Fractacular [Fractals]

By Rohan Shah, Sean Abraham, and Jon Graven

Welcome to the Fractacular Page. Below, you will find what information on what we (Sean, John, and Rohan) have worked very hard on for the past few weeks. On the left side is a set of links that redirect you to the Java programs. Please read the guide to see if you can run the programs and how to use them.

Project Details [Brief]

Definitions & Concepts

Fractals Iterated Function Systems Escape Time Fractals

Rohan's Work

For my work with fractals, I focused mainly on the Mandelbrot and Julia sets. I used the Java compiler and the integrated development envirenment (IDE) Eclipse on a Windows 7 machine. I created two separate programs, one of which draws the Mandelbrot Set and the other that has the ability to draw several Julia sets. How the two sets work is governed by the equation...

zn+1 = zn^2 + c

We essentially see if a certain starting value for z or c allow this iteration formula to be trapped in a circle of radius two or escape it and draw a color based on that. So, given a visual representation or window, for the Mandelbrot Set, z is constant (0 + 0i) whereas c changes depending on where you are in the window. For the Julia set, z changes depending on location and c remains constant.

Jon's Work

The method I used to generate my Sierpinski triangle and tetrahedron is called the chaos game, a term coined by Michael Barnsley. These fractals are known as Iterated Function Systems (IFS's) This method uses a polygon a randomly selected point inside of the polygon. Then for each iteration, a random vertex of the polygon is chosen and the next point is plotted is a chosen, and constant, fraction of the way along the line between the current point and the vertex. This is repeated a large amount of times and the emerging pattern generally takes the form of a fractal, and in the case of selecting a triangle and the fraction to be one half, an approximation of the Sierpinski triangle is formed. The jump to the tetrahedron simply involves choosing a tetrahedron as the polygon and the fraction to be one half again.

2D Sierpinski Fractal (Triangle)

3D Sierpinski Fractal (Pyramid)

Sean's Work

I focused on implementing Rohan's recursive method and Jon's work with 3D in the Cube. My work was primarily focused on the Menger Sponge and recreating it for the Cave/Cube through C++ and Syzygy.

Future Implementation

Some aspects of the project that we could have worked on:


Important Links & Documents