Fractal S

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Fractal History

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Fractal Interests can be traced back to the 19th Century, but they weren't very developed.

Gaston Julia (1893-1978)
=French mathematician
=Published a book on Fractals in 1918.
=Lived Before computers, he had to draw the sets of functions by hand.
=These types of fractals are now called Julia sets.

Benoit Mandelbrot (1924-present)
=is a mathematics professor at Yale University.
=He used a computer to explore Julia's iterated functions, and found a simpler equation that included all the Julia sets.
= This Mandelbrot set is named
after him.
=Called the "Father of Fractal Geometry"

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Fractals in Real Life

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Biologists have traditionally modeled nature using Euclidean representations of natural objects or series. They represented heartbeats as sine waves, conifer trees as cones, animal habitats as simple areas, and cell membranes as curves or simple surfaces. However, scientists have come to recognize that many natural constructs are better characterized using fractal geometry. Biological systems and processes are typically characterized by many levels of substructure, with the same general pattern repeated in an ever-decreasing cascade.
Perhaps in the future biologists will use the fractal geometry to create comprehensive models of the patterns and processes observed in nature.

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Julia Set

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The Julia set is another very famous fractal, which happens to be very closely related to the Mandelbrot set. It was named after Gaston Julia, who studied the iteration of polynomials and rational functions during the early twentieth century, making the Julia set much older than the Mandelbrot set.
The main difference between the Julia set and the Mandelbrot set is the way in which the function is iterated. The Mandelbrot set iterates z=z2+c with z always starting at 0 and varying the c value. The Julia set iterates z=z2+c for a fixed c value and varying z values. In other words, the Mandelbrot set is in the parameter space, or the c-plane, while the Julia set is in the dynamical space, or the z-plane.

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Mandelbrot Set

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Named after Benoit Mandelbrot, The Mandelbrot set is one of the, if not the, most famous fractals ever discovered. It was created when Mandelbrot was playing with the “simple” quadratic equation z=z2+c. In this equation, both z and c are complex numbers. In other words, the Mandelbrot set is the set of all complex c such that iterating z=z2+c does not diverge.
To generate/create the Mandelbrot set graphically, the computer screen acts as the complex plane. Each point on the plain is tested/substituted into the equation z=z2+c. If the iterated z stayed within a given boundary forever, convergence, the point is inside the set and the point is plotted black. If the iteration went of control, divergence, the point was plotted in a color with respect to how quickly it escaped.
When testing a point in a plane to see if it is part of the set, the initial value of z is always zero. This is so because zero is the critical point of the equation used to generate the set.

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Fractals Definition & Properties

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The Definition of a Fractal states that it is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole. What this basically means is that a fractal is an extremely complex shape that looks more or less the same at all magnifications. This basically means that if you zoomed in on any fractal it wouldn’t look any different. For instance, If you look carefully at a fern leaf, you will notice that every little leaf - has the same shape as the whole fern leaf. You can say that the fern leaf is self-similar. The same thing applies to fractals: you can magnify them an infinite amount of times and after every time you will see the same shape, which is characteristic of that particular fractal. Fractals are made by certain rules called algorithms. Also fractals have many characteristics & properties, Such as self-similarity & scale symmetry.

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Fractals as Optical Illusions

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One of the more trivial applications of fractals is their visual effect. Not only do fractals have a stunning aesthic value which basically means that they are really pretty images, but they also have a way to trick the mind such as an optical illusion.

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A Little Fractal Geometry

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Let’s say that (name) wants to be a computer Animation Programmer, using applications like Photoshop. One extremely difficult thing that (name) would encounter would be making natural landscape such as mountains, trees, and clouds. Also if someone zoomed in on the landscape, it would be fuzzy and missing a lot of detail. Well, a thing that Mathematicians invented, are Fractals. A famous quote by Benoidt Mandelbrot, "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." In fact, there is a whole branch of geometry called Fractal Geometry. The math that we are currently learning this year is called Euclidean geometry, discovered by Euclidean, uses lines, ellipses, circles, etc. However the math that we are going to be talking about today is Fractal Geometry, a whole, relatively new branch of geometry using certain rules & instructions on how to make a fractal called Algorithms.

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Scale Symmetry

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A more subtle form of scale symmetry is demonstrated by fractals. As conceived by Mandelbrot, fractals are a mathematical concept in which the structure of a complex form looks exactly the same no matter what degree of magnification is used to examine it. A coast is an example of a naturally occurring fractal, since it retains roughly comparable and similar-appearing complexity at every level from the view of a satellite to a microscopic examination