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## Eric Vandendriessche’s work

**Minutes of the meeting of the 26/05/05 on Eric Vandendriessche’s work**

Were present : Agathe Keller, Alain le Mignot and Eric Vandendriessche.

The discussion starts freely on the different paths by which you can obtain a figure when playing a string game. Eric Vandendriessche explains that in certain regions of the world people know several paths that lead to a same figure, when in other places, on the contrary, the recalling of the path and that of the figure goes together. Agathe Keller draws an analogy with the algorithmic working in the mathematical traditions of China and India : to have several procedures to obtain a same result is linked with the search for proofs, a way to verify that a result is correct.

Alain le Mignot asks what is the status of string games in traditional societies. Eric Vandendriessche explains that for the time being he doesn’t know much. He also wonders if the context is really interesting in an ethnomatical perspective. This is a real question. He thinks that the songs and stories linked to string games can sometiomes help to memorize them. However, sometimes the songs are voiced after the figures are made. Eric adds that anthropologists where interested in string figures because they though it was a way of proving transmissions. The question however is if sub-procedures or common figures that can be found in different areas of the globe proove an exchange or result rather from the inner constraints of the game.

Agathe Keller explains that once again analogies can be drawn with the situation in the History of Mathematics. In the scholarly tradition of India, sûtras are composed with mnemonic trics like puns to help one to memorize algorithms. Furthermore, the study of mathematical exchanges, when we cannot find a text translated from one language to another, raises similar questions. When one finds similar problems dealing with irrationals in India and China, does this mean that there was an exchange or that such irrationals appear because the they always end by appearing when one deals with right triangles. Eric replies that indeed comparaisons of « came string games » are always in fact with games that are similar but not exactly the same, just as in the history of mathematical transmissions.

The discussion then breaks of on the question of evaluating if string games are associated to the idea of knowledge by those who practice it. Agathe Keller returns to the words used by Eric in his DEA to qualify the intellectual processes linked with string game. Is the word « traces », a good choice ? She feels that it suggests that the intellectual work was seperated from the gestures linked to the string games, and prefers the word « expression » which would suggest that the movement and the thought go together. Again and again the discussion comes back on how to speak of knowledge : there’s a practical knowledge, a know-how, which imoplies a knowledge of the system without a theorization of it. For Eric Vandendriessche, it is impossible to decide for the momment if those who invent new figures, who are « good » at it, have a theory of their practice. We talk about music and language that can be theorized as a system but practiced by very creative people who know the system but do not think in theoretical terms. The notion of motif or of musical phrase with its rythm and its syntax can express the way a string game can at the same time be a process and a set of topological/geometrical objects.

Agathe Keller asks Eric Vandendriessche to specify how he links the fact that a string game can be seen as a process or as a statical geometrical object. Eric Vandendriessche evokes mathematical theory linking string games with knot theory. A group of japaneses computor scientists have obtained a caracterisation of the object « string game » using a polynome of strings, derived from the Vaughan Jones polynome. They have shown that for each « stable » step of a string game (what Eric calls a « normal position ») one can associate a characterising polynome. Then one needs to study the transformations such an polynome undergoes under the action of different elementary operations. A string game can then be described as a series of polynomes. Eric Vandendriessche underlines that such a caracterisation of the object « string game » is based on the description of a certain number of static states, and thus cannot really describe movement.

To study a corpus of string games, or to compare corpuses, Eric Vandendriessche has elaborated a symbolical language enabling one to describe the long procederes of a string game in one formula. He shows us a program that he has written, which enables him to find the frequence of an elementary operation or a sub-procedures. Agathe Keller remarks that such formulas could enable the creation of an online data base where everybody, anthropologists and amators of the ISFA, « International String Figures Association » could enter their corpus. Agathe Keller wonders if other attempts at symbolising string games have already been carried out. Eric Vandendriessche replies that a japanese amateur uses such a language to share string figues with ISFA members. Agathe Keller says that one should prepare and prospect well before creating a data base in order to find which code to yse. It would really be stupid not to benefit from somebody else’s work because of a difference in the system used.

Eric evokes developments in knot theory that could be interesting for a mathematical description of string games. Recently the american mathematician Louis Kauffman has published a certain number of results around « tangles » (french « enchevêtrement »). This object is constructed from a sphere. A first string crosses the sphere. The second does the same but after having tangled itself inside the sphere, a third one tangles itself around the two others, etc. until the nth string. An « n-tangle » is created. The different steps of a string game can be identified to a n-tangle. The study of a 2-tangle has given interesting results to biologists to modelize different possible recombinations of DNA when it has been cut by enzymes and reglued at other places. There is a theorem that was proved first by John Conway, then differently by Louis Kauffman. A certain class of 2- tangles is in a bi-univocal relation with a set of rattionals to which the infinite point is added. One can show that each 2-tangle of this class (called rational tangles) one can associate a continuous fraction. Eric Vandendriessche explains that string games can be seen sometimes as a sorth of arithmetics of torsions.

We regret that Sophie with her knowledge of weaving strings is not here to react.