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.

### Contents

1. MSRI and the Klein quartic William P. Thurston; 2. The geometry of Klein's Riemann surface Hermann Karcher and Matthias Weber; 3. The Klein quartic in number theory Noam Elkies; 4. Hurwitz groups and surfaces A. Murray Macbeath; 5. Eightfold way: the sculpture Helaman Ferguson with Claire Ferguson; 6. From the history of a simple group Jeremy Gray; 7. On the invariants of SL2(Fq) acting on Cn, for 2n±1 Allan Adler; 8. On Hirzebruch's curves F1, F2, F4, F14, F28 for Q(√7) Allan Adler; 9. On the order-seven transformation of elliptic functions Felix Klein.

### Prize Winner

2002 JPBM Communications Award