I am currently a student at Purdue University studying Astronautical Engineering and Mathematics. I want to use this blog to share cool thoughts, interesting projects, and fun things I've learned. Enjoy!
P.S. I especially welcome comments and criticism from any experts in the fields I talk about. While I consider myself fairly knowledgeable about some things and will try to provide sources, I'd rather learn where I might be wrong and discuss it openly.
The Mandelbrot set is determined by the set of complex numbers which do not diverge when the equation
is iterated. This equation diverges once the magnitude of "z" is greater than 2. The number of iterations before "z" diverges is found at each point in the complex plane and assigned a color based on the number of iterations.
The Julius sets are generated from the Mandelbrot equation when
where "c" is a complex number which determines the different Julius set and does not vary over the complex plane like it does for Mandelbrot.
I decided it would be cool to make a program that could display various fractals at different resolutions and regions in the complex plane. Here are some cool fractals I plotted:
A couple of years ago, I had to make a catapult as part of an elective in high school. This project got me thinking about how someone might try to predict the performance of one of these contraptions before they committed to building it. I did some research and found out about dynamic systems and the Lagrange equations. Here, I derive these equations and relate them to the topic of functional derivatives and the action integral. I also show how they can be applied to a theoretical trebuchet using Matlab!
In doing a little research, I found a webpage by Donald Siano where he has written a more robust piece of software for making trebuchet calculations. He also has a paper in which he describes the algorithms he has used, which are based on the Lagrange equations. His paper details the algorithmic treatment of more complex trebuchets in much the same way I have treated a simple one. A link to his paper is:
Back when I was learning Linear Algebra, I always found it ridiculously cool that one could reduce a wide range of recursive sequences to discrete functions by representing them as repeated applications of a matrix. The matrix could then be diagonalized and an analytic solution found. One cool example was finding the discrete function that describes the Fibonacci sequence. I decided to explore it a little deeper and share it with you. It turns out that the Fibonacci sequence is intimately related to the Golden Ratio; in fact, it's one of its eigenvalues! Moreover, there are continuous and complex extensions to the sequence.
This is an essay I wrote as part of my application for a NASA scholarship. I should be hearing back about it next month. Funny story though, at first I read the instructions and though it said the maximum length was 10,000 words; so, I was about half way there when I realized that it actually said 10,000 characters. Such a me thing to do lol suddenly I had quite a bit of revising to do! I really didn't want to cut my paper by more than half cause I hit some really interesting points, but that's the way it goes.
So, here is the short version:
...and the long one: (spelling and polishing might not be the best 'cause I had to shorten it lol)
Although, the above technical challenges are relevant, the most pressing problem facing Aeronautics is best explained by Neil DeGrasse Tyson:
"If you want to build a ship, don't drum up people to collect wood and don't assign tasks and work, but rather teach them to long for the endless immensity of the sea."
-Antoine de Saint-Exupery
If you have not already, watch Carl Sagan's Cosmos. It is well worth it.
The first think I'd like to post is really the culmination of years of work in high school. While I am continuing to pursue the topics of CFD and hybrid rocket propulsion, these are the first papers I really have to show for it. Along these lines, I have developed numerous related Fortran and Matlab codes which I will be giving you access to as well. In fact, having this blog might even motivate me to turn many of my scripts and functions into GUIs or executable files. Let me know if you would like this.
Upon visiting Purdue and taking his Fluid Mechanics class my first semester, I became friendly with Dr. Steven Collicott and shared these papers with him. Passing the one about gas dynamics along to Dr. Gregory Blaisdell, I was offered a research position this summer as a part of the Summer Undergraduate Research Fellowship! Of course, I happily accepted the offer and will be studying the application of high order Weighted Essentially Non-Oscillatory (WENO) schemes in conjunction with centralized compact methods for the Large Eddy Simulation (LES) of supersonic turbulent flows. I will definitely be keeping you posted about this, since it is just that awesome.
