ickeeuklidiska

Ickeeuklidiska refers to geometries that do not satisfy Euclid's fifth postulate, the parallel postulate. In these geometries the familiar results of Euclidean geometry change in ways that depend on the curvature of space. The term is commonly used in Swedish-language contexts to describe the broader family of non-Euclidean geometries.

Historical development began in the 19th century when mathematicians such as Nikolai Lobachevsky and János Bolyai independently developed hyperbolic geometry, illustrating a space of constant negative curvature. Carl Friedrich Gauss studied related ideas and laid groundwork, while Bernhard Riemann’s work on manifolds and curvature extended the concept beyond constant curvature, forming the basis of modern differential geometry. Beltrami later provided concrete models showing that non-Euclidean geometries could be realized on curved surfaces.

Main types and models: Hyperbolic geometry features triangles with angle sums less than 180 degrees; elliptic geometry (often called spherical geometry) has triangle angle sums greater than 180 degrees. Hyperbolic geometry is commonly visualized with models such as the Poincaré disk or the Klein model, whereas elliptic geometry corresponds to the geometry on the surface of a sphere. In both cases, the parallel postulate is replaced by statements about how lines behave under curvature.

Applications: Ickeeuklidiska geometries underpin modern physics, particularly general relativity where spacetime is curved. They also play roles in computer graphics, cosmology, navigation, and architectural design, where curved spaces or surfaces are relevant representations.