Mathematica as a Design Tool
On of the great disappointments of earning a degree in engineering is the later realization that of the four years of math that you were forced to endure has very few applications in the social realm. What is even more disheartening is the realization that you own expensive software capable of working with mathematical concepts that seemed terribly impractical in class and even more so now. Fortunately, some of us disheartened engineers now tinker in web design and are afforded the opportunity to resurrect the impractical. My copy of Mathematica lay dormant until I had an urge to explore "mathematical spaces" one night. Mathematical spaces is simply the pseudo-reality of seemingly tangible objects that have come from representations that serve no other purpose than to aid the human mind in understanding. We graph to understand, and in doing so, new worlds have become familiar to us. Immediately, I began to work these mathematical objects into my design work.
One of the benefits of packages like Mathematica is that it allows you to not only graph very complex 3 dimensional models but also to export those models and graphics into formats that are readable by most graphics programs. Below is an example using Bryce and Photoshop to render the object for the web.
#1: Modeling (Mathematica)
The first trick is obviously to find a mathematical formula that produces something that looks relatively interesting. Mathematica graphing code is all over the web, so if your math skills need a refresher course just Google around until you find something. Some suggested searches: Klein bottles, Trefoil knots, or Riemann Surfaces.
One you have something of interest, you can export the model using the very obvious Export command. I have chosen to export into DXF, an AutoCad drawing file format, so that the model can be manipulated in a 3d graphics package before a final rendering is made. Additionally, the Export command allows you to export into raster and vector graphics formats if you aren't interested in doing any 3d manipulation.
<< Graphics`Shapes`
bih = ParametricPlot3D[
#2: 3d Rendering (Bryce)
With all the data points for my model contained in the file gr.dxf, I import that file into my 3d graphics package of choice. In this case, I am using Bryce only because it happened to be handy but any package should work. Rumor has it that there are even PovRay add-ons to import Mathematica models. I also often use 3d Studio Max for this stage as it tends to render better than Bryce (as it should given the enormous price difference). One word of warning: Bryce tends to crash if you attempt to import anything significant.
One of the nice things about importing into a 3d graphics package is that I can now manipulate the object that I created. In this case, I simply made three copies and translated them up the vertical axis. Finally, I select a material to apply to the object. In this case, a simple orange material will suffice. Then I render a bitmap image that I can easily work with in Photoshop.
#2: Final Touches (Photoshop)
Finally, Photoshop can be used to touch up the image. In this case, Bryce rendered the drop shadow a little too cleanly so I put the shadow into a layer of its own and applied a Guassian filter to soften the edges.
QED.
Mathematica as a design tool is really nothing new and there are some, like Michael Trott, who have taken it to a new level. However, for obvious reasons, it still tends to be the design tool only for total nerds. As a result, almost all of the mathematic art on the web is rendered in a hideous marble texture. For most web design, the arbitrary shapes of mathematics are poorly suited for anything but eye candy, but damn good eye candy it is.