By András Prékopa, Emil Molnár
"From not anything i've got created a brand new various world," wrote János Bolyai to his father, Wolgang Bolyai, on November three, 1823, to enable him recognize his discovery of non-Euclidean geometry, as we name it this present day. the result of Bolyai and the co-discoverer, the Russian Lobachevskii, replaced the process arithmetic, opened the way in which for contemporary actual theories of the 20 th century, and had an effect at the background of human culture.
The papers during this quantity, which commemorates the 2 hundredth anniversary of the beginning of János Bolyai, have been written by means of best scientists of non-Euclidean geometry, its historical past, and its purposes. a number of the papers current new discoveries in regards to the lifestyles and works of János Bolyai and the background of non-Euclidean geometry, others take care of geometrical axiomatics; polyhedra; fractals; hyperbolic, Riemannian and discrete geometry; tilings; visualization; and purposes in physics.
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Extra info for Non-Euclidean Geometries: Janos Bolyai Memorial Volume
Gauss formulated the concept of parallels, which, undoubtedly, is similar to the parallelism of Bolyai and Lobachevskii. As has already been mentioned, in his reply of 1819 to Gerling (actually, to Schweikart) Gauss gave the upper bound of the area of the hyperbolic geometrical triangle without a proof. In his letter of July 12, 1831, to Schumacher he presented the circumference of the hyperbolic geometrical circle without proof. This can be obtained by formal manipulation from a corresponding formula of the spherical circle with radius r by substituting ir for r.
Einstein pointed out that if light, coming from a distant star, passes by the Sun, its path, compared to the Euclidean line, will deflect slightly due to the gravitational effect of the Sun. Therefore we see the star in a position different from its actual one. The position where we see it may be obtained if the line of the shaft of light is extended in the reverse direction, as shown in Figure 4. The phenomenon can be justified through observation only when there is a solar eclipse because a t other times we are unable to observe a star in the neighborhood of the Sun due to the strength of its light.
All of them have a common perpendicular with I . The geometry corresponding to this case is called hyperbolic. At this point we remark that Bolyai's lines are not lines in the everyday sense even though we visualize them as such in Figure 2. Lines in the Bolyai-Lobachevskii geometry may be any geometrical objects satisfying the axioms. Figure 2. Parallel lines. Bolyai developed the absolute geometry that is independent of the 5th postulate. The theorem stated below belongs to the absolute plane geometry.
Non-Euclidean Geometries: Janos Bolyai Memorial Volume by András Prékopa, Emil Molnár