{"id":"non-euclidean-geometry","domain":"geometry","type":"Concept","title":"Non-Euclidean Geometry","slug":"non-euclidean-geometry","url":"/mathematics/geometry/non-euclidean-geometry/","summary":"Rejecting Euclid's parallel postulate yields consistent alternative geometries (hyperbolic, elliptic). Discovered independently by Lobachevsky, Bolyai, Gauss c.1829-1832. Later realized physically in general relativity.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Bonola, R. (1955). Non-Euclidean Geometry: A Critical and Historical Study. Dover."},{"tier":1,"citation":"Greenberg, M.J. (2007). Euclidean and Non-Euclidean Geometries. 4th ed."},{"tier":2,"citation":"Misner, C.W., Thorne, K.S. and Wheeler, J.A. (1973). Gravitation."},{"tier":3,"citation":"Weeks, J.R. (2001). The Shape of Space. 2nd ed."}],"relationships":[{"type":"rejects_postulate_from","target":"euclidean-geometry","label":"Euclidean Geometry"},{"type":"built_on","target":"euclids-elements","label":"Euclid's Elements — parallel postulate"},{"type":"physically_realised_in","target":"noethers-theorem","label":"Noether's Theorem — general relativity context"},{"type":"related_to","target":"circle","label":"Circle — Gaussian curvature"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.4"}