Ray-traced Limit Sets

This gallery contains some animations I created in 2005 showing ray-traced convex hulls of limit sets of families of Kleinian groups. They were created with the open-source program POV-Ray.

I'm certainly not the first person to make pictures of limit sets and their 3-dimensional convex hulls using ray-tracing; for other similar galleries of this type, see:

• Yair Minsky's pictures
• Jeff Brock's gallery
• Yasushi Yamashita's gallery

Maskit Rotation

This animation shows the limit set of a single group near the boundary of the Maskit slice--with circles replaced by spheres--rotating in front of a background page of mathematical text.

Sample frame: 512x512, 1024x1024 Animation: 400 frames, 25 fps 512x512 xvid, 3Mbit/s, size 5.9M 1024x1024 xvid, 7Mbit/s, size 13M

Apollonian Twist and Rotation

This is a variant of the classic twisting animation including the Apollonian gasket where the entire limit set rotates in front of a fixed background. (Compare to Apollonian Twisting 1, Apollonian Twisting 2.)

Sample frame: 512x512, 1024x1024 Animation: 400 frames, 25 fps 512x512 xvid, 3Mbit/s, size 5.7M 1024x1024 xvid, 7Mbit/s, size 14M

Tangential Approach

This animation shows a one-parameter family of limit sets in the Maskit slice approaching the boundary "tangentially" (but not quite reaching it). This type of approach can lead to differing algebraic and geometric limits. The path in the Maskit slice is actually a segment on a carioid whose cusp is at the Apollonian gasket point.

Sample frame: 512x512, 1024x1024 Animation: 253 frames, 15 fps 512x512 xvid, 3Mbit/s, size 1.6M 1024x1024 xvid, 7Mbit/s, size 8.8M

Apollonian Twisting 1

This is the classic twisting animation including the Apollonian gasket. It corresponds to a horizontal line in the Maskit embedding of the Teichmüller space of the punctured torus.

	Glass Sample frame: 512x512, 1024x1024 Animation: 297 frames, 20 fps 512x512 xvid, 3Mbit/s, size 2.5M 1024x1024 xvid, 7Mbit/s, size 13M
	Chrome Sample frame: 512x512, 1024x1024 Animation: 300 frames, 20 fps 512x512 xvid, 3Mbit/s, size 614K 1024x1024 xvid, 7Mbit/s, size 1.8M

Apollonian Twisting 2

This is a twisting animation including the Apollonian gasket, similar to "Apollonian Twisting 1" but following a path that dips down on each side of the prominent cusp of the Maskit slice. This produces an interesting though somewhat jerky animation.

Sample frame: 512x512, 1024x1024 Animation: 300 frames, 20 fps 512x512 xvid, 3Mbit/s, size 1.1M 1024x1024 xvid, 7Mbit/s, size 3.4M

Rendering details

Software and Hardware

The animations were rendered with POV-Ray 3.5 on a linux machine with dual AMD Opteron 250 processors and 2Gb of RAM. Per-frame render times ranged from several minutes to about one hour. The POV-Ray scene files were simple wrappers for the output from lim2pov to set up the lighting, backgrounds, materials, etc.

Backgrounds

Some of the limit sets are rendered with pages from the mathematical writings of Fricke, Klein, Poincaré, etc., in the background. These scanned pages were taken from the archive of historical mathematical monographs at Cornell.