Fractals – a way to see infinity

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Ever wonder why a river, looks like a tree, looks like a leaf? IBMer Benoit Mandelbrot did. His quirky way of looking at the world led to the discovery of fractal geometry. Fractals have made it possible to mathematically explore the kinds of rough irregularities that exist in nature. Clouds are not perfect spheres, mountains are not symmetric cones, and lightning does not travel in a straight line. “In the whole of science, the whole of mathematics, smoothness was everything. What I did was to open up roughness for investigation,” Mandelbrot said.

So what exactly is a fractal? A fractal is a never-ending pattern. They are created by repeating a simple process over and over. Instead of focusing on rigid geometry, Mandelbrot created new equations in shapes that were rough and jagged. “I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.”

You can see how fractals work in the video below.

In this final interview shot by filmmaker Erol Morris, Mandelbrot shares his love for mathematics and how it led him to his wondrous discovery of fractals. His work lives on today in many innovations in science, design, telecommunications, medicine, renewable energy, film (special effects), gaming (computer graphics) and more.

Fractals in cinema and graphic design

After Loren Carpenter, co-founder of Pixar Animation Studios, read Benoit Mandelbrot’s Fractals: Form, Chance, and Dimension, he began experimenting with fractals to make his computer graphics look more realistic. This technique gave rise to software programs now widely used across the computer graphics industry to create special effects, including fictitious landscapes and imaginary worlds—such as the Genesis planet sequence in Star Trek II: The Wrath of Khan and the damaged Death Star in Return of the Jedi.

Fractal image compression

Fractal compression converts images consisting of random information into fractal code—saving only a small, representative amount of information that is later used to re-create the original image. Since the fractal image is now code instead of pixels, file size is drastically reduced and the image can be scaled to any size without losing its sharpness.

All my life, I have enjoyed the reputation of being someone who disrupted prevailing ideas"Fractals in biology

Fractal geometry is being used in the biological sciences to accurately model the human lung, heartbeats and blood vessels, neurological systems and countless other physiological processes. Doctors and researchers are now using the mathematics behind fractal geometry to build models that they hope will identify microscopic patterns of diseases and abnormalities earlier than ever before.

Fractals in the stock market

In light of the financial crisis that began in 2007, many market theorists are turning to the teachings of Benoit Mandelbrot, whose fractal approach to price variation revealed that a crash was not as improbable as forecast by conventional wisdom. Mandelbrot dismissed the theory of efficient markets as simplistic and overly generalised, commenting that the world is not tidy or infinitely stable—turbulence is a natural unavoidable force. These views are discussed further in Mandelbrot’s 2004 book The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward, co-written by Richard L. Hudson.

Fractals in climate science

Scientists have recently shown that the distribution of large branches to smaller branches in a single tree exactly replicates the distribution of large trees to smaller trees in an entire forest. Research is currently underway to use this information to measure how much carbon dioxide a single forest is capable of processing. From there, scientists will be able to apply their findings to every forest on earth, quantifying how much carbon dioxide the entire world can safely absorb.

Benoit Mandelbrot, the father of Fractals

Today, we have merely scratched the surface of what fractal geometry can teach us. Weather patterns, stock market price variations and galaxy clusters have all proven to be fractal in nature, but what will we do with this insight? Where will the rabbit hole take us? The possibilities, like the Mandelbrot set, are infinite.

Benoit Mandelbrot was an intellectual jack-of-all-trades. While he will always be known for his discovery of fractal geometry, Mandelbrot should also be recognised for bridging the gap between art and mathematics, and showing that these two worlds are not mutually exclusive. His creative approach to complex problem solving has inspired peers, colleagues and students alike, and instilled in IBM a strong belief in the power of perspective.

Curated by: Anne Nicholson, Brand and Content A/NZ

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