Fig. 23. Fractal? clouds over the Channel
A fractal flight
"Das Wandern ist des Müllers Lust, das Wandern, das Wandern" .
Franz Schubert & Wilhelm Müller, "Die schöne Müllerin".
Paris
I returned extremely satisfied to Paris after this week in Geneva and decided to find more experimental data on saccade amplitudes during a free search. Imagine my joy, and relief, when my colleague Ariane Lévi-Schoen kindly gave me the experimental answer in the format of computerized hyperbolic distributions of saccades as a function of their amplitude! She provided me with 30 nice computer histograms collected in different experiments on vision by her team. The eye movements samples were automatically collected at small intervals of amplitude. I remember that I was uncertain, and scared too, about some few contradictory data for very small amplitudes (near 1û) in some records, but I was relieved when she told me that these cases were sampling artifacts. This was again confirmed by herself during the last seminar on the topic I gave in Paris in December.These beautiful distributions supported my idea that eye movements also follow (as words do) an hyperbolic function (in this case of frequence/amplitude) at least during the limited amounts of time allowed for eye recording in the laboratory. But I was confident that this was a general phenomenon in all settings and for all individuals. I was truly excited! I have found what I was looking for after so many days, the experimental data that could fit into a psychophysical model, a general description for a particular complex motor behavior: the hyperbolic function, so common in so many empirical fields! I started to imagine new interpretations following Mandelbrot's ideas about scaling fractals.
Fig. 23. Fractal? clouds over the Channel
London
With these findings the 14th of november I took the train La flèche d'argent to London, to meet my friend the mathematician Eduardo Ortiz at Imperial College. I wanted to understand better "the probabilistic machine of hyperbolic curves" that was certainly hidden under the saccadic tree. I remember well that curious trip by train - plane across the Channel - train. I took some pictures of the wonderful clouds over the sea. Again the intriguing shape of fractals objects. Do the clouds have an edge? I asked that question some months before to a nephew who flew with me among the clouds in a small plane over the Argentine Pampas. This question ¿tienen bordes las nubes? has remained as a family joke since.
Ortiz was very kind and helpful. We met at South Kensington. He had already read my first version of the fractal interpretation of eye movements and was fascinated with the new fractal world and its wide applications. He offered to make the computer calculations, if needed, in further eye movement experiments. I was excited when I read in J. R. Pierce Symbols, signals and noise (1961) that Mandelbrot observed that as a child grows the power function exponent, in a sample of his vocabulary, decreases from values around 1.6 to values around 1.1. I supposed that a similar developmental study could be carried out during infancy in the field of eye movements, and so I told Mandelbrot in a letter dated March 18th, 1980. Also I planned to analyze the value of D for different settings and individuals, but unfortunately time ran short and I never had the leisure nor the equipment to produce the needed experimental work. I always hoped somebody would catch the fractal idea and produce a real advance in the field of human perception. Not having this opportunity myself I choose to write this essay as a modest contribution to the study of scientific creativity instead.
The first version of this essay on the psychogenesis of an idea was written in French in July 1981 with the title: La température du regard. Réflexions sur une démarche scientifique, and sent to Fernando Vidal, then a young Argentine psychologist graduated from Harvard, now the celebrated author of Piaget before Piaget, (Harvard University Press, 1994) a most outstanding book in the history of ideas. Incidentally Fernando was my first contact with Mandelbrot, when both were at Harvard. The first letter from Fernando reported Mandelbrot's interest in my research.
5). Mandelbrot has told him that he had already thought about the similarity between saccades and fractals. This was very encouraging for me, indeed! And I am glad to testimony this clear example of intellectual honesty and modesty of the great mathematician because Mandelbrot never told me by letter or in person that he had been the first to imagine the fractal nature of saccades. This reminds me of remarkable synchronies in so many macrodiscoveries. But some were quite disturbing, as in the impressive history of non euclidean geometries. Gauss boldly wrote to Johann Bolyai's father, that he had already discovered what his son tried to prove: "to praise it would amount to praise myself; for the results to which he is led, coincide almost exactly with my own meditations"! Fernando helped me a lot to edit this Fractal Story, as I charged him with the onus of being the documentalist of several of my scientific confidences, preprints, books and papers since those good old times. I was silently preparing him, perhaps in spite of himself, to become an expert in the psycho-history of some standard microdiscoveries as well as in big science!
Fig. 24. With Fernando Vidal
In London I decided to send Mandelbrot an English version of my fractal model. I wrote him a letter on the train to Oxford, where I met my junior colleague from the Centro de Investigaciones Filosóficas, Ana Mustapic. We had lunch together at the cosy Oxonian restaurant "La Sorbonne" and visited the famous Uccello "The Hunt", at the Ashmolean Museum. A fantastic "Gibsonian perspective" with a powerful "attractor" in the middle of the visual field. The art of perspective has always been one of my favorite studies, but this is another story (III). By chance I also met Ezequiel Gallo, a distinguished Argentine historian in the train, returning to London. We were then cautiously optimistic for our country in the near future. How wrong we were ! I remember an Argentine military march the next day at the change of the guards at Buckingham Palace. It was only three years before the Falklands-Malvinas war! Who could at that time make such a terrifying prediction? This war engaged me in the field of peace studies for a couple of years. But this is another story (IX) related to Lawrence Kohlberg and Harvard. I spent some splendid days in London at the Barbican apartment of an old friend and colleague of mine, Claudio Agué, a disciple of prof. Eysenck and a remarkable pianist. Now he became my partner in a computer asssisted program for selection of personnel and shiftwork in large organizations. But this is a most recent story (X), the "ABCD story", an acronym from the names of the four partners, Agué, Battro, Cardinali, Denham.
Paris
On my return to Paris I received an invitation to participate in a Symposium on ocular movements at the Maison des Sciences de l'Homme. Several international experts of the first rank were attending the meeting. I understood immediately that I wasn't on the same "longueur d'onde". At one point I asked to make a comment and I stood up in order to draw on the blackboard the hyperbolic curve of the frequency of saccadic amplitudes and explain my idea of a power function and the fractal model underlying it. It was premature indeed, and I think nobody took notice of my proposal. Most of them were interested in rigorous experiments and I was speaking about a rough model with still poor data. Even Mandelbrot at that time was quite unknown to most psychologists in Paris! I must say that I have had the same sad impression that "no one is a prophet in his own land", when I discussed some applications of René Thom's catastrophe theory on visual illusions at the XXI Congrès International de Psychologie in 1976 (with R. J. A. Rozenstraten and A. Santos Andrade: "A visual catastrophe. The reversal of the Oppel Kundt illusion in the open field"). At this congress Piaget celebrated his 80th birthday!
I received also an unexpected invitation from our Unesco embassador, the philosopher Víctor Massuh, to participate at a meeting in Kathmandu! I had met Massuh some years before when he came to the CIF to discuss his interesting book about violence. I greatly welcomed this opportunity. The meeting was about alternative sciences and epistemologies with the grandiose title of Meeting of experts on philosophical investigation of the conditions for endogenous development of science and technology (10-14 december 1979). I wrote a paper "Vision sociale et projet scientifique: la contribution de la psychologie de la connaissance" (1979, Unesco, 79/Conf. 613) and there I went to New Delhi via Konstanz and Rome. It is time now to see on a map (fig. 25) the many trips that have been involved in this fractal story. I agree that this was not a standard way of doing a standard scientific research; so many flights and stopovers, a fractal path in itself!
Konstanz
At the beautiful University Konstanz (Sozialwissenschaftliche Fakultät, Fach Psychologie, prof. Dr. Heinz Walter) I gave a lecture about scale, visual catastrophes and perception of large open spaces (am Freitag, dem 30 November 1979). I lectured in English but I tried to answer the questions in German. It was a challenging experience, not very successful, I imagine, for my audience.
Fig.25. The highly irregular path over America, Europe and Asia, during my fractal search.
Rome
"Urbem, quam dicunt Romam, Meliboee, putavi stultus ego huic nostrae similem" . Virgilio, "Egloga I".
In Rome I had the pleasure and the honour of meeting Pope John Paul II in a general audience at the Vatican. I met him again in 1994 in an private audience given to the members of the Pontifical Academy of Sciences, where I was invited to give a lecture on "artificial intelligence and intellectual prosthesis" (October 1994). This meeting had been organized by professor Jérôme Lejeune, the famous geneticist who discovered trisomy 21, the triplet of chromosomes producing the Down syndrome. Lejeune was an expert on "maladies de l'intelligence" at the University of Paris and he became a very close friends since our participation as Vatican observers to the Ecumenical Congress on Faith, Science and the Future at MIT in July 1977.
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Fig. 26. Jérôme Lejeune in Paris |
During that congress we visited with Lejeune the Artificial Intelligence Lab, where Seymour Papert and José Valente were just starting their pioneering work with computers and disabled persons, an activity that I still practice since 1979. Later we did some work together in Paris using Logo with mentally retarded children. But this is another story (V). Lejeune, the great Catholic humanist and savant died in Paris on Easter Sunday 1994. That day I was praying for him in Azul, Argentina, at the Trappist Monastery. But this is another story, the story of my Faith (XI).
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In Rome I was particularly "scale sensitive" and I returned to the Pantheon to enjoy the view of the oculus of the dome from the floor. The diameter of the oculus is 8.30 m and the average adult estimation is about 90% larger, as I have observed some years before (with Graciela Zar de Quintas "Estudio sobre la constancia de tamaño en el Panteon de Roma" Estudos Cognitivos, São Paulo, 1976, 1) I also took some pictures to show the disparity of size in the trompe l'oeil perspective produced by Borromini's at the famous gallery of Palazzo Spada. This is another story (III).
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Fig. 27. A scale illusion in Palazzo Spada |
Fig. 28. Father Jorge Mejía, now Archbishop, in Rome
I had time enough at the airport during the long hours of waiting to write my "fractal discovery", a kind of preview of this essay. As a matter of fact the plane was delayed until next day, December 8, and we were forced to stay at a hotel near Fiumicino. Instead of making a fuss about this delay I remember how happy I was to have more time to write down my notes and souvenirs. I was enjoying a lot my explicit decision to make detailed annotations about my personal path towards a microdiscovery, as I enjoy it now, some 14 years later! I continued to write my souvenirs without interruption during the flight to New Dehli, the day after. I still keep the manuscript of 26 pages I wrote in Spanish. I met with a delay my colleagues in Nepal where I spent some unforgettable days.
Kathmandu
This Himalayan interlude was the beginning of a new friendship with Mario Bunge, the renowned Argentine philosopher of science settled in Canada, whom I knew from a Piagetian Symposium in Geneva. The Tribhuvan University of Kathmandu sponsored the Unesco's meeting.
We returned in the same plane to Paris and we kept for several years a very rich correspondance. He was so gentle as to quote and illustrate some of my works on visual alleys and Riemannian visual geometries in one of his books The Strategy of Inquiry. When Bunge received my paper, in French, about the psychological process of my fractal research he kindly supported my "dual experiment". He told me that this was the first time he read a "two track" scientific paper, one concerning facts and models, the other related to the discovery process itself and he was very encouraging.
Fig. 29. Cycling in Kathmandu.
Paris
Fraisse called me to the Hôtel Lenox, rue de l' Université, to invite me to make a presentation about my findings at the Laboratoire de Psychologie Expérimentale. The stay in Paris was at its end and I was preparing my trip back to Buenos Aires. I gave my last talk at George Noizet's seminar, not a traditional Monday morning but a Tuesday evening, because of a strike of young researchers, I believe. I think it caused some impact among my good friends and colleagues, it was amusing and had plenty of anecdotes. I only wrote an aide mémoire on the back of a large yellow envelope with the label "Régie Nationale des Usines Renault". I still have the schema of my exposé on this envelope. It contained the first computer graphics I could obtain of a family of hiperbolic functions and log/log transformations for decreasing exponents -D, a very helpful tool indeed for my empirical work on saccadic distributions. These graphs, in particular, were produced at the famous Usines Renault by a physicist who rented me his apartment in Paris and helped me with the calculations.
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