Pythagorean Theorem - Generalizations - Non-Euclidean Geometry - Hyperbolic Geometry

Hyperbolic Geometry

Main article: Hyperbolic geometry See also: Gaussian curvature

For a right triangle in hyperbolic geometry with sides a, b, c and with side c opposite a right angle, the relation between the sides takes the form:

where cosh is the hyperbolic cosine. This formula is a special form of the hyperbolic law of cosines that applies to all hyperbolic triangles:

with γ the angle at the vertex opposite the side c.

By using the Maclaurin series for the hyperbolic cosine, cosh x ≈ 1 + x2/2, it can be shown that as a hyperbolic triangle becomes very small (that is, as a, b, and c all approach zero), the hyperbolic relation for a right triangle approaches the form of Pythagoras' theorem.

Read more about this topic:  Pythagorean Theorem, Generalizations, Non-Euclidean Geometry

Other articles related to "hyperbolic geometry, geometry, hyperbolic":

Universal Hyperbolic Geometry
... Wildberger, termed universal hyperbolic geometry, based on rational trigonometry, his reformulation of the main metrical properties from Euclidean geometry (quadrance and spread) ... completely dual when transferred to a projective (hyperbolic) one ... in the metrical properties derived ('hyperbolic quadrance' and 'hyperbolic spread') reflect this complete point-line correspondence from the non-metrical point of view ...
Gyrovector Space
... A gyrovector space is a mathematical concept for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry, using so-called gyrovectors that have addition based ... it was realized that Einstein's velocity addition law could be interpreted in terms of hyperbolic geometry (see History of special relativity) ... Different models of hyperbolic geometry are regulated by different gyrovector spaces ...
Marilyn Vos Savant - Controversy Regarding Fermat's Last Theorem
... Wiles' proof should be rejected for its use of non-Euclidean geometry was especially contested ... Specifically, she argued that because "the chain of proof is based in hyperbolic (Lobachevskian) geometry", and because squaring the circle is considered a "famou ... She was criticized for rejecting hyperbolic geometry as a satisfactory basis for Wiles' proof, with critics pointing out that axiomatic set theory (rather ...
Boundedly Generated Group - Free Groups Are Not Boundedly Generated - Hyperbolic Geometry
... As is well known from the theory of dynamical systems, any orbit (gk(z)) of a hyperbolic element g has limit set consisting of two fixed points on the ... easy to choose α so that fα equals one on a given hyperbolic element and vanishes on a finite set of other hyperbolic elements with distinct fixed points ... Dynamical properties of hyperbolic elements can similarly be used to prove that any non-elementary Gromov-hyperbolic group is not boundedly generated ...
History - Creation of Non-Euclidean Geometry
... decisive steps in the creation of non-Euclidean geometry ... Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on hyperbolic geometry ... Consequently, hyperbolic geometry is called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry ...

Famous quotes containing the word geometry:

    I am present at the sowing of the seed of the world. With a geometry of sunbeams, the soul lays the foundations of nature.
    Ralph Waldo Emerson (1803–1882)