# Math/Critical Thinking Puzzle – 8/30/12

Here’s a nifty little math puzzle I came up with recently.

Preliminary Information

There are three sets of parallel lines on a 2D surface (plane). Each set contains an infinite number of lines, and each line is infinite. The parallel lines in each set Continue reading

# An Iterative Solution for Pi Using Pythagorean’s Theorem

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

# Spherical Configuration of Rounded Gears – A Derivation

Introduction

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

## Pendulum Wave Demonstration With Matlab

### Video

I’ve decided to build a pendulum wave and write an Instructable to document the process.

In the meantime, here’s a matlab demonstration showing how I hope it will behave. Matlab function is posted after the video.

``` %Created by Eric Cox - https://ericboy.wordpress.com - 5/17/2012```

function [] = PWT(n,f,df,t)

%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

%Plot pendulum wave
plot(x,y,’o’,’MarkerEdgeColor’,’k’,…
‘LineWidth’,2,…
‘MarkerFaceColor’,’b’,…
‘MarkerSize’,11);

%Define axis
axis([0 n+1 -1.5 1.5]);

%Increment loop time
t_n=t_n+1/30;

%Animation delay
pause(1/30);

end

# Spherical Gear Configuration

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! 🙂

Here is the rendering I have so far: 