Hyperbolic geometry is a non-Euclidean geometry where the parallel postulate does not hold, allowing for multiple parallel lines through a given point not on a line. It features unique properties such as the sum of angles in a triangle being less than 180 degrees and the concept of hyperbolic space, which models hyperbolic surfaces and spaces with constant negative curvature.

Hyperbolic Geometry

Hyperbolic geometry

Non-Euclidean geometry is a branch of mathematics that explores geometries based on axioms fundamentally different from those of Euclidean geometry, allowing for the study of curved spaces. It is crucial for understanding the geometry of the universe in the context of general relativity, where space can be curved by gravity.

non-Euclidean geometry

The parallel postulate is a fundamental element of Euclidean geometry, stating that through a point not on a given line, exactly one line can be drawn parallel to the given line. This postulate is crucial because altering it leads to the development of non-Euclidean geometries, such as hyperbolic and elliptic geometry, which have significant implications in fields like physics and cosmology.

parallel postulate

multiple parallel lines

sum of angles in a triangle

180 degrees

Hyperbolic space is a non-Euclidean geometric structure where the parallel postulate of Euclidean geometry does not hold, allowing for multiple parallel lines through a given point. This space exhibits constant negative curvature, leading to unique properties such as exponential growth of area and volume with respect to radius, and is used in various fields including complex analysis, relativity, and network theory.

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