COVIMETRY Texts Site Content

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Classifying and counting clearly appear as human activities that date back to the dawn of time. In his book "The Thumb of the Panda", Chapter 20, Stephen Jay Gould recalls our natural propensity to establish classifications, and attributes it to a neurological origin. He mentions that this ordering of natural complexity stems from our limited cognitive capacities. Moreover, Jorge Luis Borges, in his tale “Funes el memorioso”, takes the opposite logical path by endowing his character with an unlimited memory, which prevents him from accessing the concept of category and organizing the knowledge that he stores. For counting, he goes as far as renaming each number without any logic, destroying any idea of a number system. Modern work in cognitive sciences has reinforced the proof by providing visible evidence that our brains are already pre-wired to process digital quantities, as well as being ready to recognize faces or process speech. It is therefore not surprising to find this question of enumeration not only in Science but also in Art.

But there are two ways of counting. We can count by enumeration, as when we have to count sheep in a flock or goods in a store, or we can do it analytically, for example when we ask ourselves how many ways there are to arrange 10 guests around a table. The analysis is mainly due to the tangible absence of the set to be counted, but also to the difficulty to separate this set into categories. Funes, the character invented by Borges acts with his means. Indeed, he memorizes objects and the tiny variations of their environment without being able to dissociate them : the neighbor's dog, in the changing light of twilight, multiplies itself ad infinitum in his memory[1]. In contrast to this uncontrolled storage, our mind usually sets milestones which will mark the differences and allow the infinite variations of the environment to be enumerated by sampling. Each sample will represent a prototype of the identified classes. Claude Monet proceeds this way with the Cathedral of Rouen or with the haystacks : he sets temporal milestones which form an inventory of characteristic light possibilities (morning light, evening light, etc.).

Bringing the infinite range of possibilities within the limits of a set of known points in order to better master it is a typical human will that we find in the Malagasy divination called Sikidy. Sikidy uses a finite number of binary digits combinations stemming from an initial choice made by the person calling on the diviner, the only person able to interpret the sequences appearing after. Should we see in this very ancient practice one of the origins of the enumerations from which the theory of probability will derive? Why not, if we admit that divination is a search to master chance. Be that as it may, it is indeed with all the necessary mathematical rigor that Pascal and Fermat founded the theory of probability by tackling the problem posed by the Chevalier de Méré. To do that, they first enumerated by analysis all the possible cases. Then, by leaning on the inventory of these possibilities, they were able to prove the correctness of the intuition of the Chevalier, used to the games known as " of chance ”. In our time, combinatorial analysis has become a very important field of mathematics, attracting the greatest mathematicians of our time (Paul Erdös or Terence Tao, for example). In the artistic field, more precisely in painting or sculpture, we have seen with Monet that the idea of inventory is sometimes underlying, and that it is rather a question of setting characteristic points of reference in an infinite set of possibilities. In contemporary art we find this tendency coupled with the use of randomness, whether in François Morellet, Vera Molnar or Manfred Mohr. Moreover, with Morellet, this randomness is also encountered in the face of an admittedly finite set, but whose exhaustive exploration would require a duration much greater than that of a human life (“Random Distribution of 40,000 Squares").

Another frequently found inventory type in contemporary art is categorical arrangement. The reason for making that collection is often subjective, and its scientific counterpart may well be the species inventory. The display of the results, in a natural history museum or in an art gallery, is moreover quite similar. For example, Tony Cragg carefully arranges plastic objects according to their color, and Herman de Vries aligns, with as much care and touching  thoroughness, branches, leaves, or even tools. In the same way as researchers do, Berndt and Hilla Becher have photographed, with the same angle and the same light, dozens of blast furnaces or water towers to then present them as scientific collections. For her part, Joa Zak scrupulously notes daily the sequence of her movements between the rooms of her home, to record it on an abstract grid in a single synthetic table reminiscent of sociological analysis.

The inventory supported by a combinatorial analysis is found in the 18th century in the famous work of Fathers Truchet and Douat. While its title clearly displays its practical purpose, it offers a superb analysis of the possible combinations starting from a binary pattern (a square divided by its diagonal into two colored halves) repeated and juxtaposed. But this kind of work did not leave the field of decorative arts until the 20th century with the appearance of geometric abstraction. However, presentation as an inventory is not so common. Indeed besides Morellet's partial inventories, the presence of such works in large museums is quite rare. Nonetheless, Sol Lewitt's "Incomplete Open Cubes" had descendants, like the works by Blinky Palermo where the colored permutations are strictly observed ("Dance a slow dance", for example). Rigor and completeness are also found in works by Knut Navrot and José Bréval. The first one lays down very simple rules and sets the goal of completely exhausting the combinatorial possibilities by presenting its results either grouped ("Limit 64 67, 1 2 3 4"), or isolated ("Attempt to exhaust possible displacements on a limited area ”). With José Bréval, the approach is occasionally similar, but his inventory of triangles ("105 triangles") with their arrangement into categories is fully in line with an inventory objective.

Some objective inventors, like Joa Zak, have opted for a synthetic, but not rigid, presentation of combinatorial possibilities. Danja Tekic, for instance, exhibits works divided into moving parts that can be positioned by the viewer ("Ars combinatoria"). Viktor Hulik also often appeals to the viewer, to whom he offers the possibility of rotating certain parts of his works on their axis, thus modifying the global aspect ("Geo-mover"). Roger Vilder, on the contrary, adopts a technical solution (motorization) which immediately brings him into the category of kinetic art. However, if he puts his compositions in motion, it is indeed in order to exhaust the possible combinations thanks to a continuous movement that sweeps through all the positions during a cycle that can be sometimes very lengthy.

My contribution to the "Covimetry" project attempts to summarize both mathematical and artistic aspects of the inventory, applied to a project which I have been working on for a long time. It is precisely the equitable division of the square by straight horizontal and vertical lines that are drawn between already existing lines. Exploring the possibilities of these rules can be done by hand, but beyond a division into 7 parts, it becomes necessary to build a formal model usable by the machine. It is this model that I present next to its graphic result : on one hand the formal expression, and on the other the corresponding design. By doing so, I hope to remind people that the mask that hides our face also hides the reflection of our brain. And, in an overly simplified way, our brain is itself divided into two hemispheres, one meant to deal with logical tasks, and the other with artistic tasks. But these two hemispheres, logic and art, communicate with each other constantly. This makes inventory a specifically human activity.

[1]Borges also examines the question of the categorization of the Universe in "El idioma analitico de John Wilkins"