Non-Euclidean Geometries (eBook)

History.- The Revolution of János Bolyai.- Gauss and Non-Euclidean Geometry.- János 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.