The Eightfold Way: The Beauty of Klein's Quartic Curve (Mathematical Sciences Research Institute Publications, Series Number 35)

The German mathematician Felix Klein discovered in 1879 that the surface that we now call the Klein quartic has many remarkable properties, including an incredible 336-fold symmetry, the maximum possible degree of symmetry for any surface of its type. Since then, mathematicians have discovered that the same object comes up in different guises in many areas of mathematics, from complex analysis and geometry to number theory. This volume explores the rich tangle of properties and theories surrounding this multiform object. It includes expository and research articles by renowned mathematicians in different fields. It also includes a beautifully illustrated essay by the mathematical sculptor Helaman Ferguson, who distilled some of the beauty and remarkable properties of this surface into a sculpture entitled "The Eightfold Way." The book closes with the first English translation of Klein's seminal article on this surface. Cover The Eightfold Way - The Beauty of Klein's Quartic Curve ISBN 0521004195 Contents MSRI and the Klein Quartic A Mathematical Sculpture by Helaman Ferguson The Geometry of Klein's Riemann Surface 1. Introduction 2. Triangle Groups and Platonic Surfaces 3. Two Platonic Surfaces of Genus Two 4. The Hyperbolic Description of Klein's Surface 5. Oblique Pants and Isometry Subgroups 6. Fermat Surfaces x^k + y^k + z^k =0 are Platonic 7. Cone Metrics and Maps to Tori References The Klein Quartic in Number Theory Introduction 1. The Group G and its Representation (V, rho ) 2. The Klein Quartic X as a Riemann Surface 3. Arithmetic Geometry of X 4. X as a Modular Curve References Hurwitz Groups and Surfaces 1. How I Got Started on Hurwitz Groups 2. Klein 3. Hurwitz 4. Poincaré 5. From 1893 to 1960 6. Hurwitz Groups 7. The Wider Picture 8. Conclusion References From the History of a Simple Group Plane Algebraic Curves References Eightfold Way: The Sculpture My View Ramanujan-Michelangelo Geometry-Topology Counting-Philosophy Geometry Center-MSRI Serpentine-Marble Athena-Escher Robot-Stewart Platform Location Acknowledgements References Invariants of SL2(Fq)Aut(Fq) Acting on Cn for q =2n+-1 1. Introduction 2. Group Representations and Bicycles 3. The Tricks of Felix Klein for PSL2(F7) 4. Conjectural Generators of the Bicycle of Invariants of Aut(Fq)SL2(Fq) 5. Conjectural Generators of the Bicycle of Invariants of Sp 6. Geometric Constructions 7. Applications of Contemporary Invariant Theory 8. Appendix: The Fundamental Intertwining Operator Acknowledgements References Hirzebruch's Curves F1, F2, F4, F14, F28 for Q(sqrt(7)) Introduction 1. Some Hilbert Modular Surfaces for Q(sqrt(7)) 2. Some Congruence Subgroups of Unit Groups of Orders in Quaternion Algebras 3. Modular Curves on a Hilbert Modular Surface 4. Volumes and Genera of Modular Curves on X(sqrt(7)) 5. Intersections of Modular Curves on X(sqrt(7)) 6. The Switching Involution tau 7. Intersections on the Nonsingular Model Z(sqrt(7)) of X(sqrt(7)) 8. The Symmetric Hilbert Modular Surface W 9. The Projective Plane as a Hilbert Modular Surface 10. The Ring of Invariants of G on C3 11. Orbits of G Acting on P2(C) 12. Characterization of the Hessian of Klein's Quartic 13. The Jacobian Variety of the Hessian 14. Invariant Line Bundles on the Hessian 15. Identification of the Curve F2 of Degree 12 16. Identification of the Curve F4 of Degree 18 Appendix: Matrices for Some Generators of G Acknowledgements References On the Order-Seven Transformation of Elliptic Functions 1. Classification of the Substitutions Modulo 7 2. The Function eta(omega) and its Branching with Respect to J 3. The Normal Curve of Order Four 4. Equations for the Curve of Order Four 5. The 168 Collineations in Relation to the In ection Triangle. Other Formulas 6. Construction of the Equation of Degree 168^14 7. Lower-Degree Resolvents 8. The Resolvent of Degree Eight 9. Contact Curves of the Third Order. Solution of the Equation of Degree 168. 10. The Resolvent of Degree Seven 11. Replacement of the Riemann Surface of Section 2 by a Regularly Tiled Cover 12. Explanation of the Main Figure 13. The 28 Symmetry Lines 14. Definitive Shape of Our Surface 15. The Real Points of the Curve of Order Four [Additional Remarks Concerning Some of the Literature] References Felix Klein Discovered In The 1870s That The Simple Equation X[superscript 3]y + Y[superscript 3]z + Z[superscript 3]x = 0 (in Complex Projective Coordinates) Describes A Surface Having Many Remarkable Properties, Including 336-fold Symmetry - The Maximum Possible For Any Surface Of This Genus. Since Then This Object Has Come Up In Different Guises In Several Areas Of Mathematics. The Mathematical Sculptor Helaman Ferguson Has Tried To Distill Some Of The Beauty And Remarkable Properties Of This Surface In The Form Of A Sculpture That He Entitled The Eightfold Way, Permanently Installed At The Mathematical Sciences Research Institute In Berkeley. This Volume Seeks To Explore The Rich Tangle Of Properties And Theories Surrounding This Object, As Well As Its Esthetic Aspects.--jacket. The Eightfold Way : A Mathematical Sculpture By Helaman Ferguson / William P. Thurston -- The Geometry Of Klein's Riemann Surface / Hermann Karcher And Matthias Weber -- The Klein Quartic In Number Theory / Noam D. Elkies -- Hurwitz Groups And Surfaces / A. Murray Macbeath -- From The History Of A Simple Group / Jeremy Gray -- Eightfold Way : The Sculpture / Helaman Ferguson With Claire Ferguson -- Invariants Of Sl2(f[subscript]q)·aut(f[subscript]q) Acting On C[superscript]n For Q=2n±1 ; Hirzebruch's Curves F1, F2, F4, F14, F28 For Q([symbol For Square Root]7) / Allan Adler -- On The Order-seven Transformation Of Elliptic Functions / Felix Klein ; Translated By Silvio Levy. Edited By Silvio Levy. Includes Bibliographical References. In 1879, the German mathematician Felix Klein discovered that the surface that we now call the Klein quartic has many remarkable properties, including an incredible 336-fold symmetry, the maximum possible degree of symmetry for any surface of its type. Since then, mathematicians have discovered that the same object comes up in different guises in many areas of mathematics. This volume explores the rich tangle of properties and theories surrounding this multiform object. It includes a beautifully illustrated essay by the mathematical sculptor Helaman Ferguson, who distilled some of the beauty and remarkable properties of this surface into a sculpture entitled "The Eightfold Way". The book closes with the first English translation of Klein's seminal article on this surface.