This past weekend I learned the sad news that Benoit B. Mandelbrot , the father of fractal geometry, had passed away at age 85. I was fortunate enough to have spent 14 months working for him in 1991 and 1992. I had just completed my M.Sc. under the supervision of Przemyslaw Prusinkiewicz, who had spent some time on sabbatical with Mandelbrot. This paved the way for fellow student Dave Fracchia (Ph.D.) and me to join Mandelbrot’s Fractals Project. What an opportunity! Brien Maguire, now Dean of Science, said to me then that I might be one of those people who puts his first job at the top of his resume.
Since I’ve heard the news, I’ve spent a lot of time reflecting on my experiences with Dr. Mandelbrot and that period of my life. I have many happy memories of working with him and his group. I was happy to see and hear him in a recording of his TED talk from February of this year:
In 1998, I wrote a piece for “Textshop” (which you will find further along in this post) in which I reflected on the fractal nature of my life’s journey. Now, the impact of my time with Benoit Mandelbrot is even more clear.
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| "My Life” (1997) by Daryl Hepting. This fractal was created by repeatedly transforming ever smaller copies of the letters L, I, F, and E. The transformations of LIFE represent the directions in which one is pulled through one’s life choices. Complexity and beauty arise from the collection of many simple choices. This image first appeared in Textshop. |
by Daryl Hepting
Originally published in Textshop, Winter 1998
University and education have been central to most of my life. As I continue my journey, my formal education becomes informal: I learn about my own life and get a glimpse of my big picture. No doubt, my picture is influenced by my past. I am from Saskatchewan, I have worked for Benoit Mandelbrot, and many other things. Saskatchewan has given me an appreciation for the sky and the infinite horizon. Mandelbrot has given me a means to embrace that infinity.
Mandelbrot is the father of fractal geometry and a wonderful man. His dedication, his precision, and his stories have all been inspirational to me. And they have been the opposite too, whenever I thought myself incapable of his challenges or unwilling to play at life. Fractals are his life’s noble work: he has spent many years learning to see the world through a strange new pair of glasses. And now he shares those glasses with others. The image that accompanies this text is an expression of what I have learned to see through these glasses: a simplicity that calls out from beneath worldly complexity.
What do fractals say about life? It may not be such a difficult leap to think that they might contribute something profound: history is full of examples where mathematics has been used to describe essential parts of our lives. Biologists define life by a list of properties that most living things have, most of the time. That fractals require the same style of definition only lends to their organic appeal. Fractal properties include detail at all scales, some form of self-similarity and a simple definition. Fractals were discovered in the richness of life and they describe our natural world with startling clarity. They can be seen in the record of the Nile river, the price of commodities, the trees outside my window, and the structures in my body. They bring into focus the details and repeated themes of musical scores and the structure of the universe. Before the word “fractal,” objects found to possess fractal properties were definitely considered the exception rather than the rule. My life exhibits detail, as the patterns of my life are repeated in various ways throughout it. Could the complexities of life’s patterns, like fractals, result from a very simple foundation? At some level, I know that life is simple and straightforward. I see the character of Forrest Gump as an unwitting role model: his life remained free of the complexities that others created for themselves.
When I wonder what I should do with my life, the answer I find is simply: live it! There are some days when I am not up to that imperative, some days when I would rather lie in bed and rationalize my life away. I prefer to create shades of gray, rather than to devote my life to the discipline of black and white: there are some decisions that some days I dare not make.
Escher is an artist whose work is recognized by Mandelbrot as “pre-fractal.” He employed symmetries to intricately fill planes. One of my favourite pictures, called “Angels and Devils,” is by Escher. Angels and devils are interlocked throughout and for me this expresses both positive and negative views. The angelic interpretation is that no matter how bad things look, if I live my life as an expression of my principles, my simple rules, then I can successfully navigate amidst the devils. The devilish view is that life is really as complex as it seems: sometimes a necklace of pearls and sometimes something more sinister. And perhaps this is the view I more commonly take: I see intricate structures unfolding, as my life collides with others. Yet it is easy for me to mistake these artifacts for my life, and gather further evidence of life’s complexity. Whenever I view my life in terms of my past, I mask the simplicity of life. Each day I stand at the precipice and each day might be my last. What must I do? If I defer to my past, I trace out the same pattern in every increasing detail. If I choose to act rather than react, I create a new pattern. Like the butterfly flapping its wings, somewhere, I cannot fathom the ramifications of my simple acts.
As I discern my path in life, I realize that it is truly my own path and it is here to be explored, and lived. I sometimes hold myself back by thinking about what might have been: if I hadn’t taken Chris Fisher’s geometry course or any number of other chances. Then with clarity, I focus on the present and the future: though I cannot see it clearly now, I know that it lies ahead along the intricate path I am creating by living each moment.
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