Hyperbolic Geometry

Hyperbolic geometry is a non-Euclidean geometry where, unlike in traditional Euclidean geometry, the parallel postulate is replaced. In this framework, through a point not on a given line, there are infinitely many lines that do not intersect the original line. This geometry contrasts with the familiar flat surfaces we encounter daily and allows for completely different shapes and relationships, where triangles have angles that sum to less than 180 degrees. Hyperbolic space can be visualized using models like the Poincaré disk or the hyperbolic plane, revealing fascinating properties that challenge our standard perceptions of space.