Preface

     The world of curves is a lush jungle populated by exotic objects with bizarre names: cardioids, lemniscates, limaçons, strophoids, tautochrones, etc. Mathematicians have long been fascinated by curves; from Apollonius and his conics, to Frey and his cubic – recently used in the proof of Fermat's conjecture – via Descartes and his  folium, or Bézier and his curves for vector drawing.

  As always in mathematics, two simultaneous movements complement each other harmoniously. On the one hand, the somewhat chaotic enrichment of the pool of examples risks leading to an inextricable situation in which we do not understand

Nothing anymore as the objects that present themselves seem diverse. On the other hand, the highlighting of structures makes it possible to group together apparently different objects, and offers a global intelligence of the landscape. Neither of these two movements could exist without the other.

 In the case of curves, there are many examples of these two trends. The discovery of Cartesian coordinates in the seventeenth century opened the door to a wave of innumerable examples that swept through the nineteenth century. Curves of degrees one and two have been well known since the ancient Greeks: venerable straight lines, ellipses, parabolas, and hyperbolas. By degree three, the situation became complex. Newton tried to put these cubics in order and distinguished 72 types (which would become 219 later, when Plücker took up the question again). Curves of degree four seem to be so diverse that we had to limit ourselves to studying particular examples. Beyond that, it was total darkness... Riemann's genius will illuminate all this. The use of complex numbers, projections, and more generally birational transformations, will reduce the study of cubics to a single invariant, its modulus. The concepts of Riemann surfaces, genus,  and modulus space have revolutionized the way we think about an algebraic curve. In the same way, the twentieth century brought new needs and fractal curves, for example, first appeared as isolated examples (Koch's curve, Sierpinski's curve, etc.) before becoming aware that they "exist" in nature, as in Brownian motion. The time is ripe to develop a solid theory of dimension – whether topological or fractal – which in turn sheds light on the previous examples. Theory and examples will never be dissociated in mathematics.

Once we have understood that Diocles' cissoid, Maclaurin's trisectrix, or even the astroid, are never anything but curves of genus zero, and therefore birationally equivalent, should we relegate them to oblivion? Some mathematicians think so. It is no longer important for a professional mathematician to know Cassini's ovals or the lituus, but the fact that he has become acquainted with many curves is still extremely formative. For the apprentice mathematician, the study of curves, as practiced in secondary education and in the first cycle of university, will play the same role for a long time as that of the practice of scales for the apprentice musician.

HAMZA KHELIF's book is a dictionary of curves of all kinds, in alphabetical order, as it should be.

"Spike, epicarroid, epicycloid, epitrochoid, equiangle (spiral), Cayley equipotential, equitangential, error (curve), ampersand, fractal star, starfish, stirrup, Eudoxus (kampylus of), etc."

   A veritable inventory à la Prévert:

" A tripe, two stones, three flowers, a bird, twenty-two grave diggers, a  lover, the raccoon, a lady so-and-so, a lemon, a loaf of bread, a great ray of sunshine, a groundswell, a pair of trousers,  etc."

Alphabetical order may not be appropriate for classifying curves but, just as in Prévert's inventory, improbable associations between words that have nothing to do with each other engender poetry. You have to open this book at random and wander around, jumping from curve to curve!   Just for fun. I would have liked  owning this book when I was a student.

My favorite curve? Maybe Klein's quartic. His equation

doesn't reveal much. Neither is its outline in the plan. But if we observe it in its natural environment, the complex projective plane, we admire a jewel whose

168 symmetries form the simple group .

   I would like to modestly add a curve that is not in the dictionary

and which illustrates the importance of curves for mathematicians: the Poincaré cold curve. Between November 7 and November 17, 1873 (or 1874), Henri Poincaré had a cold. As a young student at the École Polytechnique, he informed his mother by mail and the best way to do this was of course by a curve! The story doesn't say what the mother thought of her son's letter... Here is a copy of the letter, as it is found on the wonderful site www.univ-nancy2.fr/poincare :  

     There are many kinds of mathematicians. Some are discoverers of unknown territories, others are builders of majestic structures, and others are contemplatives. The mathematician HAMZA KHELIF likes to contemplate the curves encountered in his garden, but he likes to show them to his visiting friends even more.

Étienne Ghys

Lyon, 26 September 2009