page_title: Hyperbolic geometry
section_title: Physical realizations of the hyperbolic plane
hierarchical_section_title: Hyperbolic geometry / Physical realizations of the hyperbolic plane
caption_attribution_description: Hyperbolic crochet kelp garden by the Institute For Figuring at the Los Angeles County Fair. Sept 2006. The installation is part of the Fair Exchange show at the worlds biggest celebration of prize-winning animals andhome-baked wonders. A book with Instructions for making these models is available at the IFF website: &lt;a href="http://theiff.org/publications/index.html"&gt;theiff.org/publications/index.html&lt;/a&gt; photo by the IFF
context_page_description: In mathematics, hyperbolic 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.
  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, parabolic 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.
context_section_description: The hyperbolic plane is a plane where every point is a saddle point. There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature.
  By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space.
  Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved.  A particularly well-known paper model based on the pseudosphere is due to William Thurston.
  The art of crochet has been used (see Mathematics and fiber arts § Knitting and crochet) to demonstrate hyperbolic planes with the first being made by Daina Taimiņa.
  In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball" (more precisely, a truncated order-7 triangular tiling).
  Instructions on how to make a hyperbolic quilt, designed by Helaman Ferguson, have been made available by Jeff Weeks.
caption: A collection of crocheted hyperbolic planes, in imitation of a coral reef, by the Institute For Figuring
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