The Pathways of π (1999)

I had long dreamed of finding an ideal system, simple to understand, easy to realize with my customary tools (ruler and protractor), that would be capable of creating lines in a succession that was, at least in appearance, unpredictable and infinite. 

In spite of numerous sporadic attempts, such as my drawings from 1952, it was not until 1998 that I managed to make this dream a reality. 

For those who may be surprised that it took me so long to arrive at something so simple, here is a brief history of my trajectory.

In 1958, with my 6 répartitions aléatoires de 4 carrés noirs et blancs d’après les chiffres pairs et impairs du nombre π, I had found a system whose development, which can be imagined to be infinite, already used the decimals of π. But its binary principle reduced the path it created to a sort of rectilinear paved alley that would have offered no surprises and thus become boring well before it had attained infinity. 

At the time, true enough, I was interested above all in random repetitions of surfaces (squares, triangles) or straight lines. Broken lines, which would later be “the solution,” seemed to me then to contain deviationist, crypto-Expressionist, even crypto-Baroque tendencies that I categorically refused.

Indeed, I realized afterward that one of my systems’ reasons for being, and not the least of them, was to simulate and parody genuine artistic movements, those that were illustrated by genuine masterpieces, executed by genuine artists as the result of innumerable subjective and brilliant decisions. I knew, of course, that my systems were well disposed to turn simple elements into pseudo-Constructivist works, and I accepted as much, but I did not love these systems enough to let them create their own nonsense: pseudo-Impressionism, pseudo-Expressionism, etc., all the way to the pseudo-Rococo of today.

I no longer recall whom it was who said: “One advantage of systematic art is that it allows one to make things one does not like, which considerably widens the field of creation.” And anyway, the (relatively) greater success pure, hard geometry enjoys today has dulled the exclusive love I once had for it, and my interest has turned instead toward more equivocal and frivolous forms, though they are of course still systematized. 

This digression brings us to 1998, when I was at last able to realize my dream of an infinite line of unpredictable succession that generates itself on its own—all thanks to the expiration of my allergy to broken lines, a busted accordion, the decimals of the number π, and a computer in good working condition.

— My tolerance for broken lines was the first condition that enabled this gracious and piquant path toward infinity.

— The image of a damaged accordion came to me during my first attempts. An accordion in good condition presents a regular zigzag in which all the segments form among themselves equal angles that can all, together, enlarge or reduce their aperture. My zigzag, on the other hand, would be inadvisable in a serious accordion: its angles are all different from one another and fixed permanently in place.

— The degrees of these angles are determined by the sequence of digits of π after conversion. The simplest conversion is 1 = 10°, 2 = 20°, 3 = 30°, 4 = 40°, 5 = 50°, 6 = 60°, 7 = 70°, 8 = 80°, 9 = 90°, 0 = 100°. But any other value can be given to the decimals. I arbitrarily chose four equivalencies: 1 = 10°, 1 = 30°, 1 = 45°, 1 = 90°, whose path I illustrated from 13 to 50 and 3,000 decimal places. I also made two little expeditions of 3,000 decimal places with the equivalencies 1 = 179° and 1 = 180° (with 1° of difference: the explosion and the straight line).

— Of course, I must pay homage to the computer in good working condition as well as to the programmer Rémi Bréval and to my assistant Philippe Lamy, thanks to whom I was able to save a few dozen years in adding the sequence of hundreds of thousands of decimals to those I had prospected over the course of a year spent with only the aid of my ruler and my faithful protractor.

Yes, thanks to all these conditions, I can get lost along the way, with infinite jubilation, in these new paths toward infinity. And I can also share my joy with those who are connected to the Internet and to my wild imaginings: all they have to do is enter www.culture.entreelibre.fr. Rather, they will share neither my joy π nor my boredom, but keep them for themselves as they discover their own pathways, which have every chance of being uncharted, given the “almost infinite” quantity of possibilities. 

Why π? 

Yes, of course, any sequence of digits “at random” would have resulted in kinds of pathways more or less similar to those generated by the decimals of π. For example, the numbers in a telephone directory (with local area codes removed), which I often used back when my source of decimals of π did not have enough digits. 

But on the one hand, as in the fifties, I prefer that the sequences of “random” digits that I use be verifiable, that no one may accuse me of cheating in order to, for instance, make the work look “better.” And then on the other hand, I have enjoyed, in recent years, reading various works on π and its decimals (such as Le fascinant nombre π by Jean-Paul Delahaye); I have been amused, even impassioned, by the importance that those somewhat nutty philosopher-mathematicians have managed to ascribe to this series of numbers. 

The proof can be found in their relentlessness, since antiquity, in adding more and more decimals to the sequence, especially without practical reasons for doing so after it surpassed hundreds of billions. 

Some are still expecting the decimals to begin repeating themselves in a loop, for a digit to appear more often than others, for the conversion of the digits to letters to yield the meaning of π and, why not, that of life. In an American film that came out this year, whose title is simply π, the decimals of π spiced up with a few other elements cause a computer to implode after it discovers . . . God! 

So it is once again my taste for parody, for simulation, that has pushed me to take part in this adventure, as frivolous as it is systematic. Unless, perhaps, one day, one of my paths, or one of yours, reveals . . . 

Note: The pathways I have just described, for better or for worse, are the basis of my series π picturaux and are named π piquants. Maintaining the same skeleton but dressing it up with quarter-circles, I first made a π rococo; then, with full circles, came π cycles; with colors added to the skeleton, π color; with thickened and prolonged segments, π puissants; and so on. At some point soon I plan to make a catalogue raisonné of this series and then to prohibit myself from creating any new ones (other than as gifts for weddings or significant birthdays). I have done the same in the past for my Géométree, Géométrie dans les spasmes, and Steel Life series.