Chaos and Fractals: Chaos and Fractals by John Como


Benoit B. Mandelbrot: Benoit B. Mandelbrot Coined the term fractal from the latin word fractus meaning broken and irregular. Created the now famous Mandelbrot set in the late 1950’s with IBM.


What is a fractal?: What is a fractal? A fractal has a very fine structure. Too irregular to be described by regular calculus or traditional geometry. Has self-similarity. Can have a non-integer dimension. Can be described by a simple iterative formula. Can often have a natural appearance.


Simple Fractals: Simple Fractals


M.C. Escher: M.C. Escher Circle Limits IV can be considered a fractal due to its self-similarity. If you zoom in on the edge of the circle, the same picture is repeated to infinity.


Durer’s Pentagons: Durer’s Pentagons Start with a pentagon and divide it into six like pentagons (first iteration). Divide those into six pentagons etc.


Koch Curve: Koch Curve The Kock snowflake is a fractal whose area is enclosed by an infinite perimeter!


Slide8: The Kock curve area increases per iteration but converges at a finite number.


Slide9: The perimeter is infinite! N is the number of sides L is the length of one segment P is the perimeter


How to make a fractal…: How to make a fractal… Start with a recursion formula. Example: The different Julia sets are given by


How to make a fractal…: How to make a fractal… Iterate different z values for the entire complex plain. If, after a set number of iterations, the iterates do not pass a critical magnitude, the point is colored black. If the iterates do pass the critical magnitude, the point is colored corresponding to the number of iterations it went through.


Slide12:


Julia sets for different c values: Julia sets for different c values


The Mandelbrot Set: The Mandelbrot Set Acts as a ‘dictionary’ for the Julia sets.


The Mandelbrot Set: The Mandelbrot Set Same iteration formula, however it contains all the Julia sets within it by connecting them.


Slide16:


What are fractals used for?: What are fractals used for? Fractal pictures bridge the gap between our real world and the mathematical world. Are used in Hollywood Ex. Were used in Star Trek II: The Wrath of Khan for the genesis planet. Are used in video graphic design to create landscapes. Make beautiful art. (M’art’hmaticians)


Application: Application Fractals descibe this coastline.


Application: Application These fractals look like real trees.


Application: Application Fractal images of leaves and mountains.


From chaos comes order and beauty.: From chaos comes order and beauty.


Thanks to…: Thanks to… http://www.jracademy.com http://www.pen.k12.va.us/Div/Winchester/jhhs/math/lessons/calculus/escher.html http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Mandelbrot.html http://hypertextbook.com/chaos/23.shtml Falconer, Kenneth: Techniques in Fractal Geometry http://ecademy.agnesscott.edu/~lriddle/ifs/pentagon/Durer.htm http://en.wikipedia.org/wiki/Image:Cantors_dust_in_seven_iterations.png Devaney, Robert L.; Chaos, Fractals, and Dynamics http://www.fractaldomains.com/fractal-of-the-week/week.html http://www.mcgoodwin.net/julia/juliajewels.html http://aleph0.clarku.edu/~djoyce/julia/julia.html http://www.visualbots.com/tree_project.htm Barnsley, Michael; Fractals Everywhere