Thursday, February 23, 2012

Remembering Benoit B. Mandelbrot

This blog remembers the enormous mathematical contributions of Benoit B. Mandelbrot who died on Oct. 14, 2010. Mandelbrot is best known for coining the term "fractal", developing their basis in power law scaling and applying them to a broad range of natural processes, geophysics as well as astrophysics. He is best known for his books, including The Fractal Geometry of Nature.

Attached to this blog is one type of fractal I generated using a simple program. All such fractals are generated via a self-similarity property that enables us to compute them via repeated iterations. One can easily see from this that the self-similarity replicating property repeats at smaller and smaller scales. (Hint: Save the graphic, then click on it to "zoom in" at various points especially near the center, but also toward the periphery).

Two years ago, I showed how Mandelbrot's creations could be applied in cosmology, and specifically to a fractal dimension. Thus, the universe may well not be four dimensional, or five dimensional, but fractal dimensional with the dimension non-integer, say between 4 and 5. In the case of the cosmos at large, one generally wishes to examine concentric spheres of radii: R1, R2, R3....RN and assess using these radii the density of objects within, assuming a hierarchical configuration. In the most abstract sense then, the cosmos' fractal dimension will depend on regularity between successive expansions factors k, k', such that the dimension will be:

D = (log k')/ (log k)

and one uses radii: r_n = k_n r_o

where the cosmic radius r_o contains N_o objects, and r_1 contains N_1 objects, r_2 contains N_2 objects and so forth. In each rn we assign what we call "strata" and "sub-strata" for both particles-objects and cosmic space. It's important to understand at least in a general way how the boundaries apply between sphere radii. We take a prosaic example to try to illustrate the separation of substrata-strata by radii and also by fractal dimension.

To fix ideas, let the zeroth radius r_o = 1 pc or 1 parsec (3.26 light years). Let it possess N_o = 1 (1 object, say a star). Then we go to r_1 = 10 pc and N_1 = 100. Then, r_2 = 1000 pc, and N_2 = 10,000. If one truncates the data right here, then we can establish the expansion factor using the last two r-values:

r_2 = k_n x r_o = (1000) r_o = 10^3 (r_o)

r_1 = k_n x r_o = (10) r_o = 10^1 (r_o)

From this we determine:

D = log k3/ log k1 = 3

Note, however, that if r_2 = 1500 pc (same objects interior) then D = 3.17 and we have a true fractal dimension. A sketch comparing an "ideal" cosmic fractal and a real sector with a computed value (right, based on an early estimate obtained by Vaucouleurs (SCIENCE, Vol. 167, 1970, p. 1203) for the cosmos, of f_d = 1.8 is also shown.

This can actually be approximated using a similar process to what I showed, and a normalizing constant A= N_o/ r_o^D.

Many other astrophysical areas opened up with the application of fractals and I will show some of those over the next month.

Mandelbrot received 21 prizes and awards over his rich academic life, including the Wolf Prize for Physics, in 1993, the Lewis Fry Richardson Medal of the European Geophysical Society in 2000 and the Japan Prize in 2003.

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