# Reflections

## 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