Mandelbrot and Fractals

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:

 Mandelbrot

Mandelbrot Set

Mandelbrot Corner

Julius
 

Julius Set for some "c" (that I forget)

Burning Ship Fractal

Lagrange Equations and Trebuchets

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:

Trebuchet Mechanics by Donald Siano

a free GUI-based program that performs these simulations can be found on his website:

The Algorithmic Beaty of the Trebuchet

Eigenvalues and Fibonacci

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.