The Geometric Vein

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   1. Introduction

      • Chandler Davis, Branko Grünbaum, F. A. Sherk

      Pages 1-3

3. H. S. M. Coxeter: Published Works

   1. H. S. M. Coxeter: Published Works

      • Chandler Davis, Branko Grünbaum, F. A. Sherk

      Pages 5-13

4. Polytopes and Honeycombs

   1. Front Matter

      Pages 15-15

      PDF

   2. Uniform Tilings with Hollow Tiles

      • Branko Grünbaum, J. C. P. Miller, G. C. Shephard

      Pages 17-64

   3. Spherical Tilings with Transitivity Properties

      • Branko Grünbaum, G. C. Shephard

      Pages 65-98

   4. Some Isonemal Fabrics on Polyhedral Surfaces

      • Jean J. Pedersen

      Pages 99-122

   5. Convex Bodies which Tile Space

      • P. McMullen

      Pages 123-128

   6. Geometry of Radix Representations

      • William J. Gilbert

      Pages 129-139

   7. Embeddability of Regular Polytopes and Honeycombs in Hypercubes

      • Patrice Assouad

      Pages 141-147

   8. The Derivation of Schoenberg’s Star-Polytopes from Schoute’s Simplex Nets

      • H. S. M. Coxeter

      Pages 149-164

   9. The Harmonic Analysis of Skew Polygons as a Source of Outdoor Sculptures

      • I. J. Schoenberg

      Pages 165-176

   10. The Geometry of African Art III. The Smoking Pipes of Begho

       • Donald W. Crowe

       Pages 177-189

   11. Crystallography and Cremona Transformations

       • Patrick Du Val

       Pages 191-201

   12. Cubature Formulae, Polytopes, and Spherical Designs

       • J. M. Goethals, J. J. Seidel

       Pages 203-218

   13. Two Quaternionic 4-Polytopes

       • S. G. Hoggar

       Pages 219-230

   14. Span-Symmetric Generalized Quadrangles

       • Stanley E. Payne

       Pages 231-242

   15. On Coxeter’s Loxodromic Sequences of Tangent Spheres

       • Asia Weiss

       Pages 243-250

5. Extremal Problems

   1. Front Matter

      Pages 251-251

      PDF

   2. Elementary Geometry, Then and Now

      • I. M. Yaglom

      Pages 253-269

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.

• Finite
• Fundamental group
• Invariant
• Lie
• Slate
• algebra
• algebraic surface
• derivation
• group
• harmonic analysis
• mathematics
• polygon
• polytope
• presentation
• transformation

• Department of Mathematics, University of Toronto, Toronto, Canada

  Chandler Davis, F. A. Sherk

• Department of Mathematics, University of Washington, Seattle, USA

  Branko Grünbaum

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