Some problems span multi-generations, Benoit Mandelbrot solved one such problem and earned himself a well-deserving place in history, as the father of fractals. Many of these mathematical structures and their descriptions go back to classical mathematics and mathematicians of the past like Cantor, Peano, Hilbert, Koch, Sierpinski, Julia, Hausdorff, but it was Mandelbrot who extended the early topology. Fractals solve the problem of how to organize the complicated structure in an efficient way. Of course, this was not what Peano and Hilbert were interested in almost 100 years ago. It was only after Mandelbrot’s work that the omnipresence of fractals became apparent.
Mandelbrot was a visionary in his ability to connect mathematics and patterns. He too as many other great thinkers worked on ideas of aggregation, simplification, order, efficiency rules, cyclicality in errors, the interconnectedness of nature through geometrical structures. Mandelbrot illustrated that a very simple formula can generate objects that exhibit an extraordinary wealth of structure. His work encompasses mathematics, physics, economics, and diverse other fields of physical and social sciences, music, and art. He died on 14 Oct at the age of 85 after suffering from pancreatic cancer.
In 1958, he was asked to tackle the problem of line noise. The reigning joke was that the noise was generated by a “guy with a screwdriver” fiddling with some piece of connected equipment. Meanwhile, engineers sought to solve the problem by increasing signal strength to drown out the noise. But Mandelbrot’s eventually showed that the noise was both consistent and erratic, some kind of inescapable natural feature of the system that did not disappear with increased signal strength. But more remarkably he also showed that every burst of noise also contained within it bursts of a clear signal. Stranger still, found that the ratio of periods of noise to periods of clean transmission remained constant, regardless of the scale of time used to plot the phenomenon (i.e. months, days, seconds).
Mandelbrot returned home and turned his attention to analyzing cotton prices. Using records dating back to 1900, he began to perceive an astonishing pattern — one that hearkened back to his work on line noise a decade earlier. He discovered that cotton prices followed a pattern that was both erratic and regular. That is, although price changes were erratic in terms of normal distribution and no one could predict the exact amount of any particular price change, the changes themselves followed a symmetrical pattern with regards to scaling. Regardless of whether the scale of time was hourly, daily, or monthly, the curve was the same. And it had remained so for at least 60 years (the length of time covered by his records). Mandelbrot’s models were not that they generated deceitful randomness, but that they could generate graphed data whose visual pattern accurately mimicked the visual patterns created by real phenomena. Fractal Geometry was born, a simple process that could define a complex pattern. Fractal Geometry is also the geometry of chaos.
The fractal geometry he developed would be used to measure natural phenomena like clouds or coastlines that once were believed to be unmeasurable. Fractals are easy to explain, it’s like a romanesco cauliflower, which is to say that each small part of it is exactly the same as the entire cauliflower itself. It’s a curve that reproduces itself to infinity. Every time you zoom in further, you find the same curve. Applied mathematics had been concentrating for a century on phenomena that were smooth, but nature was not smooth, it was rough.
Mandelbrot developed the idea of fractals while trying to determine the length of the British coastline, when he realized a seemingly smooth shore becomes more and more detailed as you zoom in. he also described his life as a very crooked line. But despite being the inspiration for such metaphysics, Mandelbrot, when asked if fractals don’t point to a single rule underlying reality, has simply stated, “There is no single rule that governs the use of geometry. I don’t think one exists”.
Many scientists would disagree. But then disagreement also leads to a human quest of uncovering the truth. Mandelbrot did his bit to show us the truth. According to him though ‘Time’ does not run in a straight line, it changes according to price and stretches and shrinks like a balloon rubber. It speeds up and slows down. The seriousness of his work suggests that either ‘Time’ was not important enough to be featured in his work or he did not consider ‘Time’ to have a mathematical fluctuation or in words an organic life. If time had similar mathematics and roughness as Mandelbrot illustrated in nature he would have his answer why his life was like a coastline.