17 Facts About Hyperbolic geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry.

Hyperbolic plane geometry is the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate.

Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements.

Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry.

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Non-intersecting lines in hyperbolic geometry have properties that differ from non-intersecting lines in Euclidean geometry:.

Special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides.

Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self consistent, but still believed in the special role of Euclidean geometry.

Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

Hyperbolic geometry realised that his measurements were not precise enough to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the absolute length is at least one million times the diameter of the earth's orbit.

Hyperbolic geometry enters special relativity through rapidity, which stands in for velocity, and is expressed by a hyperbolic angle.

Hyperbolic geometry plane is a plane where every point is a saddle point.

Hyperbolic geometry lines are half-circles orthogonal to the boundary of the hemisphere.

Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane.

Hyperbolic geometry is not limited to 2 dimensions; a hyperbolic geometry exists for every higher number of dimensions.

Hyperbolic geometry space of dimension n is a special case of a Riemannian symmetric space of noncompact type, as it is isomorphic to the quotient.

Many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.