Alec S.
Julia Set Fractal, C = .28 + .008i
As enthusiastic about mathematics as I am, the famous Mandelbrot set is of high interest to me. The Mandelbrot set is the very first computer generated fractal discovered by the Polish-born French-American Mathematician Benoit B. Mandelbrot, and is also know for contributing to and founding the branch of mathematics called Fractal Geometry. Simply put, the Mandelbrot set is a visual representation of an iterated function on the complex plane, and represents every possible Julia set. The formula used to graph the Mandelbrot set is Z_(n+1) = Z_(n)^2 + C, in which Z is a complex number and C is a complex number as well as a constant. When it is generated, every point make its own path of iterations where the starting point is the point who's behavior is being tested and therefore is the constant for that particular chain of iterations. After each iteration, the absolute value of the number calculated for that iteration is compared to the number 2. The rule is if its absolute value is less than or equal to 2, it spirals toward a fixed point, is a part of the set, and is colored black (in this case red). If the absolute value of the number calculated for that iteration is greater than 2, it approaches infinity, escapes the set, and is colored any other color (in this case black). Over the years, mathematicians have said that the Mandelbrot set is the most complex thing humans have ever discovered, since it is basically infinite complexity, or complexity within complexity. For this screen recording, I used the program Geogebra 5, which is an interactive and educational program used for geometry, algebra, calculus, physics, and statistics. In case anyone was wondering, this only shows that the Mandelbrot set looks like at 19 iterations.
For more information, I have a few links to explain further.
Mandelbrot set - Wikipedia
en.wikipedia.org/wiki/Mandelbrot_set
The Amazing Mandelbrot Set tutorial - YouTube
www.youtube.com/watch?v=0YaYmyfy9Z4
Mandelbrot Set -- from Wolfram MathWorld
mathworld.wolfram.com/MandelbrotSet.html
Fractals The Hidden Dimension HD 108p Nova - YouTube
www.youtube.com/watch?v=wkI0y43EqHI
This particular fractal is called a Julia set, named after the French mathematician Gaston Julia. Like I mentioned with the Mandelbrot set (www.flickr.com/photos/44952145@N07/46290813045/in/datepos...), every point on the complex plane represents a Julia set fractal. The Julia set is different because for every Julia set, its corresponding C value is the constant used in every point's chain of iterations.
Julia Set Fractal, C = .28 + .008i
As enthusiastic about mathematics as I am, the famous Mandelbrot set is of high interest to me. The Mandelbrot set is the very first computer generated fractal discovered by the Polish-born French-American Mathematician Benoit B. Mandelbrot, and is also know for contributing to and founding the branch of mathematics called Fractal Geometry. Simply put, the Mandelbrot set is a visual representation of an iterated function on the complex plane, and represents every possible Julia set. The formula used to graph the Mandelbrot set is Z_(n+1) = Z_(n)^2 + C, in which Z is a complex number and C is a complex number as well as a constant. When it is generated, every point make its own path of iterations where the starting point is the point who's behavior is being tested and therefore is the constant for that particular chain of iterations. After each iteration, the absolute value of the number calculated for that iteration is compared to the number 2. The rule is if its absolute value is less than or equal to 2, it spirals toward a fixed point, is a part of the set, and is colored black (in this case red). If the absolute value of the number calculated for that iteration is greater than 2, it approaches infinity, escapes the set, and is colored any other color (in this case black). Over the years, mathematicians have said that the Mandelbrot set is the most complex thing humans have ever discovered, since it is basically infinite complexity, or complexity within complexity. For this screen recording, I used the program Geogebra 5, which is an interactive and educational program used for geometry, algebra, calculus, physics, and statistics. In case anyone was wondering, this only shows that the Mandelbrot set looks like at 19 iterations.
For more information, I have a few links to explain further.
Mandelbrot set - Wikipedia
en.wikipedia.org/wiki/Mandelbrot_set
The Amazing Mandelbrot Set tutorial - YouTube
www.youtube.com/watch?v=0YaYmyfy9Z4
Mandelbrot Set -- from Wolfram MathWorld
mathworld.wolfram.com/MandelbrotSet.html
Fractals The Hidden Dimension HD 108p Nova - YouTube
www.youtube.com/watch?v=wkI0y43EqHI
This particular fractal is called a Julia set, named after the French mathematician Gaston Julia. Like I mentioned with the Mandelbrot set (www.flickr.com/photos/44952145@N07/46290813045/in/datepos...), every point on the complex plane represents a Julia set fractal. The Julia set is different because for every Julia set, its corresponding C value is the constant used in every point's chain of iterations.