Lobachevskian

Lobachevskian is an adjective relating to the work of the Russian mathematician Nikolai Lobachevsky and, more broadly, to the non-Euclidean geometry that bears his name. In Lobachevskian geometry, Euclid's fifth postulate is replaced by a parallel postulate stating that through a given point not on a line there are infinitely many lines that do not intersect the original line. This framework is now known as hyperbolic geometry, or Lobachevskian geometry.

The Lobachevskian approach leads to a geometry of constant negative curvature. In a Lobachevskian triangle, the sum of the interior angles is always less than 180 degrees, and many standard Euclidean results fail or are modified. The area of a triangle in hyperbolic geometry is related to the deficit from a straight angle, and distance and angle measurements are defined with respect to geodesics, the straightest possible lines in curved space.

Historically, Lobachevsky published and discussed these ideas in the 1820s and 1830s, challenging the long-held assumption that Euclidean geometry was the only true geometry of space. His work, developed independently by János Bolyai and discussed by Gauss, helped inaugurate non-Euclidean geometry. In modern mathematics, Lobachevskian geometry is a foundational instance of hyperbolic geometry and is studied alongside Euclidean and elliptic geometries. The term remains common in historical or descriptive contexts, and the Lobachevsky plane is a standard model used to illustrate two-dimensional hyperbolic geometry.