Hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
For other uses, see Hyperbolic (disambiguation).
(Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.)
The hyperbolic plane is a plane where every point is a saddle point.
Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane.
A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model.
When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry.
In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky.
This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolic space for more information on hyperbolic geometry extended to three and more dimensions.
Every isometry (transformation or motion) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In n-dimensional hyperbolic space, up to n+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)
All the isometries of the hyperbolic plane can be classified into these classes:
Hyperbolic geometry in art[edit]
M. C. Escher's famous prints Circle Limit III and Circle Limit IV
illustrate the conformal disc model (Poincaré disk model) quite well. The white lines in III are not quite geodesics (they are hypercycles), but are close to them. It is also possible to see quite plainly the negative curvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.
For example, in Circle Limit III every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450°; i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In Circle Limit III, for example, one can see that the number of fishes within a distance of n from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.
The art of crochet has been used to demonstrate hyperbolic planes (pictured above) with the first being made by Daina Taimiņa,[29] whose book Crocheting Adventures with Hyperbolic Planes won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year.[39]
HyperRogue is a roguelike game set on various tilings of the hyperbolic plane.