David Akers

The Rotating Die Problem

Click here for background on the problem. Over the last several weeks, I've been working on figuring out what a cube (and/or hypercube??) looks like when spun rapidly on a vertex. I had some success figuring out the mathematics of the rotations, involving figuring out circumscribed and inscribed circles, but the process of drawing the actual image was frustratingly slow. So I decided to write a computer program to do it. What the program does is to take pictures of a cube as it rotates a full 360 degrees. It then transposes all of the frames to create one two-dimensional image which represents the distribution of matter during the rotation:

As you can see, both the outer shadow and the inner shadow are visible, as well as the transition from outer to inner. I was actually surprised to find out that the inner shadow is a simple diamond! This would mean that the radius of the inscribed circle actually changes linearly with the height of the cube. I hadn't realized this.

I guess what is nice about the program is that it can be used to rotate any type of 3-d object, not just a cube. One could also rotate a pyramid, a torus, or any other shape that can be simply defined.

The Rotating Icosahedron

The Rotating Dodecahedron

The Rotating Octahedron

The Rotating Tetrahedron

Degrees of Freedom

I want to take a little time to talk about Michael Matthews' article from last week.

The main point of his article was that we can't take the dimensionality of an activity or a movement for granted. He seemed to suggest that there is a difference between "degrees of freedom" and "dimension." Thus while human beings exist in what we think of as a three dimensional world, we are (primarily) limited to the surface of this world. While our space is three dimensional, our degrees of freedom are two dimensional. The difference between the two can perhaps best be demonstrated by one of the latest first-person perspective 3-D video games: Descent. Unlike its predecessors, Descent allows the player six degrees of freedom. The system of tunnels which the player explores contains not only "left, right, forward, and back," but also "up" and "down." This of course leads to a startling disorientation (and sometimes even nausea), since the player is not used to thinking in six degrees of freedom. Just as four dimensional objects are not easily grasped by human intuition, neither are worlds with six degrees of freedom. If only we lived in a world like Descent . . .

David Akers