Fractal Theory: Patterns and Beyond

Summary

In 1978, engineers at Boeing Aircraft in Seattle were designing experimental designs for aircafts. A young computer scientist called Loren Carpenter was helping them visualize what the planes might look like while in flight. He had a requirement of creating a landscape of mountains that the planes could fly through. But with existing techniques during that period, there was no way to do that. It was then that Loren Carpenter stumbled upon a book titled Fractals: Form, Chance and Dimension by a mathematician called Mandelbrot. In his book Mandelbrot described that many forms in nature can be described mathematically by patterns called fractals which looked like jagged and broken pieces. Using fractals, Loren was able to create beautiful landscapes through the usage of endless iterations of fractals. Fractals can be formed from any smooth looking shape and breaking it into pieces, over and over again. Infact in modern history, people like the great 19th century japanese artist Hokusai had created and used such repetitive patterns in his art. His works started with large patterns that are broken down into smaller repetitive patterns. Mandelbrot's fascination with the visual side of math began as a student. German mathematician Cantor created some of the first fractal patterns. He started with a straight line, which he would break into three parts and would part away with the middle part. He would continue with this iterative process with the repetitive patterns. Swedish mathematician Niels Fabian Helge von Koch put forth another famous pattern. He started with an equilateral triangle and on each side he would substitute two other sides that are each no longer than the original piece. He continued iterating this process infinitely many times. This Koch snowflake curve proved to be crucial to a nagging measurement problem of being able to measure the coastline of a country. In the 1940's, British scientist lewis richardson had observed that there can be great variations between different measurements of a coast line. Mandelbrot observed that the fine inundations in the Koch's curve were precisely useful in measuring the coastline. Though Mandelbrot knew he couldn't measure the length using these fractals, he figured out that he could measure the roughness of the curve. French mathematician Gaston Julia was looking at taking a simple euqation and then feeding back the result into the same equation. The series of numbers that would result through such feed back iterations led to the formation of Julia's sets. At IBM, Mandelbrot did smoething that Julia couldn't do. He used the IBM computers to run the Julia set iterations millions of times. He then plotted these sets into images and he began to get his own sets of numbers. In 1980, he created an equation of his own that combined all of the julia's sets into a single image. This led to the formation of the Mandelbrot set which quickly became the emblem of fractal geometry. Images generated through fractal geometry completely transformed the creation of special FX in hollywood movies. Extremely wonderful scenes from Star Wars 3 used fractals to generate such eye catching effects. All the inbuilt antenna's used in today's cell phones were based on a fractal geometry theorem dealing with self similarity, that lead to an immensely effective use of space and resources thereby revolutionizing the cell phone industry. Such fractal design based antennas were much smaller and were able to recieve a very wide range of frequencies. Subconscious processes like the beating of the heart, focussing of the eye, balancing of bodily motion were all found out to follow the fractal patterns. Researchers like Peter Burns from the University of Toronto are using fractal patterns to generate mathgematical models that could be used to detect cancers at very early levels by being able to differentiate benign from malignant patterns. Geoffrey West from the Santa Fe Institute began investigating biological mysteries that dealt with the mass-energy usage relationships among the biological world. Such laws are applicable all across the biological life ranging from tiny bacteria to blue whales. Though these laws were discovered in the early nineteenth century, no one was able to prove and give a reasoning for this law. Along with researchers like Brian Enquist from University of Arizona, fractal patterns were used to give an explanation to this law. Computers have always been helpful in such problems requiring endless iterations when solving them with fractals.

Part 2: http://www.youtube.com/watch?v=y-HelWSdP_w

Part 3: http://www.youtube.com/watch?v=mN37gojpdaM

Part 4: http://www.youtube.com/watch?v=WGqNZBn-HL8

Part 5: http://www.youtube.com/watch?v=OVNatGQvpBI
Keywords: Mandelbrot Set, Self Similarity, Chaos Theory, Julia Sets, Fractal Dimension, Snowflake, Fractal Antennas, Fractal Trees, Compression, ChaosPro, FractalExplorer, Fractint, QuaSZ