Definitions for hyperbolic geometry
hy·per·bol·ic ge·om·e·t·ry

Princeton's WordNetRate this definition:0.0 / 0 votes

1. hyperbolic geometrynoun

   (mathematics) a non-Euclidean geometry in which the parallel axiom is replaced by the assumption that through any point in a plane there are two or more lines that do not intersect a given line in the plane

   "Karl Gauss pioneered hyperbolic geometry"

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1. hyperbolic geometry

   Hyperbolic geometry is a non-Euclidean geometry where, contrary to the Euclidean geometry, the sum of the angles of a triangle is less than 180 degrees, and parallel lines may not maintain a consistent distance separating them. This geometry was discovered in the 19th century as an attempt to prove Euclid's fifth postulate or parallel postulate, leading to the discovery of alternative geometric systems. Hyperbolic geometry is used in areas of science such as special relativity, cosmology, and quantum computing.

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1. Hyperbolic geometry

   In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced. The parallel postulate in Euclidean geometry is equivalent to the statement that, in two-dimensional space, for any given line R and point P not on R, there is exactly one line through P that does not intersect R; i.e., that is parallel to R. In hyperbolic geometry there are at least two distinct lines through P which do not intersect R, so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid. Because there is no precise hyperbolic analogue to Euclidean parallel lines, the hyperbolic use of parallel and related terms varies among writers. In this article, the two limiting lines are called asymptotic and lines sharing a common perpendicular are called ultraparallel; the simple word parallel may apply to both. A characteristic property of hyperbolic geometry is that the angles of a triangle add to less than a straight angle. In the limit as the vertices go to infinity, there are even ideal hyperbolic triangles in which all three angles are 0°.

Matched Categories

1. • Mathematics
   • Non-Euclidean Geometry

How to pronounce hyperbolic geometry?

1. Alex

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How to say hyperbolic geometry in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of hyperbolic geometry in Chaldean Numerology is: 5

2. Pythagorean Numerology

   The numerical value of hyperbolic geometry in Pythagorean Numerology is: 5

References

1. ^ Princeton's WordNet
   http://wordnetweb.princeton.edu/perl/webwn?s=hyperbolic geometry
2. ^ ChatGPT
   https://chat.openai.com
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=hyperbolic geometry