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Mandelbrot and Akaike: from taxonomy to smooth runways (pioneering work in fractals and self-similarity)

Mandelbrot on taxonomy (from 1955; the first publication about fractals that I know of):


Searching for Mandelbrot on the blog led me to Akaike, who also recently passed away and also did interesting early work on self-similar stochastic processes.

For example, this wonderful opening of his 1962 paper, “On a limiting process which asymptotically produces f^{-2} spectral density”:

In the recent papers in which the results of the spectral analyses of roughnesses of runways or roadways are reported, the power spectral densities of approximately the form f^{-2} (f: frequency) are often treated. This fact directed the present author to the investigation of the limiting process which will provide the f^{-2} form under fairly general assumptions. In this paper a very simple model is given which explains a way how the f^{-2} form is obtained asymptotically. Our fundamental model is that the stochastic process, which might be considered to represent the roughness of the runway, is obtained by alternative repetitions of roughening and smoothing. We can easily get the limiting form of the spectrum for this model. Further, by taking into account the physical meaning of roughening and smoothing we can formulate the conditions under which this general result assures that the f^{-2} form will eventually take place.

P.S. I’ve placed this in the Multilevel Modeling category because fractals are a form of multilevel model, although not always recognized as such: fractals are hi-tech physical science models, whereas multilevel modeling is associated with low-grade fields such as education and social science. The connection is clear, though, when you consider Mandelbrot’s taxonomy model or the connection between Akaike’s dynamic model of roughness to complex models of student, teacher, and classroom effects.

I met Mandelbrot once, about 20 years ago. Unfortunately, at the time I didn’t recognize the general importance of multilevel models, so all I could really do in the conversation was to express my awe and appreciation of his work. If I could go back now, I’d have some more interesting things to ask.

I never met Akaike at all.


  1. Bill Jefferys says:

    Of course, fractals are a lot older than this; Sierpinski (1915), for example, and Peano (1890) published pioneering work on fractals.

  2. Andrew Gelman says:


    Yes, but Akaike and Mandelbrot were going further by developing models for natural (or social) phenomena. This is a step beyond mere mathematics, no?

  3. Bill Jefferys says:


    Sure, I just wanted to point out how old the mathematics was.

    I recall vividly when Mandelbrot's book came out. John Archibald Wheeler, who was then on the University of Texas faculty (at the time we didn't have a mandatory retirement age, Princeton did), had bought a copy and was showing it off to our Friday lunch group. He knew how important it was.

  4. Mark Palko says:

    At the risk of being one of those people who drags every conversation around to his favorite topic, Mandelbrot had really interesting comments on his own education here:

    <a href="” target=”_blank”>…” target=”_blank”>

    He might be the best example of how visual and symbolic thinking can complement each other.

  5. Andrew Gelman says:


    Thanks for the link. Mandelbrot's essay is fascinating (even if he maybe overdoes the whole "maverick" thing).

  6. Monte Davis says:

    > he maybe overdoes the whole "maverick" thing)

    Yes… think also of Richard Feynman's thousand-times-retold bongos, lock-picking etc. (I think some of that was backlash against the awe that many young American physicists of the 1940s felt — or felt they ought to feel — among the violin-playing, Goethe-quoting emigres from Europe. "So I'm a plain-spoken wisecracking Brooklyn kid, I'll be the super-plain-spoken wisecracking Brooklyn kid."
    It also feeds into science writers' weakness for "See, he's a wild and crazy guy, not some inhuman egghead." And as human beings will, the scientists play up to it.

  7. Probably the first fractal discovered was the Cantor Set by Cantor in 1883.

  8. Andrew Gelman says:

    Yes, I should probably amend that to "the first publication on fractals as a potentially applied model" or something like that.

  9. Bill Jefferys says:

    Just to mention some other early practical examples:

    Cantor sets appear in applied physics. KAM theory (ca. 1962, though Kolmogorov's conjecture was earlier — 1954) shows that in general, under perturbations, the orbits that persist are a Cantor set in the phase space.

    Much earlier, G. D. Birkhoff did work on homoclinic and heteroclinic points in dynamical systems, which generate chaotic motions; Saturn's moon Hyperion is believed to rotate chaotically due to this.

    Fractals also arise naturally in dissipative dynamical systems, such as the Lorentz attractor (1963).

  10. Nick Cox says:

    John Venn had fractal material in the 3rd edition of "The logic of chance" (1888). It is online at

    This is the Venn recalled by the name of Venn diagrams (which were used earlier by Euler).

  11. Bill Jefferys says:

    Speaking of Lorentz, this cartoon appeared today. Scroll down to the large version and look at the second cartoon:

  12. Nick Cox says:

    In this context for "Lorentz" read "Lorenz".

  13. Bill Jefferys says:

    Oops! Yes, it should be 'Lorenz'.

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