I Go to the Workbench

Target Audience: Fractal enthusiasts and/or UNIX types

In my email today [note: this article was begun some time ago and since put aside] was an offer from Wolfram Software, makers of Mathematica, to try out their new "Wolfram Workbench," which promises an integrated development environment and debugger for Mathematica. My curiosity aroused, I decided to go ahead with the enormous download. It came in the standard Wolfram installer format, which is to say a self-extracting shell script.

First problem: the program is bailing from an "error message" produced by tail, warning that the old notation tail -1 should be replaced by the new notation tail -n 1. As this seems easy enough to fix, I go into the script and change the offending line to use the newer convention. This leads directly to the...

Second problem: the installer is checksummed internally somehow, and my alteration to the script has lead the program to believe the download was corrupt. This is helpfully reported to me as "Not enough space on " -- with nothing following. I fix this by extracting the checksum code. The install proceeds to the...

Third problem: the installer, after showing all sorts of things being installed and taking up a good deal of disk space, tells me that the install failed with no other information.

Checking in the install directory, there seems to be a completely installed program, so on a whim I just try starting it up. It works, and I am treated to the familiar Eclipse IDE screen. That's right, I went through all that for an Eclipse plugin. Perhaps all is not lost, however, as the package promises a breakpoint debugger.

Not having any work at hand that would benefit from Mathematica, I decided to try it out with something that might make for a fun article for this site, and coded up a program that generated three-dimensional Sierpinski Gaskets. This took a little playing around with mathematica's 3D graphics primitives, which I've never looked at before, but I managed to get some code done in a reasonable amount of time.

The resulting shape surprised me somewhat. In my mind, I had invisioned that the "hole" between four tetrahedra stacked on top of one another would be tetrahedral. With a little abstract thought I could have easily seen that this would not be the case -- removing a smaller tetrahedron could not possibly impact all four faces of the larger one -- but bad intuitions die hard. Even once I saw the picture, it was difficult to visualize the shape.

I wound up asking my high school buddy Konstantin, who has in the mean time received his B.S. in Physics, what the shape was. After a bit of reasoning he was able to figure it out. Instead of revealing the answer immediately I'd like to give the readers out there a shot at it; I'll post some answers next week or thereabouts. If you want the code, by the way, you can download the Sier.m source file; the syntax is MyGasket[n], where n is the number of levels of recursion you want. Here's an example when n = 5:

For now, I'll just leave you to look at the pretty pictures. I was going to write up some stuff about fractals and the topological concept of dimension, but I've been a bit too busy at the moment. In the mean time, check out the following on Wikipedia:

• Haussdorff-Besicovich Dimension
• (Lesbegue) Topological Dimension

For you math majors out there, try working out these dimensions for the limiting form of the shape above. It's a bit harder that I thought it'd be at first. Answers to come later.