Non-Euclidean geometry

Non-Euclidean geometry studies consistent geometric systems that reject Euclid’s parallel postulate, including geometries where parallel lines behave differently or are absent.

Euclidean geometry: Geometry satisfying Euclid’s postulates, including the claim that exactly one parallel passes through a point outside a line. Its parallel postulate is the one non-Euclidean systems replace.
Hyperbolic geometry: A geometry in which infinitely many lines through an external point do not intersect a given line. It replaces the unique-parallel rule with infinitely many parallels.
Euclid: An ancient Greek mathematician whose Elements organized geometry into definitions, postulates, and propositions. His parallel postulate became the focus of centuries of attempted proof.
General relativity: Einstein’s theory of gravity, in which spacetime curvature is related to matter and energy. Its spacetime geometry is non-Euclidean in the presence of gravity.
Consistency proof: A mathematical argument showing that a formal system does not derive a contradiction. Relative consistency arguments showed Euclidean and hyperbolic geometry could coexist as formal systems.
Parallel postulate: An axiom governing whether and how many lines through an external point do not meet a given line. Changing this axiom produces the principal non-Euclidean geometries.
Elliptic geometry: A geometry in which every pair of straight lines eventually intersects. It removes parallel lines rather than multiplying them.
Proclus: A fifth-century Greek philosopher and mathematician who wrote influential commentaries on Euclid’s Elements. His critique helped preserve the early history of parallel-postulate investigations.
Riemannian geometry: The study of smooth manifolds equipped with inner products on their tangent spaces. It provides the broad framework for many curved geometric spaces.
Felix Klein: A German mathematician who unified geometries through transformation groups and classified them in an influential program. His work clarified how non-Euclidean geometries relate to symmetry and invariance.

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