Hyperbolic geometry

Summary

Hyperbolic geometry is the non-Euclidean geometry in which, through a point off a line, there pass at least two lines that do not meet it. It can be realised on a surface of constant negative Gaussian curvature. In it, the angles of a triangle add up to less than $\pi$, and the area of a triangle is determined by its angles. It appears throughout mathematics, from complex analysis and number theory to the topology of surfaces and 3-manifolds.