The parallel postulate leads to the familiar theorems about triangles, circles, and polygons. In the 19th century, mathematicians questioned Euclid's postulates and developed non-Euclidean geometries by Altering the parallel postulate. Hyperbolic geometry features many lines through a point that do not meet the given line, yielding constant negative curvature; elliptic geometry, associated with spherical surfaces, has no parallel lines and positive curvature. These developments showed that consistent alternative geometric frameworks could model different notions of space.

Other geometries extend or modify fundamental ideas. Affine geometry preserves straightness and parallelism under affine transformations but not lengths or angles. Projective geometry focuses on incidences and properties invariant under projection, such as cross-ratio, ignoring distances and parallelism. Riemannian geometry equips a smooth manifold with a metric, enabling measurement of lengths, angles, and curvature, and forms the mathematical basis of general relativity. Differential geometry studies smooth manifolds with such structures. Algebraic geometry investigates solutions of polynomial equations and the shapes they define, linking geometry to algebra. Computational and discrete geometry study geometric problems in finite settings and with algorithms, essential in computer graphics and robotics.

Geometries collectively provide versatile tools for science and engineering, shaping how space, form, and measurement are understood across disciplines.