Linked from

The 47 pages that link to Non-Euclidean geometry, each with the reason it gives.

• Euclidean geometryCompared with: Its alternatives show that Euclidean geometry is not the only consistent account of space.

• Projective geometryCompared with: Its focus on metric structure differs from projective invariants and incidence.

• Pythagorean theoremCompared with: On curved spaces, the Euclidean side-length equation generally fails for geodesic triangles.

• Euclid's ElementsCompared with: Its development ended the assumption that Euclid's geometry was the only coherent geometry.

• EuclidCompared with: Its emergence showed that Euclid’s parallel postulate was not forced by the other geometric assumptions.

• AxiomRelated: Its development showed that changing a postulate can yield coherent alternative geometries.

• Coordinate geometryCompared with: Its curved spaces require coordinate systems beyond the plane's usual Cartesian model.

• Hyperbolic geometryNarrower topic: Hyperbolic geometry is one of the two principal classical non-Euclidean geometries.

• Right triangleCompared with: On curved surfaces, triangle angle sums and familiar right-triangle relations can differ.

• Analytic geometryCompared with: Coordinate models can describe geometries whose rules differ from the Euclidean plane.

• Felix KleinRelated: Klein’s framework placed these alternatives alongside Euclidean geometry through their symmetry groups.

• GeometryCompared with: Its alternatives to Euclidean parallelism show that familiar spatial rules are not inevitable.

• Synthetic geometryRelated: Its development showed that synthetic geometry could proceed from axioms other than Euclid’s.

• Parallel postulateNarrower topic: Rejecting or altering the postulate opened the path to alternative geometries.

• Interior angleCompared with: On curved surfaces, polygon angle sums can differ from their Euclidean values.

• Foundations of mathematicsRelated: Its discovery showed that geometrical axioms need not describe one uniquely necessary system.

• Line (geometry)Compared with: Its lines can obey different parallel and intersection rules.

• SuprematismRelated: Malevich’s later theories invoked non-Euclidean space to imagine forms beyond ordinary perception.

• Axiomatic methodCompared with: These systems contrast with Euclidean geometry while retaining axiomatic derivation.

• Cosmic horrorRelated: Lovecraftian descriptions use unfamiliar geometry to suggest spaces beyond ordinary perception.

• Kazimir MalevichRelated: Malevich used the idea of non-Euclidean space to discuss forms freed from ordinary perspective.

• Elliptic geometryNarrower topic: Elliptic geometry is one of the two classical non-Euclidean geometries.

• Nikolai LobachevskyNarrower topic: Lobachevsky’s work was a foundational development in this broader family.

• Hyperbolic planeNarrower topic: The hyperbolic plane is one of its two principal constant-curvature forms.

• DeconstructivismRelated: Curved and distorted spatial ideas help explain some of the movement’s departures from conventional geometry.

• Foundations of GeometryRelated: Its development showed that changing a foundational postulate can yield coherent alternatives.

• Geometric proofCompared with: Changing a geometric postulate can change which claims are provable.

• Philosophy of mathematicsRelated: Its development challenged the assumption that geometry describes one necessary mathematical space.

• Hilbert's axiomsCompared with: Changing the parallel axiom yields geometries outside Hilbert's Euclidean system.

• János BolyaiNarrower topic: Bolyai’s work became one of the foundational developments in this broader family.

• PseudosphereNarrower topic: The pseudosphere offers a spatial demonstration of one non-Euclidean geometry.

• Erlangen programRelated: Its emergence challenged the idea that geometry had only one coherent form.

• History of geometryRelated: Its emergence showed that coherent geometries need not obey Euclid’s parallel postulate.

• Johann Heinrich LambertRelated: Lambert explored the consequences of denying Euclid’s parallel postulate before non-Euclidean geometry was established.

• Hyperbolic triangleNarrower topic: Hyperbolic triangles are one concrete manifestation of geometry beyond Euclid’s postulates.

• Lovecraftian horrorRelated: Distorted spaces give physical form to realities that human perception cannot comfortably contain.

• Pasch's axiomRelated: Comparisons reveal that rejecting parallelism need not reject order axioms like Pasch's.

• Tarski's axiomsCompared with: Its models show how changing the parallel assumption produces geometries beyond Tarski's Euclidean target.

• Exterior angle theoremCompared with: Triangle angle sums can differ from 180 degrees, so the theorem’s Euclidean proof does not transfer unchanged.

• Hausdorff paradoxRelated: Both challenge geometric intuition, though the paradox concerns set-theoretic decompositions rather than altered geometry.

• Hilbert's theoremNarrower topic: The theorem addresses whether a hyperbolic geometry can appear as a complete surface in Euclidean space.

• Playfair's axiomNarrower topic: Such geometries expose the consequences of replacing or rejecting Playfair's axiom.

• Axiom (mathematics and logic)Related: Their development showed that geometric axioms need not be self-evident truths about space.

• Daina TaimiņaNarrower topic: Hyperbolic geometry, the subject of her models, is one branch of this broader family.

• Geometry (configuration)Compared with: It allows configurations whose distance and angle rules differ from flat Euclidean space.

• Hjelmslev's theoremNarrower topic: Hjelmslev’s theorem belongs to the consequences of rejecting Euclid’s parallel postulate.

• Pasch's theoremCompared with: Changing the parallel postulate does not by itself settle which order axioms, including Pasch's, are assumed.