Victoria Fractal Tree

August 27, 2006


Provided and copyright by: Tom McGuire
Summary author: Tom McGuire

The photo above shows a mature eucalyptus tree near Hamilton, Victoria, Australia, that appears to have replicating patterns. Fractals are a branch of mathematics discovered by Benoît Mandelbrot, a Polish mathematician educated in France. In looking at complex natural phenomena, Mandelbrot realized that some of them could be modeled mathematically with relatively simple operations. Reiteration produces visible objects of unlimited complexity and scale-independent similarity. This can be observed with either orders of magnitude reduction, or magnification. Fractals can also be characterized as geometry of intermediate dimensions; thus a fractal pattern on a plane may have a dimensionality of 1.4. The resulting object is certainly more complex than a line, but doesn't fill a surface like a plane of two dimensions.

I had the pleasure to meet Dr. Mandelbrot at the Thomas J. Watson IBM Research Center in suburban New York while I was involved in educational outreach at that facility. A research collaborator. Richard Voss, (the father of one of my students) has created artistic and natural looking landscapes using 100% computer-generated shapes. Those with a dimensionality of approximately 2.7 appear the most natural to human observers. Why nature seems to have this particular fractal signature is now known. In fact, the judgment is subjective. Moving from the mathematical to human perceptions of our world is a fascinating field of investigation and speculation.

Fractals are sometimes called the interface between order and chaos, and are closely related to investigations of deterministic chaos. In 1994 the LTCM partnership used the Black-Scholes model to generate substantial earnings for major investors until it appeared that the self-renewing chaos of the market overcame their system within four years, rendering their large-scale methods futile.

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