In keeping with the mathematics theme established in the previous installment of this series (on Buckminster Fuller), Part III is about Benoit Mandelbrot. It is impossible to ignore the “geodesic”, forward-looking genius of Benoit Mandelbrot. Like Fuller before him, Mandelbrot used geometry to identify and educate us about the nature of infinity. Mandelbrot’s elucidation of “fractals” may have given the human race a much closer look at nature’s grand design.
Benoit B. Mandelbrot was born November 1924 and died on the 14th of October, 2010. A mathematician born in Poland but raised in France, Mandelbrot spent much of his life living and working in the United States. Starting in 1951, Mandelbrot worked on problems and published papers in mathematics and applied math, information theory, economics, and fluid dynamics. He became convinced that two key themes - fat tails and self-similar structures - ran through a multitude of common problems in those fields.
Perhaps Mandelbrot's most famous contribution is the M-set. Mandelbrot discovered the M-set in 1980; this discovery has been widely discussed in books such as The Fractal Geometry of Nature by Mandelbrot and Chaos by James Gleick and in scientific magazines (for example see the beautiful pictures and excellent summary in the July 1985 issue of Scientific American).
I am by no means a mathematician. I’ve always been humbled by the complexities of higher mathematics, more of a right brained guy I guess. Mandelbrot’s discovery of the “M-set” may well be a look in to the true fabric of Mother Nature, and sure enough, Mom speaks math.
For those who are mathematically inclined, here is a brief outline of how the M-set is created: start with the expression z -> z^2 + c; choose two complex numbers z and c; solve the expression z^2 + c to get a new value of z; put the new z into the z^2 + c term and compute another z value; continue this process on a computer for much iteration. Color coding the rate at which different values of c cause z to either (1) shoot off to infinity, (2) stabilize in the realm of finite numbers, or (3) go to zero creates the visual embodiment of the “m-world”. One of the many wonders of this infinitely complex “world” is that it can be created by just a few simple lines of computer code that are repeated recursively. From these little algorithmic loops comes the most rococo universe that anyone has ever seen. No matter how many times you magnify the M-set to infinity, it continues to expand. And you can see the M-set everywhere in nature. Mandelbrot found a mathematical formula to describe a “fractal” (a term he invented to describe the M-set) – in which each part mimics the pattern of the whole.
“Fractal geometry is not just beautiful, but useful – for modeling turbulence, financial systems, the distribution of galaxies. It underpins the physics of disorder and chaos theory. My whole career became one long, ardent pursuit of the concept of roughness,” Mandelbrot wrote in an essay on receiving the 1993 Wolf prize for physics. “Fractal geometry plays two roles. It is the geometry of deterministic chaos, and it can also describe the geometry of mountains, clouds, and galaxies.”
Science and geometry have always been partners and have progressed in lockstep throughout history. In the 17th century, Johannes Kepler found that he could represent the orbits of the planets around the Sun by ellipses. This inspired Isaac Newton to explain these elliptical orbits by devising the law of gravity. Similarly, the back-and-forth motion of a perfect pendulum is represented by a sine wave. Simple dynamics used to be associated with simple geometrical shapes. This kind of mathematical picture implies a smooth relationship between an object's form and the forces acting on it.
Indeed, Galileo put forth the notion that "the great book of nature is written in mathematical language", adding that "its characters are triangles, circles and other geometrical figures, without which one wanders in vain through a dark labyrinth." Mandelbrot added: "(Certain) phenomena need geometries that are very far from triangles and circles. They require non-Euclidean structures and a new geometry called “Fractal” geometry."
A lot of math stuff I know, but I found that just learning a little of this math through researching Mandelbrot’s discovery is well, for lack of better term, simply mind blowing. For most of us this kind of mathematical language is often cosmically abstruse and uninteresting; but here with the M-set it is deeply compelling, and it has a certain universal appeal. Mandelbrot’s discovery of fractals shows that an otherwise drab mathematical formula can produce an apparently self-replicating, infinite design that is everywhere around us. When asked whether fractals point to a single rule underlying reality, Mandelbrot simply stated:
"There is no single rule that governs the use of geometry. I don't think one exists." In addition, according to Mandelbrot, "The beauty of geometry is that it is a language of extraordinary subtlety that serves many purposes."
Perhaps if we take away the symbolic, mathematical aspect of the M-set and fractal geometry, it becomes a little simpler to understand. First, get a microscope that kicks ass. My point here and the core amazing trait of the M-set is that no matter how many times you magnify it, it just keeps going on to infinity. This is where we can learn lessons for nature and art (for those of us who don’t want to go the math route). In a PBS Nova documentary called “Hunting the Hidden Dimension” one prominent Ph.D. made a comment that summed up this art-math notion of the M-set quite nicely:
“Math is really quite close to art...They just speak different languages....”
“How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" is a paper first published in Science in 1967. In this paper Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not invent it until 1975. In late 1982, Mandelbrot published his visionary work The Fractal Geometry of Nature. This work explains how fractals can explain a variety of natural forms and structures ranging from trees to coastlines.
What Mandelbrot’s brilliant discovery gives us is a new appreciation of the natural world and the beautiful art we see within it. Many things previously called chaotic are now known to follow subtle fractal laws of behavior. This reflects what we observe in both small details and macroscopic patterns of life in all their physical and mental varieties as well. It all centers around what’s called in Mandelbrot’s math world “iteration”.
Here is Webster’s definition of "iteration". It is short and sweet, and this article just won’t allow the space to go into it more:
the action or a process of repeating; as a : a procedure in which repetition of a sequence of operations yields results successively closer to a desired result b : the repetition of a sequence of computer instructions a specified number of times or until a condition is met - compare recursion.In other words, it continually repeats itself... A formula continues to repeat itself in both math and nature. Fun to get your head around.... Just take a close look at the fern hanging inside or outside of your home. You’ll start to see. Look up a picture of a Nautilus (the sea mollusk, not Nemo’s awesome ship) or a simple snail; the M-set is truly everywhere. For us old hippies, it’s everywhere in our tie-dyes and art. It’s quite hard for anyone paying attention to miss. The beautiful tapestries and rugs of Persia and India have the M-set in them as well. The M-set is in winter snow. Before catching that snowflake on your tongue, take a close look at it.
The M-set gives us poetry as well. I felt that closing with this thought was comfortable and soothing and wouldn’t leave most of us dizzied like we'd just received an intensive crash course in advanced geometry. The great poet William Blake penned in 1803 these beautiful lines in his great work “Auguries of Innocence”:
To see a world in a grain of sand, And a heaven in a wild flower, Hold infinity in the palm of your hand, And eternity in an hour.