The fractal connection
"So, naturalists observe, a
flea hath smaller fleas that on him prey; and these have smaller
fleas to bit'em; and so proceed ad infinitum".
Jonathan Swift.
It
is interesting for the purpose of this journey to trace back the
"fractal source" itself. As I have said my first contact with
Mandelbrot's ideas occured during the elaboration of my Ph.D. thesis
in Paris. Piaget's ideas in perception were in the main focus of my
interest at that time and Mandelbrot has been a member at Piaget's
Center of Genetic Epistemology and the author of some interesting
reflections about cognitive equilibrium in one of the first volumes
of the Etudes d' Epistémologie Génetique. Mandelbrot
reappeared in my intelectual life only eighteen years later! The
encounter happened through the reading of one of the famous
"Mathematical Games", of Martin Gardner's fame, in The Scientific
American of April 1978 ("White and brown music, fractal curves and
one-over-f fluctuations"), a permanent source of intriguing new ideas
for many persons of my kind. The whole article, and particularly
Mandelbrot's fractal curves somehow made a click in my mind. In fact
at that time I was fascinated by the problem of "scale" and "scaling"
and performing a wide series of experiments in the large scale of big
open spaces in Brazil, but this is another story (III).
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Figure 8. Paul Fraisse at the Simpósio de
Estudos Cognitivos, Gramado, 1977
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Political and economical
reasons forced me in 1972 to search for work outside my
country. Some Brazilian universities accepted my proposal
and I began commuting every month by plane from Buenos
Aires, where I lived with my wife and our three children, to
different towns in Brazil. This decade was, in many senses,
one of the most productive times of my scientific life. I
edited a Journal called Estudos Cognitivos and directed a
large group of psychologists Grupos de Estudos Cognitivos in
several universities in Porto Alegre, Curitiba, São
Paulo, Rio de Janeiro, Araraquara, Rio Preto and Campinas.
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The influence of my training in
Paris and Geneva was evident in this enterprise and I had the
pleasure of hosting Paul Fraisse and Bärbel Inhelder in some of
our cognitive symposia in Brazil. I got to have more than 30
assistants and colleagues working under my direction in the
perception of open spaces and to have published around 20 papers on
the subject.
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Unesco gave me in 1975
the opportunity of working in the Man and Biosphere Project
(MAB), under the direction of the great architect and urban
planner Kevin Lynch at the Massachusetts Institute of
Technology. We studied how small groups of young adolescents
used and valued their spatial environment in different towns
in Argentina, Australia, Mexico and Poland. This first
international study was edited by Lynch and published as
Growing Up in Cities (MIT Press, 1977).
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Figure 9. Bärbel Inhelder at the
Simpósio de Estudos
Cognitivos,
Rio de Janeiro, 1979
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I decided to continue this kind of
work in Brazil too, with my friends the Argentine architect Eduardo
J. Ellis and the Brazilian psychologists Eny Caldeira and Orlene
Capaldo. A book, (still unpublished, but available in a copy at the
MIT Library) La imagen de la ciudad de los niños , offered to
Kevin Lynch, was the result of this long research. The last pages of
the book were dedicated to the problem of scale and to the idea of a
"fractal scaling" in the cognition of places. This was the first time
I could apply Mandelbrot's fractal theory in an explicit form in one
of my works.
The new idea was that the mental
representation and reconstruction (via maquettes or drawings) of a
very big space follow the same process as in the image of a small
part of it. For example a child makes exactly the same use of Kevin
Lynch's landmarks, paths, frontiers, etc, in the reproduction of a
large garden or of a small playground. This idea of applying the
concept of "self-similarity" to the image of urban spaces occured to
me when at MIT preparing the MAB project with K. Lynch. I remember I
was invited to discuss some of my experimental findings in a seminar
of the Department of Urban Studies. Only some hours before the
meeting I was busy at the library in the search of Mandelbrot's new
book on Fractals (published in 1977). I needed it badly but it wasn't
at the MIT Coop and I called Harvard Coop for help (this is in my
travelling notes, believe it or not !). I remember the blue cover of
the first edition and the search for the second and modified edition
in Paris, where nobody seemed to know it, one year later. I think my
first presentation of "scaling fractals in the cognition of open
spaces" at MIT was quite naïve, and perhaps some people smiled
at my intent. But it pushed me to continue the (hard) study of
fractals much further.
