Geometry of hyperbolic space

Hyperbolic space is a type of non-Euclidean geometry where, unlike flat (Euclidean) space, parallel lines diverge and the angles of a triangle sum to less than 180 degrees. Imagine a saddle-shaped surface that curves away from itself in multiple directions. This space has constant negative curvature, meaning it stretches out faster than flat space as you move away from a point. This unique geometry results in larger areas and volumes for shapes than in flat space and has applications in areas like physics, complex networks, and the visualization of abstract mathematical concepts.