Non-Euclidean Geometries -

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Résumé :
From nothing I have created a new different world, wrote J?nos Bolyai to his father, Wolgang Bolyai, on November 3, 1823, to let him know his discovery of non-Euclidean geometry, as we call it today. The results of Bolyai and the co-discoverer, the Russian Lobachevskii, changed the course of mathematics, opened the way for modern physical theories of the twentieth century, and had an impact on the history of human culture. The papers in this volume, which commemorates the 200th anniversary of the birth of J?nos Bolyai, were written by leading scientists of non-Euclidean geometry, its history, and its applications. Some of the papers present new discoveries about the life and works of J?nos Bolyai and the history of non-Euclidean geometry, others deal with geometrical axiomatics; polyhedra; fractals; hyperbolic, Riemannian and discrete geometry; tilings; visualization; and applications in physics.

Sommaire:
History.- The Revolution of J?nos Bolyai.- Gauss and Non-Euclidean Geometry.- J?nos Bolyai?s New Face.- Axiomatical and Logical Aspects.- Hyperbolic Geometry, Dimension-Free.- An Absolute Property of Four Mutually Tangent Circles.- Remembering Donald Coxeter.- Axiomatizations of Hyperbolic and Absolute Geometries.- Logical Axiomatizations of Space-Time. Samples from the Literature.- Polyhedra, Volumes, Discrete Arrangements, Fractals.- Structures in Hyperbolic Space.- The Symmetry of Optimally Dense Packings.- Flexible Octahedra in the Hyperbolic Space.- Fractal Geometry on Hyperbolic Manifolds.- A Volume Formula for Generalised Hyperbolic Tetrahedra.- Tilings, Orbifolds and Manifolds, Visualization.- The Geometry of Hyperbolic Manifolds of Dimension at Least 4.- Real-Time Animation in Hyperbolic, Spherical, and Product Geometries.- On Spontaneous Surgery on Knots and Links.- Classification of Tile-Transitive 3-Simplex Tilings and Their Realizations in Homogeneous Spaces.- Differential Geometry.- Non-Euclidean Analysis.- Holonomy, Geometry and Topology of Manifolds with Grassmann Structure.- Hypersurfaces of Type Number 2 in the Hyperbolic Four-Space and Their Extensions To Riemannian Geometry.- How Far Does Hyperbolic Geometry Generalize?.- Geometry of the Point Finsler Spaces.- Physics.- Black Hole Perturbations.- Placing the Hyperbolic Geometry of Bolyai and Lobachevsky Centrally in Special Relativity Theory: An Idea Whose Time has Returned.