Constructing and coloring fractal graphs

1. Constructing and coloring fractal graphs

2. Exploring fractal graphs Mandelbrot Sets

3. Mandelbrot Sets The closer you magnify, other fractal graphs emerge. http://www.softlab.ntua.gr/miscellaneous/mandel/mandel.html

4. Julian and Mandelbrot Sets Side by side comparisons of both sets. http://aleph0.clarku.edu/~djoyce/julia/explorer.html

5. Mathematics needed in constructing fractal graphs Three main ideas in understanding construction: -functions -graphs -imaginary numbers Fractal graphs are graphs of different types of functions.

6. Julian Set f(x) = f(x)² + c Mandelbrot Set f(x) = f(x - 1)² + c Recursion Law x = coordinates of point, c = complex numberIn this equation, c, a complex number (contains an imaginary number). It can be of any value and the result will be a different Julian set.The letter x stands for the coordinates of the point. The coordinates are special because they deal with imaginary numbers.

7. Julian Set f(x) = (2 + i)² + (1 + i) = (2 + i)(2 + i) + (1 + i) = 4 + 2i + 2i + i² + 1 + I = 5 + 5i – 1 = 4 + 5i Mandelbrot Set f(x) = (2 + I - 1)² + (1 + i) = (1 + i)(1 + i) + (1 + i) = 1 + i + i + i² + 1 + i = 2 + 3i – 1 = 1 + 3i Lets do the mathSubstitute the following complex numbers.x = 2 + I, c = 1 + i

8. How is color selected? First you need a point to color. Let's take the point (2 + 1i). For our c value, we'll use (1 + 1i). Remember, the c value can be any complex number. Remember, if you run a set of coordinates through a function, the result is a new set of coordinates. 4 + 5i or 1+3i are new sets of coordinates. The work shown above represents one iteration. We continue to run each new set of coordinates through the function, “positive feedback loop”, until we can prove that the point will a.) leave the graph or b.) never leave the graph (the rule is after 200 iterations, if the point is still on the graph, it will never leave.) This is how a color is selected. If the point leaves after one iteration, it is assigned a color. Every point after, that leaves the graph after one iteration, is that same color. All points that leave after two iterations will be assigned a different color, and so on. Every point that never leaves the screen is assigned one color, usually black. After doing this process for each and every point of the graph, the result could look something like this Julian set.

9. Final Construction To construct a fractal on a graph we need about 303,200 points with 200 iterations per point. Graphs of fractals portray natural structures of everyday life. i.e. clouds, plants, landscapes, etc. Many of these fractal graphs are utilized in many technological setting: landscapes and backdrops for movies, video games, etc.

10. Web Resources http://www.uen.org/themepark/patterns/fractal.shtml http://oak.kcsd.k12.pa.us/~projects/fractal/index.html http://mathforum.org/alejandre/workshops/fractal/fractal3.html http://local.wasp.uwa.edu.au/~pbourke/fractals/fracintro/ http://math.rice.edu/~lanius/frac/ http://www.softlab.ntua.gr/miscellaneous/mandel/mandel.html