Benoit Mandelbrot's eyes are closed for now, but he has left us in this world with eyes more open. The Jewish mathematician from France will always be remembered for his wonderfully elegant yet majestically powerful gift to mankind  a mathematical discovery, a new geometry of infinity. Mandelbrot's notion of "fractals" has launched an entirely new genre of art, provided a tool for unraveling the greatest complexities of nature, modeled with mastery the meanderings of the stock market, and perhaps most miraculously of all, inspired bored high school students to realize that math is indeed cool. Put most simply, a fractal is an object in which every part resembles the whole. Nature is full of things that have a fractal structure. The fern frond at left looks quite realistic but it is actually just a graph of a mathematical formula called a Mandelbrot set. Nonetheless the manmade version is uncannily similar to a real fern, such as the one pictured right. Simulations such as these have been beautifully rendered for all kinds of natural objects, ranging from clouds to riverbeds, and from seashells to broccoli buds. Often the simulations are indistinguishable from the real thing. The uncanny ability of mathematics to model nature at every scale was so wondrous in the eyes of Albert Einstein that he commented that the only thing unintelligible about nature is that it is intelligible. "Atheistic" Physicist Paul Davies wonders if our mathematical insights into nature are just products of our imagination or actually part and parcel of the phenomena we study. If math describes nature so well, how could nature just happen to line up with our mathematical musings? He concludes that it cannot be that we are imposing our mathematical thinking on the world, reading math into nature but rather we are reading math out of nature. He demonstrates that we are actually detecting the abstract concepts embedded in the orderliness of the myriad things around us. But where did that math come from? Where were those mathematical notions before they became embedded in stuff? Could our puny human minds be somehow communing with the Mind of Gd? Our Sages say, "Gd looked into the Torah and created the world,"[1] that the Torah is the divine blueprint of Creation. Are there fractals in the Torah, selfsimilar patterns at different scales? The Shabbos concept is a fine example. Thus we have seven days of the week, seven weeks in the Omer period, seven years of the Shmita cycle, and seven Shmitos followed by a Jubilee year. The same sabbatical theme is even expressed on a cosmological scale with the seven creation days on one end the seventh millennium on the other. Here is another Torah fractal. In one of his talks,[2] the Lubavitcher Rebbe describes how the letters of the essential Divine name  Yud / Hey / Vav, and Hey  are expressed in the four spiritual worlds, the four elements of the global ecosystem, and the four dimensions of the human nature. For example the "Vav" compares to the world of Yetzira, the plant kingdom, and the emotions. The "Vav" even looks like a plant, fixed and rooted, like personality traits that are relatively fixed in people, and like the world of Yetzira in which entities are fairly well defined and formed, albeit in the spiritual realm. Fractals have another property that is at once attractive and infuriating, and that is the fact that they are infinite. The unfolding of selfsimilar patterns is an unending process, limited only by the amount of computer time you can throw at expanding the pattern, as this video portrays. Mandelbrot asks a deceptively simple question: How long is the coastline of Great Britain? On the surface of things, anyone with a map, a string and a ruler should be able to come up with an answer, but however detailed the map, the answer is only approximate. As you zoom in there is always more and more detail so if you want an accurate answer, you must reckon it's something approaching infinity! Here is what they call a Koch Snowflake, a simple geometric example of an infinitely long edge to a shape of fixed size. Take a triangle and put a smaller triangle on the middle of each edge. Then put smaller triangles on those smaller edges, and keep going. The result looks like this, but if you zoom in, it goes on forever.
Mandelbrot's gift opened a window for us into the Mind of Gd, using algebra and geometry to convey infinity in a nutshell, as it were. His mathematical infinity and the infinite thematic variations in nature both all derive from the infinity of Torah which is one with the Mind of the Infinite One Above. Mandelbrot's story is a fractal itself. His discovery is but one element in the grand story unfolding of how in our times science is progressively discovering more about the synergy of Gd, man and the nature of things. That process itself has been mapped out and scheduled in the Torah, in the Zohar and in Chassidus Chabad as a clear indication of the imminent redemption.[3][4] We are now entering the age of infinite revelations from both Torah and science, culminating in the complete revelation of Moshiach NOW! There is what to see. We just need to keep our eyes open. [1] Zohar 1:134a, 2:161a [2] Likkutei Sichos, Vol.6, the second Yisro. [3] http://www.chabad.org/library/article_cdo/aid/112700/jewish/MoshiachandScience.htm [4] http://www.chabad.org/library/article_cdo/aid/113102/jewish/Appendix1FaithScienceConvergenceExplained.htm
Dr. Aryeh (Arnie) Gotfryd, PhD is a chassid, environmental scientist, author and educator living near Toronto, Canada. To contact, read more or to book him for a talk, visit www.arniegotfryd.com or call 4168589868
