#35: The Eightfold Way: The Beauty of Klein's Quartic Curve
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MSRI Publications  Volume 35
The Eightfold Way: The Beauty of Klein's Quartic Curve
Edited by
Silvio Levy
Felix Klein discovered in the 1870's that the simple and elegant
equation
x^{3}y +
y^{3}z +
z^{3}x
(in complex projective coordinates)
describes a surface having many remarkable properties, including
336fold 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.
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 Murray Macbeath (Riemann
surfaces).

A historical overview by Jeremy Gray (reprinted).

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
orderseven transformation of elliptic functions''.