The Geometric Vein: The Coxeter Festschrift

Description

Geometry has been defined as that part of mathematics which makes appeal to the sense of sight; but this definition is thrown in doubt by the existence of great geometers who were blind or nearly so, such as Leonhard Euler. Sometimes it seems that geometric methods in analysis, so-called, consist in having recourse to notions outside those apparently relevant, so that geometry must be the joining of unlike strands; but then what shall we say of the importance of axiomatic programmes in geometry, where reference to notions outside a restricted reper­ tory is banned? Whatever its definition, geometry clearly has been more than the sum of its results, more than the consequences of some few axiom sets. It has been a major current in mathematics, with a distinctive approach and a distinc­ ti v e spirit. A current, furthermore, which has not been constant. In the 1930s, after a period of pervasive prominence, it appeared to be in decline, even passe. These same years were those in which H. S. M. Coxeter was beginning his scientific work. Undeterred by the unfashionability of geometry, Coxeter pursued it with devotion and inspiration. By the 1950s he appeared to the broader mathematical world as a consummate practitioner of a peculiar, out-of-the-way art. Today there is no longer anything that out-of-the-way about it. Coxeter has contributed to, exemplified, we could almost say presided over an unanticipated and dra­ matic revival of geometry. Front Matter....Pages i-viii Introduction....Pages 1-3 H. S. M. Coxeter: Published Works....Pages 5-13 Front Matter....Pages 15-15 Uniform Tilings with Hollow Tiles....Pages 17-64 Spherical Tilings with Transitivity Properties....Pages 65-98 Some Isonemal Fabrics on Polyhedral Surfaces....Pages 99-122 Convex Bodies which Tile Space....Pages 123-128 Geometry of Radix Representations....Pages 129-139 Embeddability of Regular Polytopes and Honeycombs in Hypercubes....Pages 141-147 The Derivation of Schoenberg’s Star-Polytopes from Schoute’s Simplex Nets....Pages 149-164 The Harmonic Analysis of Skew Polygons as a Source of Outdoor Sculptures....Pages 165-176 The Geometry of African Art III. The Smoking Pipes of Begho....Pages 177-189 Crystallography and Cremona Transformations....Pages 191-201 Cubature Formulae, Polytopes, and Spherical Designs....Pages 203-218 Two Quaternionic 4-Polytopes....Pages 219-230 Span-Symmetric Generalized Quadrangles....Pages 231-242 On Coxeter’s Loxodromic Sequences of Tangent Spheres....Pages 243-250 Front Matter....Pages 251-251 Elementary Geometry, Then and Now....Pages 253-269 Some Researches Inspired by H. S. M. Coxeter....Pages 271-277 Some Problems in the Geometry of Convex Bodies....Pages 279-284 On an Analog to Minkowski’s Lattice Point Theorem....Pages 285-288 Front Matter....Pages 251-251 Intersections of Convex Bodies with Their Translates....Pages 289-296 An Extremal Property of Plane Convex Curves—P. Ungar’s Conjecture....Pages 297-317 Front Matter....Pages 319-319 Polygons and Polynomials....Pages 321-333 Algebraic Surfaces with Hyperelliptic Sections....Pages 335-344 On the Circular Transformations of Möbius, Laguerre, and Lie....Pages 345-353 The Geometry of Cycles, and Generalized Laguerre Inversion....Pages 355-378 Inversive Geometry....Pages 379-442 Absolute Polarities and Central Inversions....Pages 443-464 Products of Axial Affinities and Products of Central Collineations....Pages 465-470 Normal Forms of Isometries....Pages 471-476 Finite Geometries with Simple, Semisimple and Quasisimple Fundamental Groups....Pages 477-484 Motions in a Finite Hyperbolic Plane....Pages 485-493 Front Matter....Pages 495-495 Generation of Linear Groups....Pages 497-509 On Covering Klein’s Curve and Generating Projective Groups....Pages 511-518 A Local Approach to Buildings....Pages 519-547 Representations and Coxeter Graphs....Pages 549-554 Coinvariant Theory of a Coxeter Group....Pages 555-559 Two-Generator Two-Relation Presentations for Special Linear Groups....Pages 561-567 Groups Related to F a,b,c Involving Fibonacci Numbers....Pages 569-576 Front Matter....Pages 577-577 Convex Polyhedra....Pages 579-582 Front Matter....Pages 577-577 Non-Hamilton Fundamental Cycle Graphs....Pages 583-584 Some Combinatorial Identities....Pages 585-592 Binary Views of Ternary Codes....Pages 593-598