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The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines. [12] By formulating the geometry in terms of a curvature tensor, Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative ...
Geometry of Complex Numbers is an undergraduate textbook on geometry, whose topics include circles, the complex plane, inversive geometry, and non-Euclidean geometry. It was written by Hans Schwerdtfeger , and originally published in 1962 as Volume 13 of the Mathematical Expositions series of the University of Toronto Press .
There is some minor argument on whether Saccheri really meant that, as he published his work in the final year of his life, came extremely close to discovering non-Euclidean geometry and was a logician. Some believe Saccheri concluded as he did only to avoid the criticism that might come from seemingly-illogical aspects of hyperbolic geometry.
Consequently, hyperbolic geometry has been called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before, [ 64 ] though he did not publish.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric ...
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, Euclidean and hyperbolic.
One example: oriented (i.e., reflections not included) elliptic geometry (i.e., the surface of an n-sphere with opposite points identified) and oriented spherical geometry (the same non-Euclidean geometry, but with opposite points not identified) have isomorphic automorphism group, SO(n+1) for even n. These may appear to be distinct.
Within contemporary geometry there are many kinds of geometry that are quite different from Euclidean geometry, first encountered in the forms of elementary geometry, plane geometry of triangles and circles, and solid geometry. The conventional meaning of Non-Euclidean geometry is the one set in the nineteenth century: the fields of elliptic ...
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