The German mathematician Felix Klein discovered in 1879 that the surface that we now call the Klein quartic has many remarkable properties. 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 and a beautifully illustrated essay by the mathematical sculptor Helaman Ferguson.

The book closes with the first English translation of Klein's seminal article on this surface.

PREFACE
MSRI and the Klein Quartic

On November 14, 1993, a marble and serpentine sculpture was unveiled at the Mathematical Sciences Research Institute in Berkeley, an event that marked one of the ways in which MSRI has been reaching out beyond its traditional role. The work had been commissioned from the famous mathematical sculptor Helaman Ferguson, thanks to a generous donation from Mitsubishi Electric Research Laboratories (MERL) made for the purpose. This sculpture, and the mathematical object that lies behind it, are the subject of this book.

Felix Klein discovered in 1878 that a certain surface, whose equation (in complex projective coordinates) he gave very simply as x3y + y3z + z3x = 0, has a number of remarkable properties, including an incredible 336-fold symmetry. He arrived at it as a quotient of the upper complex half-plane by a modular group the group of fractional linear transformations whose coecients are integers and that reduce to the identity modulo 7. Since then, the same structure has come up in di erent guises in many areas of mathematics. ...

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Edited by Silvio Levy
Cambridge University Press

• Interdisciplinary approach within mathematics
• Exposition by first-rate mathematicians of an object of great beauty
• The work of a mathematical sculptor, with illustrations

Contributors
William P. Thurston, Hermann Karcher, Matthias Weber, Noam Elkies, Helaman Ferguson, Claire Ferguson, Jeremy Gray, Allan Adler, Felix Klein