Notice anything special about the background image? It's part of a Penrose tiling.
This page contains original graphics I created during my study of Math 149B at U.C. Davis.
Much of the material goes well with a hobby I enjoy: computer graphics.
The software I use is available for nearly every platform:
POV-Ray 3.0
Background
At the time, we were studying boundary identification. A simple example is taking a piece of paper and identifying the left edge with the right edge. The surface becomes continuous along this edge and you end up with either a cylinder or Mobius Strip. You can further the process by identifying the remaining two edges of the paper. You then end up with either a torus or Klein Bottle. Try visualizing this process with a piece of binder paper. But note that you can only go so far with the Klein Bottle.
Cube Gluing
In class we considered what would happen if we glued opposite faces of the cube.
A cube has three pairs of opposite faces, and thus there are three "phases" to the
gluing process. I have animated the first two phases.
- Phase I - glue the green faces
- Phase II - glue the yellow faces
- Phase III - glue the blue faces
(350K gif)
Cube Gluing Animation
(714K gif, cyclic)
Cube Gluing Animation
(343K fli)
As the animation shows, the first two phases are relatively straight forward. Why
didn't I animate the third phase? Phase III means gluing blue to blue - the
inside surface of the thick torus to the outside surface such that
the surface normals point toward each other. This isn't possible in three
dimensions.
"Circle worth of Tori"
Let us consider what the third phase might look like in some higher dimension.
Professor Mulase used the expression "circle worth of circles" to describe
the set of configurations of a simple mechanical linkage. This set - a circle
worth of circles - is a torus. In other words, each appropriate slice of the torus
is a circle.
Now consider an appropriate slice of the "torus" created after Phase II. Each
slice (infinitlely thin) is a disk with a hole in it. For such a slice we must
glue the outer edge with the inner edge. What shape is this? It is again a torus
and, therfore, Phase III may be realized as a process which generates a "circle
worth of tori." Can you imagine a surface for which there are an infinite number
of slices yielding an infinite number of tori?
Klein Bottle
I don't know what posessed me, but I just had to try modeling
a Klein Bottle. Click the image to see
a larger version. Perhaps during summer I will persue my idea to
animate a ball rolling along the surface of the bottle.
Q: Why did the chicken cross the Mobius strip?
A: To get to the other ... er, um ...
Available Software and Source
If you are using a PC from home, I recommend downloading the FLI animation rather
than the GIF. You can play it full screen with
Autodesk Animator (82K DOS)
A Windows version is also available:
Autodesk Animator for Windows
The best shareware FRACTAL program I know of is
WinFract. This
is what I used to generate the background for this page and my homepage.
Source code for Cube Gluing Animation:
cube_glue.pov and
cube_glue.ini
Other links
Homepage
I don't know what posessed me, but I just had to try modeling
a Klein Bottle. Click the image to see
a larger version. Perhaps during summer I will persue my idea to
animate a ball rolling along the surface of the bottle.
