Last edited 12dec15 by wnewber2@illinois.edu

Fractals and the Sierpinski Tetrahedron
By: Wyatt Newberry

# Abstract

The definition of the term fractal is often a topic of debate amoung mathematicians, so for the sake of this project a fractal can be considered to be a set with a dimension that is a fraction. My project's goal is to briefly compare methods of fractal generation, and to create a 3 Dimensional version of the Sierpinski Triangle. This is known as the Sierpinski Tetrahedron. Here's a picture and link to the project: Sierpinski Tetrahedron RTICA

The two methods of fractal generation I've compared are The Chaos Game and recursion. The Chaos Game is an algorithm developed by Michael Barnsley in 1988 that involved declaring specific points, randomly selecting on, then randomly selecting another point from the originally selected point and moving a specificed distance towards the second randomly selected point. While this generally doesn't generate many sensible figures, with certain combinations of points and distances, this method can generate fractals. For example, for the Sierpinski Triangle one needs 3 points, and the distance to be 1/2. The other method is recursion, which is method of problem solving where the solution can be determined by solving smaller instances of the same problem. This is the method I used to generate my final RTICA.

## Latest update

12/15/2015:

Finished final documentation and presentation slides both of which are linked at the bottom of the page. Project is complete.

## Progress

• 10/15/2015:

Project officially approved as of yesterday. I've started to re-write my final proposal. Also, I've started to look at some of the hippochet fractal programs in C/open gl and soon hope to translate a few programs into JS/HTML.

• 10/18/2015:

Finished a version of the Sierpinski triangle in Javascript/HTML. This was done with just the HTML canvas. I haven't used any libraries yet. Sierpinski Triangle in JS/HTML

• 10/22/15:

Completed draft for my proposal: Proposal Draft

• 10/26/2015:

With the help of Graham's THREE.JS tutorial, I was able to get my first THREE.JS program completed. This program adds a tetrahedron to the scene in THREE.JS, and has an event listener that is used to detect mouse movement. If the mouse is moved, the object is rotated. I used the wireframe model because it's easier to use than solids and lightsources. Wireframe tetrahedron in THREE.Js

• 10/30/15:

Finished and turned in second draft of Proposal. Also, with Graham's help I implemented a chaos game version of the Sierpinski Triangle using HTML/Javascript. Chaos Game Sierpinski Triangle

• 11/6/15:

Began working on PowerPoint and notes for presentation. Also working with Graham on the coordinate system in THREE.JS.

• 11/15/2015:

Finished and gave seminar on 11/13/2015. Seminar Working on final project and documentation as of now.

• 11/20/2015:

Finished proof of concept and final project is nearly complete. I still need to do documentation/narrative. Proof of concept

• 11/30/2015:

Finished final version of RTICA. Documentation still needs to be finished. Added some embellishements including zooming in and out, changing the complexity of the RTICA, and toggling rotation. Sierpinski Tetrahedron RTICA

• 12/15/2015:

Finished final documentation and presentation slides both of which are linked at the bottom of the page. Project is complete.

• ## Documentation

• Pre-Proposal
• Proposal
• Seminar Slides
• Final Documentation
• Final Presentation Slides