Hyperbolic geometry is a non-Euclidean geometry that rejects Euclid's fifth postulate (the parallel postulate).

Instead of the Euclidean postulate that states 'through a point not on a given line, exactly one parallel line can be drawn,' hyperbolic geometry posits: 'through a point not on a given line, there are at least two distinct lines parallel to the given line.'

This leads to a geometry with a constant negative curvature, analogous to the surface of a saddle or a Pringle chip.

The altered parallel postulate results in several striking differences:

• Parallel Lines: Through a point not on a given line, infinitely many lines can be drawn that do not intersect the given line.
• Sum of Angles in a Triangle: The sum of the angles in any triangle is always less than \(180^\circ\). The larger the area of the triangle, the smaller the sum of its angles.
• Similar Triangles: If two triangles have the same angles, they must also have the same size (they are congruent). There are no similar but non-congruent triangles in hyperbolic geometry.
• Distance: Distances 'stretch' as you move away from a central point in some models, meaning points that appear close together might be infinitely far apart.
• Circles: Circles in hyperbolic space are not perfectly round when embedded in Euclidean space; they may appear distorted.

While hyperbolic space cannot be perfectly embedded in Euclidean 3D space, several models allow us to visualize and work with it:

• Poincaré Disk Model: The space is represented by the interior of a disk, and lines are represented by circular arcs perpendicular to the boundary of the disk, or diameters. Distances are distorted near the boundary.
• Poincaré Half-Plane Model: The space is represented by the upper half of a plane, and lines are represented by semicircles whose centers lie on the boundary line, or vertical rays.
• Klein Disk Model: Similar to the Poincaré disk, but lines are represented by straight line segments within the disk. Distances are also distorted.
• Hyperboloid Model: Represents hyperbolic space as a hyperboloid (a surface of constant negative curvature) embedded in a higher-dimensional Minkowski space.

Though seemingly abstract, hyperbolic geometry has practical and theoretical applications:

• Cosmology: Used in some models of the universe to describe the overall curvature of space.
• General Relativity: Provides a mathematical framework for understanding spacetime, especially near massive objects.
• Computer Graphics: Used in certain visualization techniques and fractals.
• Art and Design: Inspired Escher's tessellations and various artistic representations of infinite space.
• Cryptography: Some theoretical cryptographic systems explore properties of hyperbolic groups.
• Topology: Fundamental in the study of surfaces and 3-manifolds.