Non-Euclidean Geometries

Preface xi

Andr´as Pr´ekopa, Emil Moln´ar

Part I History

The Revolution of J´anos Bolyai 3

Andr´as Pr´ekopa

Gauss and non-Euclidean geometry 61

Jeremy Gray

J´anos Bolyai’s new face 81

Elem´er Kiss

Part II Axiomatical and Logical Aspects

Hyperbolic Geometry, Dimension-Free 97

Walter Benz

An Absolute Property of Four Mutually Tangent Circles 109

H.S.M. Coxeter

Remembering Donald Coxeter 115

Asia Ivic Weiss, William Weiss

Axiomatizations of hyperbolic and absolute geometries 119

Victor Pambuccian

Logical axiomatizations of space-time. Samples from the literature 155

Hajnal Andr´eka, Judit X. Madar´asz and Istv´an N´emeti

vii

viii NON-EUCLIDEAN GEOMETRIES

Part III Polyhedra, Volumes, Discrete Arrangements, Fractals

Structures in Hyperbolic Space 189

Robert Connelly

The Symmetry of Optimally Dense Packings 197

Charles Radin

Flexible Octahedra in the Hyperbolic Space 209

Hellmuth Stachel

Fractal Geometry On Hyperbolic Manifolds 227

Bernd O. Stratmann

A volume formula for generalised hyperbolic tetrahedra 249

Akira Ushijima

Part IV Tilings, Orbifolds and Manifolds, Visualization

The Geometry of Hyperbolic Manifolds of Dimension at least 4 269

John G. Ratcliffe

Real-Time Animation in Hyperbolic, Spherical, and Product Geometries 287

Jeffrey R. Weeks

On spontaneous surgery on knots and links 307

A.D. Mednykh, V.S. Petrov

Classification of tile-transitive 3-simplex tilings and their realizations 321

E. Moln´ar — I. Prok — J. Szirmai

Part V Differential Geometry

Non-Euclidean Analysis 367

Sigurdur Helgason

Holonomy, geometry and topology of manifolds with Grassmann structure 385

Neda Bokan, Paola Matzeu, Zoran Raki´c

Hypersurfaces of type number 2 in the hyperbolic four-space 407

Oldˇrich Kowalski, Masami Sekizawa

How far does hyperbolic geometry generalize? 427

J´anos Szenthe

Geometry of the point Finsler spaces 445

Lajos Tam´assy

Contents ix

Part VI Physics

Black hole perturbations 465

Zolt´an Perj´es

An Idea Whose Time Has Returned 487

Abraham A. Ungar