This summer I spent some time thinking about animation and visualizing algebraic and geometric information. I have a longer to-do list than accomplishments but I have made some progress.
One of pilot project ideas was to take a 5-variable polynomial ideal and:
1) compute the Groebner fan using Sage and Gfan,
2) intersect it with a hyperplane (so now we're down to 4 dimensions)
3) slowly rotate the resulting polyhedral complex in 4 dimensions, rendering it using Tachyon/Sage
4) animate the resulting set of frames.
For step 4, I initially wanted to use Blender, but that was really overkill for what I needed and I didn't want to figure out how to get Sage and Blender using the same copy of Python (although someone should). In the end I used ffmpeg to get my movie.
Check out my current best effort.
My next goal in this direction is to do something with Sage's @interact command and JMol to highlight pieces of the fan, since the movie isn't really informative (more art than math I think).