Constructing and coloring fractal graphs

Exploring fractal graphs

Mandelbrot Sets

Mandelbrot Sets

The closer you magnify, other fractal graphs emerge.

http://www.softlab.ntua.gr/miscellaneous/mandel/mandel.html

Julian and Mandelbrot Sets

Side by side comparisons of both sets.

http://aleph0.clarku.edu/~djoyce/julia/explorer.html

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.

Recursion Law

Julian Set

f(x) = f(x)² + c

Mandelbrot Set

f(x) = f(x - 1)² + cx = coordinates of point, c = complex number

In 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.

Lets do the math

Substitute the following complex numbers.x = 2 + I, c = 1 + i

Julian Setf(x) = (2 + i)² + (1 + i)

= (2 + i)(2 + i) + (1 + i)= 4 + 2i + 2i + i² + 1 + I

= 5 + 5i – 1= 4 + 5i

Mandelbrot Setf(x) = (2 + I - 1)² + (1 +

i)= (1 + i)(1 + i) + (1 + i)= 1 + i + i + i² + 1 + i

= 2 + 3i – 1= 1 + 3i

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.

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.

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