Definition
Non-Euclidean geometries are geometries in which Euclid's parallel postulate (through a point not on a line, there is exactly one line parallel to the given line) is false. Hyperbolic geometry (Bolyai, Lobachevsky, 1820s-30s) allows infinitely many parallels through the point. Elliptic geometry (Riemann, 1854) allows none. Both are internally consistent. Their existence shows the parallel postulate is independent of Euclid's other four axioms.
Why it matters
How it works
Hyperbolic geometry is most accessibly modeled on the Poincaré disk: a disk in the Euclidean plane where "lines" are circular arcs perpendicular to the boundary. The model satisfies Euclid's first four axioms but multiple "lines" pass through a point without intersecting a given "line." Elliptic geometry is modeled on the surface of a sphere with antipodal points identified: great circles are "lines," and any two distinct "lines" intersect. Both models are constructed within Euclidean mathematics, demonstrating relative consistency.