Beauty From Anarchy

Mathematicians often bury the beauty of the nature using abstract mathematical symbols, anything from alpha to omega. Traditionally, they love to describe smooth and definite patterns with their equations and shy away from complex and irregular patterns. In that sense, Benoit Mandelbrot, who died on Thursday at the age of 85, was a radical mathematician. He went on to explore the hidden beauty that lies beneath the seemingly chaotic systems.

An engineer can model a wonderful structural design with the common geometric shapes such as lines, circles etc., and with software they can generate a 3D model of the same. A mathematician can create a set of equations that would describe this model. Ask them to model a cauliflower or broccoli? They would probably have stumbled, at least until the work of Mandelbrot became an established branch of mathematics known as Fractal Geometry.

Fractals are like a whole world at a smaller scale embedded inside a world of increasing complexity. They can be generated by iterations of the smallest entity over large number of times. In other words, the fundamental equation that represents the smallest link could repeat endlessly to create a complex system. They could be described as the finite holding infinite. From snowflakes to shoreline, the seemingly irregular shapes harbor a beauty beneath their intricate system. The iterations can have fractal dimensions instead of the usual whole number dimensions.

The Fractal Geometry of Nature is a mathematical text published in 1983 by Mandelbrot that addresses many of the mathematical puzzles that involve fractals. He began originally as a researcher at IBM and later became a faculty at Yale University. Fractal geometry ideas are used in a variety of fields from geology to cosmology and from medicine to stock market. It explores underlying geometry and mathematics in varieties of systems that have been ignored largely by the conventional mathematicians.

"When the weather changes, nobody believes the laws of physics have changed. Similarly, I don't believe that when the stock market goes into terrible gyrations its rules have changed." Benoit Mandelbrot.

Though exact predictions of the behavior of such systems are still in its infancy, it is not doomed as once thought. Fractals reveal the fact unpredictability and complexity is part of the nature, but they are not undesirable.

The image created by the Mandelbrot set is one of the most popular and stunning image in all of mathematics. Lurking under this image is the elegant mathematics that many are not used to. Fractals are the finest examples of where math and art superimpose each other. Computer imaging has created many intriguing fractals besides the natural ones. Fractal art continues to thrive with still images and animations.

The ability of fractals to derive order from disorder is the heart of chaos theory, a new and exciting science. Though chaos implies disorder and unpredictability such as the weather system, the scientists glimpse a hope here as chaos could be, after all, deterministic.
To demonstrate the relation between the outcome and its severe dependence on initial conditions, it's been said that a butterfly flapping its wings in South America can affect the weather in central Park.But, it's also been said that even accidents follow fixed laws. Perhaps, those laws are veiled underneath the clouds, mountains and coastlines.

 

Comments

[SeekingClarity]

05:03 PM on 10/24/2010

Around 1782, René Just Haüy, the originator of the science of crystallography, dropped a crystal of calcite and realized that the broken fragments displayed identically characteristic crystal faces. This structure reflects fundamental electromagnetic geometry of the chemical compound itself. There is profound simplicity in our everyday complexity, without which the universal laws” of science would have no application (they would not be “laws” were they not widely applicable). The same patterns appear again and again. Possibly fractal geometry will one day give way to some even more powerful mathematical method, but even if so, Mandelbrot uniquely demonstrated the very surprising power of simple patterns organized by those same simple patterns beneath much of the complexity we see.

[Andras Pellionisz]

02:57 PM on 10/21/2010

"Clouds Are Not Spheres, Mountains Are Not Cones" said Mandelbrot - And "The Genome is Not Composed of Sections of Genes and Junk", says FractoGene, inspired by his genius. The beauty is in the mathematical elegance of compressing complexity of non-living and living Nature into astonishingly simple, and software-enabling applications. Understanding of the misbehavior of markets is utterly important. So is the FractoGene explanation of Recursive Genome Function (see Google Tech Talk YouTube), using fractal iterations by which the fractal genome governs growth of organelles (e.g. brain cells, cardiac coronaries), organs (e.g. lung, kidney) and organisms (e.g. cauliflower romanesca). There is order in the disorder of diseases; e.g. cancers are visibly fractals. Father of fractals was killed by a pancreatic fractal. Do we just stand by, or do something about it?

[Essem]

03:52 AM on 10/25/2010

Very interesting that fractals can apply to the genome....and even specify Mandelbrot's particular demise....fractal exploration of the genome would seem to pose infinite possibilities, yet you suggest the fractal approach narrows the search by identifying where a fractal may have broken with its own consistency!