The isotropy of eye movements
Next day I had a meeting with two experimental psychologists, André Bullinger and J. L.Kauffman at the University. We discussed some technicalities about the "stability" of eye movements. This time I wasn't speaking about lengtht of saccades (amplitude A) but of directions in the visual space (angle a). At that point I knew that the number of saccades decreased monotonically as a function of their length (amplitude). But what happened with their directions? I was in trouble because I didn't understand well the mathematics implied in Mandelbrot's discussion on "stable distributions". I didn't know how to find some consistent data about the distribution of the direction of saccades in visual space. It was pure chance for me that my colleagues had already obtained remarkable computer starlike graphics showing that the eye moves isotropically enough, reaching any point of the visual field in any direction! When every saccade is geometrically translated to a common origin, instead of making a leap from her last stopover (and thus pursuing the long and broken path that represents the actual trail of a real eye movement) a quite regular star is obtained as in figure 19. That suggests that there are no privileged angles for saccades. Isotropy implies that every angle (direction of sight) has the same probability. This is of course only true within physiological limits, but in the long run, with thousands of saccades, the star will stretch its rays in all the directions of sight. To me this property seemed essential to the fractal model. Now (1994) we can run a computer simulation of saccades as long as we wish based on the hyperbolic distribution of amplitude and the isotropic distribution of directions (fig. 14).
But equal probability of directions for saccades should not be confused with independence (non-correlation) between saccades. This distinction was emphasized by Mandelbrot himself, who in a letter dated in January 21, 1983 told me: "'my' jumps are 'uncorrelated', which is essential in their study while 'your' jumps, when they are large, are very correlated in direction". He proposed a method to overcome this problem (playing with angles smaller than some prefixed value). I regret to say that I never could collect sufficient experimental data to this purpose. But as a first approximation the starlike results which suggest a stable probabilistic distribution for a large sample of saccades, were satisfactory enough, at least for that stage of my research. A systematic bias in some direction , on the contrary, would be difficult to explain in my model.
Fig. 18 A star of saccades (every movement is translated to a common origin)
That evening, after so much exciting news I felt very tired. I went to dinner at the fashionable "Steak House" near my hotel, alone. I dreamt of saccadic stars and power functions. On Thursday night I wrote the first short version of my microdiscovery, La température du regard. Microgenèse d'une idée, this time at the "Café et Restaurant de l' Hôtel de Ville", a place of so many meetings with my friends Jacques et Laura Vonèche, Ariane Etienne, Carlos Valiente Noailles, Fernando and Brigitte Vidal, Maria Kodama (J. L. Borges' wife) in the following years. I can even quote my menu: "entrecôte aux pommes vapeur, trois "décis" de vin Gamay, glace de chocolat et un ristretto"! The manuscript has documented some "tâches de vin" too. I have also written in my log that I had two neighbours at the restaurant, one British subject reading a Simenon novel, and a Swiss reading La Suisse (this setting has very high probability in any Genevan restaurant indeed!). Then I went to see Camus Le malentendu au "Théâtre de Poche". I wondered if my power function wasn't also a terrible malentendu...This kind of mixed feelings is very common during the process of discovery, I think. Having reached a harbor, everything has to be put painfully in order again.
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