Non-Euclidean Geometries: János Bolyai Memorial Volume (Mathematics and Its Applications Book 581)

"From nothing I have created a new different world,” wrote János 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 200^th^ anniversary of the birth of János 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 János 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. __Audience__ This book is intended for those who teach, study, and do research in geometry and history of mathematics. Cultural historians, physicists, and computer scientists will also find it an important source of information. Front Matter....Pages 1-1 The Revolution of János Bolyai....Pages 3-59 Gauss and Non-Euclidean Geometry....Pages 61-80 János Bolyai’s New Face....Pages 81-93 Front Matter....Pages 95-95 Hyperbolic Geometry, Dimension-Free....Pages 97-107 An Absolute Property of Four Mutually Tangent Circles....Pages 109-114 Remembering Donald Coxeter....Pages 115-118 Axiomatizations of Hyperbolic and Absolute Geometries....Pages 119-153 Logical Axiomatizations of Space-Time. Samples from the Literature....Pages 155-185 Front Matter....Pages 187-187 Structures in Hyperbolic Space....Pages 189-196 The Symmetry of Optimally Dense Packings....Pages 197-207 Flexible Octahedra in the Hyperbolic Space....Pages 209-225 Fractal Geometry on Hyperbolic Manifolds....Pages 227-247 A Volume Formula for Generalised Hyperbolic Tetrahedra....Pages 249-265 Front Matter....Pages 267-267 The Geometry of Hyperbolic Manifolds of Dimension at Least 4....Pages 269-286 Real-Time Animation in Hyperbolic, Spherical, and Product Geometries....Pages 287-305 On Spontaneous Surgery on Knots and Links....Pages 307-319 Classification of Tile-Transitive 3-Simplex Tilings and Their Realizations in Homogeneous Spaces....Pages 321-363 Front Matter....Pages 365-365 Non-Euclidean Analysis....Pages 367-384 Holonomy, Geometry and Topology of Manifolds with Grassmann Structure....Pages 385-405 Hypersurfaces of Type Number 2 in the Hyperbolic Four-Space and Their Extensions To Riemannian Geometry....Pages 407-426 Front Matter....Pages 365-365 How Far Does Hyperbolic Geometry Generalize?....Pages 427-444 Geometry of the Point Finsler Spaces....Pages 445-461 Front Matter....Pages 463-463 Black Hole Perturbations....Pages 465-485 Placing the Hyperbolic Geometry of Bolyai and Lobachevsky Centrally in Special Relativity Theory: An Idea Whose Time has Returned....Pages 487-506 "From nothing I have created a new different world,” wrote János 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 200 th anniversary of the birth of János 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 János 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. Audience This book is intended for those who teach, study, and do research in geometry and history of mathematics. Cultural historians, physicists, and computer scientists will also find it an important source of information. "The papers in this volume, which commemorates the 200th anniversary of the birth of Janos 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 Janos 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. Audience This book is intended for those who teach, study, and do research in geometry and history of mathematics. Cultural historians, physicists, and computer scientists will also find it an important source of information."--Publisher's website