Pattern formation is a beautiful subject that has strong links to both applied mathematics and the natural sciences. The purpose of this project is to use a multimedia web environment to visualize the quantitative principles underlying reaction-diffusion models of pattern formation.

Inherent in the study of dynamical systems is the idea of change or evolution through time. That is why concepts like fixed points and stability are some of the first things that are taught in introductory dynamical systems courses. Most classes use phase planes to try and highlight the fact that we are studying systems that change over time. However, this same idea can be transmitted more naturally through movies and animations, which allow the viewer to see the systems evolve over time. As a result, the combination of short movies, animations with audio accompaniments, images and written notes will help convey ideas in pattern formation more intuitively, and can be used to supplement an introductory dynamical systems course.

This website was created in such a way that it could be read from beginning to end like any set of course notes, or a chapter in a textbook. Students with a general interest in pattern formation, and some mathematical background should be able to go to the website and gain a better understanding of the mathematical, physical, and historical ideas behind models of pattern formation.

In addition, the interactive nature of the materials can be used as teaching aids in a dynamical systems course in order to supplement the lectures. The integration of the website to a classroom environment can be accomplished in several ways, for example:

- Students can read through the relevant pages, so that they arrive to class with a grasp on the intuitive ideas and can understand the mathematical analysis more readily.
- Flash videos can be played at the beginning of the class
- The animations can be used by instructors to supplement lecture slides, and students can refer to them and listen to the audio accompaniments when they are home to refresh their memory.
- The Matlab demos and the embedded code can help students understand how the animations are created and serve as a primer to the equally interesting and fundamental field of numerical methods

This website will hopefully aid instructors to connect the mathematical concepts to the physical phenomena of pattern formation, and get students interested with the highly interactive materials presented.

I was born and raised in Mexico City. I got my bachelors from Princeton University where I designed my own major in Applied Math and Neuroscience. Now, I’m at the University of Oxford studying how brain circuits get built during development using computer simulations.

This project was created for the DSWeb 2013 contest and will be part of my PACM (Program in Applied and Computational Mathematics) certificate project.

*“And these little things may not seem like much but after a while they take
you off on a direction where you may be a long way off from what other people have
been thinking about.”*
-Roger Penrose

This project would have been impossible without the guidance and support of Dr. Philippe Trinh who agreed to advise me on this project after I took his Differential Equations course at Princeton. Not only did he help me improve my mathematical writing and increase my awareness of color swatches and typography, but he also let me use a section of his teaching website, “The Shape of Mathematics”, for my project. His continued encouragement and support motivated me to get past the roadblocks, and find my way in spite of the (many) changes in direction that this project underwent.

I am grateful to Professor Howard Stone. In addition to being my contact at Princeton when Dr. Trinh was at Oxford, his passion for teaching is an inspiration, and his advice was key in shaping the direction of this project. I would like to thank my academic advisor Professor Philip Holmes for suggesting this project.

The materials on this website were shaped and inspired by my experience in courses at Princeton that highlighted both the mathematics and the applications of dynamical systems including MAT 323, MAT 350, MOL 215, and MOL 410.

The project started in the summer of 2012 at Princeton University and continued during my exchange visit to Oxford University in January-June of 2013. I would like to thank St Edmund Hall, Oxford, for their hospitality during this time.