The eightfold way. The beauty of Klein's quartic curve (Q2732066)

The articles of this volume were reviewed individually within the 1999 hardback edition (see Zbl 0941.00006).NEWLINENEWLINENEWLINEFelix Klein discovered in 1878 that the Riemann surface defined by the equation \(x^3y+y^3 z+z^3x=0\) in the complex projective plane has a number of very remarkable properties, including 168 holomorphic automorphisms -- the maximum possible for any compact Riemann surface of genus 3 with respect to Hurwitz's bound. Thus the Klein surface represents a closed orientable real surface with 336 symmetries, together with many other geometric peculiarities. Felix Klein exhibited this surface as a quotient of the upper complex half-plane by some modular group, and it was this particular structure that revealed the fascinating geometry of this object. Since then, Klein's quartic has come up in different guises in various areas of mathematics.NEWLINENEWLINENEWLINEThe mathematical sculptor Helaman Ferguson has created a work of art, in which he distilled some of the beauty and remarkable properties of Klein's surface. His marble and serpentine sculpture, \textit{the eightfold way}, was unveiled in November 1993 at the Mathematical Sciences Research Institute (MSRI) in Berkeley, where it has found its permanent exhibition place since then. At the base is a two-color stone mosaic, representing the uniformization of Klein's surface by the hyperbolic Poincaré model. Rising out of the central tile, a seven-sided black column cups the artist's Carrara marble rendition of the surface, which highlights its tetrahedral symmetry. The name \textit{the eightfold way} is explained by the ridges and grooves that crisscross the otherwise smooth hand-polished surfaces, and which represent the same tesselation, after the surface has folded over itself. The overall effect of this fascinating mathematical sculpture is that of ``\dots a symphony of elegant counterpoint -- as if Gothic tracery and Alhambra tilings were united in one work'', as Claire Ferguson put it into words.NEWLINENEWLINENEWLINEThis volume under review seeks to explore and explain the rich tangle of properties and mathematical theories surrounding the sculpture, and that from both the geometrical and the esthetic point of view. It contains:NEWLINENEWLINENEWLINE(1) A text written by William Thurston to explain the sculpture to a wide public at the time of its inauguration in 1993;NEWLINENEWLINENEWLINE(2) a broad overview of the role of Klein's quartic in mathematics, with contributions by H. Karcher and M. Weber (geometry), N. Elkies (number theory), and A. M. Macbeath (Riemann surfaces); NEWLINENEWLINENEWLINE(3) a historical synopsis by J. Gray;NEWLINENEWLINENEWLINE(4) a richly illustrated essay by the sculptor, H. Ferguson;NEWLINENEWLINENEWLINE(5) a mathematical discussion of related surfaces by A. Adler, with new results and exposition of old ones;NEWLINENEWLINENEWLINE(6) the first English translation of Felix Klein's seminal article ``On the order-seven transformation of elliptic functions'' [cf. \textit{F. Klein}, Math. Ann. 14, 428-471 (1879; JFM 11.0297.01)].NEWLINENEWLINENEWLINEThe nine articles in this volume are highly enlightening for mathematicians, graphic artists attracted by geometry, and historians of mathematics. Thus the book under review must be seen as a great document of the interaction between art and mathematics.