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Fractals

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Providence Holley

on 3 June 2014

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Transcript of Fractals

Fractal Generating Programs
Without fractal generating programs, it is inevitable that we would have not been as progressed in the study of fractals as we are today. Today fractals are generally computer based as it allows for more complexity unachievable with the human hand. With the human hand, you can create only a few iterations, but with a computer, you can create thousands upon thousands. Some fractal generating programs include Mandelbulb (http://mandelbulb.com/), Apophysis (http://www.apophysis.org/), Electric Sheep (electricsheep.org), Fractint (www.fractint.org), Sterling (soler7.com/Fractals/Sterling2.html), Spangfract (www.fractalforums.com/spangfract) , UltraFractal (http://www.ultrafractal.com/), XaoS (xaos.sourceforge.net/english.php), and Terragen (http://planetside.co.uk/products/terragen3). These fractal generating programs generally create only 2-dimensional fractals, not 3-dimensional fractals
Examples
Fractal Zoom Video
Characteristics
All fractals contain self-similarity, but what does that mean? Think of it as detailed patterns being iterated or repeated infinitely so that one iteration will stem from another iteration. One analogy that can be used when speaking of self-similarity in fractals is imagining zooming in or out with a camera infinitely, but finding no deviation from the original image because the pattern repeats itself infinitely. Also, you can imagine it as a mathematical inception.You could end up finding little distortions of the full set because not all fractals are exact. Some types of self similarity include exact self similarity, in which the fractal is perfectly identical at all scales, quasi self-similarity, when it is approximately the same and there is a slight deviation from the the original set in the form of distorted copies, statistical, when it is randomly generated, qualitative, in a series of time, and multifractal, when there are multiple fractal dimensions or scaling rules. Another key component of a fractal is having a fractal dimension not equal to it regular dimension such as this line is one-dimensional, this surface is two-dimensional, this cube is three-dimensional. A fractal dimension is the theoretical dimension of a fractal. Most fractals have fractal dimensions that fall between integers and their normal dimensions. A fractal could have a normal dimension of 1 but have a fractal dimension approaching that of a 2D figure. Fractal dimensions allow for fractals to exist in a dimension of superposition , being both things at once.This is what separates fractals from regular geometric figures, their fractal dimensional scaling. Regular geometric figures scale by the product of their dimension and the number its lengths are being increases by. For example, is you were to double all of a square's lengths, the area would scale by 4 because the square is two dimensional and you doubled the lengths. However, if you do this with a fractal, it would scale by a non integer. Another key factor of fractals is that they cannot be measured in terms of traditional Euclidean geometry because they behave in an infinitely iterated manner. The last component is a fine or detailed structure that repeats infinitely at increasingly smaller scales.
Since the popularization of fractals by Mandelbrot in the 1980s, interest in application of fractals in different fields took off. There are numerous applications for fractals today that we can't live without. These include, but are not limited to fractal structures in antennaes to allow for more reception in less space, microchips with fractals designs to allow for faster speed in less space, fractal transistors, fractal heat exchangers, digital imaging, urban growth, classification of histopathology slides, fractal landscape or coastline complexity, enzyme/enzymology (Michaelis-Menten kinetics), generation of new music, signal and image compression, creation of digital photographic enlargements, fractal in soil mechanics, computer and video game design, computer graphics, organic environments, procedural generation, fractography and fracture mechanics, small angle scattering theory of fractally rough systems, T-shirts and other fashion, generation of patterns for camouflage, such as MARPAT, digital sundial, technical analysis of price series, fractals in networks, medicine, neuroscience, diagnostic imaging, pathology, geology, geography, archaeology, soil mechanics, seismology, search and rescue, and technical analysis. Scientists can also calculate the amount of carbon a forest can take in by measuring the amount of carbon in one leaf of one tree and using fractal branching patterns to calculate the amount of carbon in the tree and then the entire forest; this helps in global warming studies. In medicine, cardiologists have seen that the heartbeat of a patient is actually a fractal, not the steady beat we thought it to be. They think that they can detect heart problems earlier by comparing a normal steady, fractal heartbeat to that of an irregular, non-fractal, heartbeat. Also they think that they can compare the normal, fractal, branching pattern of blood vessels to cancerous, non-fractal blood vessels.


Fractals
History
The term "fractal" was first coined by Benoit Mandelbrot in 1975, it was derived from the the Latin for broken or fractured, fractus, however the use of fractals stem farther back. In the 17th century, the mathematician Gottfried Leibniz wrote about the idea of repetition and recursion. Unfortunately, fractals were forgotten until the 19th century. In 1872, Karl Weierstrass described functions of continuity but indiscernibility when describing it with Euclidean geometry in a graph. Then in 1883, Georg Cantor created the first fractal which were subsets of a line. At the time they were called mathematical monsters. In 1904, Helge von Koch gave a geometric definition of a fractal, as he felt the previous definitions were too abstract. Then, in 1915 Waclaw Seirpinski created his famous triangle and carpet. The next milestone came in 1918, when Gaston Julia described fractal behavior with complex numbers and iteratve funtions in the form of plugging a number into a equation and plugging the outcome in the equation over and over again. Then in March 1918, Felix Hausdorff allows room for non-nteger dimensions and fractal dimensions by extending the reach of dimensions. Finally in 1938, Paul Levy describes the idea of self-similar curves. Many artists ,such as the famous 19th century Japanese painter Katsushita Hoksai, used fractals in their artwork. Also fractal clothing took off with sophisticated computer programs that could create complex fractals patterns. After Mandelbrot solidified the field of fractal geometry, interest in fractals took off.
Examples of fractals in nature
You may think that fractals are confined to the minds of curious mathematicians and computers, but in all actuality, nature has thought up this brilliant idea billions of years before us. You see, nature needed a fast effective code to get things done and there is noway you could create individual codes for every single thing, so Mother nature thought up of fractals. This way there could be a single code for everything. Organisms could do things faster and more efficiently. Some examples of fractals in nature include, but are not limited to, river networks, fault lines, mountain ranges, craters, lightning bolts, coastlines, mountain goat horns, animal coloration patterns, Romanesco broccoli, pineapple, heart rates, heartbeat, earthquakes, snow-flakes, crystals, blood vessels and pulmonary vessels, ocean waves, DNA, various vegetables (cauliflower & broccoli), soil pores, and psychological subjective perception.
An example of the endless repetition of fractals and how mind-blasting they are
Just what are they?
Believe it or not, fractals are everywhere, they are the ultimate mother of all patterns, whether it be Romanesco broccoli, the branching patterns of trees and your veins, the beating of your heart, snowflakes, the essence of your entire being, your DNA, etc. The list is endless, literally. Just what are they, though? Benoit Mandelbrot, the father of modern fractal geometry , said that a fractal is a gagged repeating form. Mathematically, a fractal is a set that displays self-similar patterns. Though not all mathematicians agree on what exactly a fractal is, they have come to a consensus that all fractals have the basic characteristics of self-similarity, an irregular relationship between the fractal itself and its space, fractal dimensions, an incredibly fine or detailed structure, and a habit of annoying Euclid, by which I mean the traditional laws of Euclidean geometry do not fit accordingly with fractals in terms of describing them. What does that all mean though?
Applications
Works Cited:

"Fractal"
Wikipedia. 19 May. 2014. Web. 29 May. 2014
NOVA. "Hunting the Hidden Dimension," NOVA website, http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html (accessed May 29, 2014).
Sierpinski Traingle
Sierpinski Carpet
Koch Snowflake
Cantor Set
Julia set
Mandelbrot set
Dragon curve
Uniform Mass Center Triangle Fractal
Pythagorean Tree
Golden ratio fractal
Coastline of Britain fractal
Milky Way Galaxy fractal
Mountain fractal
Drainage system of rivers fractal
DNA fractal
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