Pythagoras, “Father of the Shortcut”, is honored every time someone diagonally traverses their campus lawn rather than taking perpendicular sidewalks to class.

Using Pythagorean’s Theorem and High School-level algebra, I will show you a simple derivation I found for an iterative expression of pi. (Yes, it was once again during a long, boring summer internship with nothing to do.)

Before we begin, we must define pi so that we will know how to find it. Pi (π) is a constant which relates the diameter of a circle to its circumference. In other words, if we know both the diameter and circumference of a circle, dividing the latter by the former will yield pi. Continue reading →

While sitting at my desk one day, I wondered how to model a spherical configuration of rounded gears (SCORG). Where does each gear need to be positioned? How should each gear be sized in such a way as to share the same diametral pitch with all other gears? How does the size of each gear affect the overall dimensions? Crappy sketches

%This function plots pendulum wave motion as seen from the top view.
%
% n is the number of pendulums, f is the frequency of the first pendulum
% in the wave, df is the incremental change in frequency between pendulums
% (added), t is the time to display the animation

%set initial time to zero.
t_n=0;

%preallocate lengths of matrices.
y=(1:n);
x=(1:n);

while (t>t_n)

%for each value in x,y matrices, populate with value at current t.
for j=1:n
x(j)=j;
y(j)=sin(2*pi*(f+(j)*df)*t_n+pi/2);
end

7/27/12 – Update: I have posted the process and calculations for creating a spherical gear configuration here.

For a while now, I’ve thought about how awesome it would be to have a spherical configuration of bevel gears with a surface radius of curvature equal to that of the sphere composed. I have made great progress in figuring out the math. If I find time, I might post the calculations in the next month or so. I haven’t added the teeth yet, but that will come sometime after finals.

I spoke to the dean of mechanical engineering at my school, and he gave me permission to build a solid model using the department’s rapid prototyping facilities. So, solid model coming summer 2012! 🙂