Hyperbolic geometry | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-06

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Hyperbolic_geometry


00:02:19 1 Properties
00:02:28 1.1 Relation to Euclidean geometry
00:04:02 1.2 Lines
00:05:22 1.2.1 Non-intersecting / parallel lines
00:07:33 1.3 Circles and disks
00:10:49 1.4 Hypercycles and horocycles
00:12:45 1.5 Triangles
00:14:18 1.6 Regular apeirogon
00:15:57 1.7 Tessellations
00:16:21 2 Standardized Gaussian curvature
00:17:34 2.1 Cartesian-like coordinate systems
00:19:29 2.2 Distance
00:23:05 3 History
00:24:53 3.1 19th century developments
00:27:50 3.2 Philosophical consequences
00:29:26 3.3 Geometry of the universe (spatial dimensions only)
00:31:16 3.4 Geometry of the universe (special relativity)
00:32:31 4 Physical realizations of the hyperbolic plane
00:34:14 5 Models of the hyperbolic plane
00:35:25 5.1 The Beltrami–Klein model
00:36:39 5.2 The Poincaré disk model
00:37:55 5.3 The Poincaré half-plane model
00:39:50 5.4 The hyperboloid model
00:41:17 5.5 The hemisphere model
00:45:32 5.6 The Gans model
00:46:25 5.7 The band model
00:47:03 5.8 Connection between the models
00:49:05 6 Isometries of the hyperbolic plane
00:51:35 7 Hyperbolic geometry in art
00:53:41 8 Higher dimensions
00:54:03 9 Homogeneous structure
00:58:09 10 See also



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SUMMARY
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In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R.
(compare this with Playfair's axiom, the modern version of Euclid's parallel postulate)Hyperbolic plane geometry is also the geometry of saddle surfaces and pseudospherical surfaces, surfaces with a constant negative Gaussian curvature.
A modern use of hyperbolic geometry is in the theory of special relativity, particularly Minkowski spacetime and gyrovector space.
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.
Hyperbolic geometry can be extended to three and more dimensions; see hyperbolic space for more on the three and higher dimensional cases.