Spherical… Polygons.

I’ve been working on some equations to quantify distortion due to projecting planar objects onto spherical surfaces.

For instance, suppose you wanted to calculate the surface area of a polygon projected onto a sphere, or the surface area of Idaho. How would you do this? Is the area of the polygon in flatland proportional to the  area of the polygon on a sphere? Nope, it’s not that simple. It even involves a little calculus.

Equations and examples (hopefully) coming soon.

Pendulum Wave Derivation and Bill of Materials

I posted a while ago that I had worked out some of the math for a pendulum wave. Attached are PDF files detailing the derivation of my specific variation of the pendulum wave and the corresponding bill of materials.

Pendulum Wave Tutorial (PDF)

Pendulum Wave Bill of Materials (PDF)

There is small problem with the bill of materials. On page 6 (arm dimensions), I have only one of the two arm types shown. Looking from the assembly drawing on the first page, it should be clear that the cut and grooved side of the arm alternates for each pendulum. The first and last arms should have inward-facing grooves.

Plane Puzzle – My Solution

This post is in response to a puzzle posted by a friend of mine.

The puzzle is as follows:

“If a plane is placed on a giant treadmill, and the treadmill is programmed to automatically match the plane’s forward speed (in the opposite direction), can the plane take off?”

Before I begin to solve the solution, here are a few assumptions: Continue reading

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

Origami Buckyball

I’ve begun work on a rather largish buckyball. Below is a section that will end up composing about 1/12 of the finished thing. There will be 12 sections of 4 colors (yellow, blue, green and red), or 3 sections of each color. I’m not exactly sure how I will connect these sections; I suppose I’ll find out when/if the time comes.

Yes, that is an Escher in the background.

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


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

Continue reading