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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 ...
Saccheri is primarily known today for his last publication, in 1733 shortly before his death. Now considered an early exploration of non-Euclidean geometry, Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw) languished in obscurity until it was rediscovered by Eugenio Beltrami, in the mid-19th century.
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.
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 .
Journey into Geometries is a book on non-Euclidean geometry. It was written by Hungarian-Australian mathematician Márta Svéd and published in 1991 by the Mathematical Association of America in their MAA Spectrum book series.
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 ...
Non-Euclidean geometry models proposed by Klein (left) and Poincaré (right) In 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On the So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley–Klein metric.
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