The Eightfold Way

Felix Klein discovered in the 1870s that the simple and elegant equation x3y+y3z+z3x=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 distilled some of the beauty and remarkable properties of this surface in the form of a sculpture that he entitled The Eigltfold 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. It contains: the text written by William Thurston to explain the sculpture to a wide public at the time of its inauguration; a broad overview of the position of the Klein quartic in mathematics, with articles by Hermann Karcher and Matthias Weber (geometry), Noam Elkies (number theory), and A. Murray Macbeath (Riemann surfaces); a historical overview by Jeremy Gray; a richly illustrated essay by the sculptor, Helaman Ferguson; an exploration of related curves by Allan Adler, with new results and exposition of old ones; the first English translation of Klein's seminal article, "On the order-seven transformation of elliptic functions".