The God Equa,on. Otherwise Known as MANDELBROT SET

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Tabitha Fox
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1 The God Equa,on Otherwise Known as MANDELBROT SET

2 For a Summer Unit I studied The Art of Photography, as photography is a tool I use to take in my environment. Capturing landscapes, buildings, laneways, inner city scapes and country scapes, viewing contrasts and indentifying similarities help me to place myself and connect with environments I travel through or live in. I am often amazed at visual similarities and this year after taking many long country-drives, upon returning to my backyard I started photographing a section of an outside brick wall that reminded me of the landscapes I had returned from. Not the colour as much as the geographical grids if seen from an aerial view. The idea of sameness feeds my constant reexamining of identity and connection to place. The idea that we are entitled to an area of land or country that excludes others from belonging, the repeating conflicts that this perception perpetuates has me trying to understand and search for a peaceful solution, if only for myself. In identifying the mini landscape on the brick wall triggered an interest in reading up on The Mandelbrot Set, which I didn t know much about till this semester though I ve often heard reference to it.

4 Geometry: branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogues! The idea of repetition, similarities, etc have activated scientists for centuries, and the Mandelbrot Set, though discovered in 1980, is based on simple mathematics, based on adding and multiplying but up to millions and millions of time which is why it wasn t discovered until the age of the computer and its ability to multiply infinitely.

6 In the late 1800 s the term mathematical monsters was coined to describe a realization where from a straight line, if you take out the middle 1/3 and the line is split into 2, and this process is repeated, instead of resulting in nothing, as was theorized, mathematicians found an inainite number of lines.

8 Leading on from this simple theory, in1917 a French mathematician Gaston Julia published papers about complex numbers. Using a simple equation, repeatedly, instead of calculating out infinitely as Mathematical Monsters did, he fed the equations back through a feedback loop ; using the result of the equation each time to feedback through the loop, he determined would result in a set. This theory is known as The Julia Set.

10 The next significant discovery leading forward is known as The Koch Curve. It is a mathematical curve and one of the earliest fractal curves described. Also referred to as the snowflake, it has a finite area bounded by an infinitely long line. Mathematician Benoit Mandelbrot, Polish born, French and American national, identified this fractal as relevant to the measurement of coastlines, when in the 1940 s it was observed that coastlines had great size variations when measured with different tools. For instance when measured with a 1mile yardstick the coastline was smaller than when measured with a 1foot yard stick. The smaller the measuring tool, the greater the length of the coastline, as smaller indentations could be measured with smaller tools, the accuracy of measurement increased the area measurement. This quandary set off the light bulb in Mandelbrot s head, who saw that the smaller indentations in the Koch Curve was precisely what was needed to measure coastlines, he saw coastlines as fractals, and that they needed to be measured in roughness, using fractal dimension and self similarity. The idea that coastlines were the same as a snowflake, made up of infinite lines opened up a fresh way of looking at the living world and changed views of the universe.

12 In 1980 and with the use of the computer Mandelbrot decided to actualize the Julia Set, that is he graphed the different equations identified by Gaston Julia and from this got hundreds of very rough pictures, which he then combined into a single image. He observed all kinds of junk around the bug as he called it, as it resembled almost any living thing, and seems familiar to all of us. The image obtained when graphed on paper is recognized as the symbol of the Mandelbrot Set (blue diagram). One original set is 6 inches across in size, and geometrical shape and icon embodying how the universe works. Exploring further by continually magnifying the image Mandelbrot identified sets within that looked like a Julia Set as well as mini Mandelbrot Sets. No matter how much he magnified the image, it keep emerging, infinitely smaller. The 6 turned into an infinite number of sets, and no matter how much they are magnified you still see new patterns and new images emerging, much greater than the universe. The end set is infinitely complex.

14 Mandelbrot coined the term Fractal to describe the fragmented, broken fractured, irregular shapes he saw. h=ps://youtu.be/56gzv0od6du?t=13m17s The colours around the black define the different areas or variations. What is amazing is that these patterns have been reflected in art and architecture for centuries before the Mandelbrot Set was identified. Seen in patterns in nature from the circulatory system of the heart, electrical circuitry in the brain, hallucinatory patterns, paisley patterns, patterns in stain glass windows. It has familiarity; the question asked has been How much is from chance and how much is already recognized by the brain? Even in a unknown world, in practice you can not determine the future, to create something in which everything is free.

The importance of beings fractal

$The importance of beings fractal$ The importance of beings fractal Prof A. J. Roberts School of Mathematical Sciences University of Adelaide mailto:anthony.roberts@adelaide.edu.au May 4, 2012 Good science is the ability to look at things