If you would like to use POV-Ray to visualize 3-dimensional geometrical objects, I highly recommend Friedrich A. Lohmueller‘s beautiful tutorial on Analytical Geometry with POV-Ray. Other POV-Ray tutorials by Friedrich A. Lohmueller can be found here.
[ image courtesy of Friedrich A. Lohmueller ]
In my humble opinion, this is how Analytical Geometry should be taught. Descartes had no access to 3-d graphics. We do. Why not take advantage of the technology?
Harmonographs are mechanical devices that employ coupled pendula (aka: “pendulums”) to control the movement of a pen relative to a drawing surface (e.g., a sheet of paper). The pen’s motion can produce some strikingly beautiful curves.
I had never heard of harmonographs until I read Mike Croucher’s post: Simulating Harmonographs. Instead of building a physical apparatus, Mike simulated harmonographs with Mathematica. The resulting drawings are indeed most aesthetically pleasing:
[ image courtesy of Mike Croucher ]
Drawings produced with harmonographs remind me of the drawings I used to make with the Spirograph toy when I was a kid. They also remind me of the Lissajous curves I liked to visualize on the oscilloscope when I was an undergraduate student and spent countless hours in the electronics laboratory.
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Some links:
Here’s an excellent video on how to turn a sphere inside out. The animation is about 20 minutes long, but well worth watching.
Hat tip: Nuclear Phynance
Possibly related:
Here’s a very sexy gallery of algebraic surfaces. One of my favorite surfaces is the one defined by equation 
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Possibly related:
If you are interested in minimal surfaces, you might want to take a look at the minimal surfaces page at Indiana University. In particular, I suggest you take a look at the gallery and the archive. Enjoy!
[ image courtesy of Matthias Weber ]
Other galleries worth taking a look at:
Some scientific papers:
Some articles on this topic:
