Preface

From curves to surfaces

Ten years ago, I wrote the preface to the “garden of curves” and here I am faced with the “garden of surfaces”. Two thousand pages of erudition, ten years of work! Is HAMZA KHELIF a Benedictine? The dictionary you have before your eyes is a mine of information, a summary of more than two thousand years of mathematics, like an invitation to travel into the world of geometry where “all is order and beauty, luxury, calm and voluptuousness”.

Is geometry virtual or, on the contrary, does it inform us about the concrete world around us? It cannot be denied that twenty-first century mathematics has an increasingly marked tendency towards an increasingly unbridled abstraction. Most contemporary articles on geometry do not contain figures, simply because the objects they study “live” in spaces of dimensions greater than three, whose geometry is not Euclidean, or even Riemannian. Shouldn't we go back to the sources, draw concrete surfaces, or even sculpt them, print them in 3D, to be able to touch them?

It may not be useless to return to the etymology of the word "surface". Unsurprisingly, it comes from Latin, with the prefix "sur", which obviously means "above", and "face" whose meaning is much more general than "face". In Latin, "facies" evokes a form, in the broadest possible sense, even if it is often the form of the human body, and more particularly that of the face. This is why the definition of "surface" found in the Littré is "Exterior, outside of a body. The entire surface of the body". This is interesting : a body, a form, or an object, presents itself to us through its surface which is its visible part. The interior of the object is inaccessible to our senses. This is why surfaces play a vital role in geometry : they are what inform us about the world around us. This is also how Diderot’s encyclopedia presents things : “In bodies, the surface is everything that appears to the eye. The surface is considered to be the limit or the exterior part of a solid.” In this sense, surfaces are the most important objects in geometry. Curves, on the other hand, only appear as derived objects : we only perceive them as apparent contours of surfaces : in a way, curves are the edges of the edges of bodies!

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So what use is a dictionary of surfaces? In my opinion, the traditional role of dictionaries has profoundly changed because of the Internet. The somewhat magical power of a dictionary, whether stored on a shelf or on a computer disk, is that it offers a random walk. Open it at random, you will surely find a surface you did not know existed, but above all the alphabetical order will have surprises in store for you. The surfaces that are close in the dictionary, but probably very different geometrically, will encourage you to visit other pages that, themselves, will make you visit others. The number of possible walks is certainly finite but it will be enough for a whole life! To paraphrase Raymond Queneau and his "One Hundred Thousand Billion Poems", the dictionary of surfaces "is after all a kind of machine for making poems, but in limited number; it is true that this number, although limited, provides reading material for nearly two hundred million years (reading twenty-four hours on twenty-four)".

To conclude, I would like to add a surface to this dictionary. It is a work by the Swiss artist Max Bill, dating from 1951 and entitled "Tripartite Unity". Since this is a surface with a boundary, I will let my reader determine whether this boundary is connected, whether the surface is orientable, and what its genus is…

It only remains for me to thank HAMZA KHELIF, a great lover of geometry, for having written this book, which will enable a hundred thousand billion journeys into the world of surfaces.

ÉTIENNE GHYS

Permanent Secretary of the French Academy of Sciences

Paris, April 25, 2020