n-gons

This book, a translation of the German volume__n-Ecke__, presents an elegant geometric theory which, starting from quite elementary geometrical observations, exhibits an interesting connection between geometry and fundamental ideas of modern algebra.

This book, a translation of the German volume n-Ecke, presents an elegant geometric theory which, starting from quite elementary geometrical observations, exhibits an interesting connection between geometry and fundamental ideas of modern algebra in a form that is easily accessible to the student who lacks a sophisticated background in mathematics. It stimulates geometrical thought by applying the tools of linear algebra and the algebra of polynomials to a concrete geometrical situation to reveal some rather surprising insights into the geometry of n-gons. The twelve chapters treat n-gons, classes of n-gons, and mapping of the set of n-gons into itself. Exercises are included throughout, and two appendixes, by Henner Kinder and Eckart Schmidt, provide background material on lattices and cyclotomic polynomials.

(Mathematical Expositions No. 18)

Contents 7 Authors' preface 11 Translator's preface 13 Summary of contents 15 Introduction 17 Part 1. Cyclic classes and cyclic mappings 29 1. Cyclic classes of n-gons 29 2. Cyclic mappings of n-gons 48 3. Isobaric cyclic mappings 67 4. Averaging mappings 76 PART II. The main theorem 85 5. Idempotent elements and Boolean algebras 85 6. The main theorem about cyclic classes 97 PART III. Boolean algebras of n-gonal theory 109 7. Idempotent-transfer. Residue class rings of principal ideal domains 109 8. Boolean algebras of the n-gonal theory I 120 9. Boolean algebras of the n-gonal theory II 132 PART IV. Atomic decompositions 147 10. Rational components of an n-gon 147 11. Complex components of an n-gon 157 12. The real components of an n-gon 165 Appendices 183 1. Lattices 183 2. Cyclotomic polynomials 195 List of symbols and notations 203 Index 205