Jos Leys emailed me to let me know about Dimensions, a computer-generated movie illustrating dimensions 2, 3, and 4. The trailer is here. The whole movie can be downloaded for free at dimensions-math.org. (It is also available on DVD.)
The movie was rendered using POV-Ray, and is joint work of Jos, Etienne Ghys, and AurÃ©lien Alvarez.
Update. John Armstrong has downloaded and watched the whole movie, and has posted a lengthy review.