Introduction
This gallery contains 3-dimensional ray-traced convex hulls of limit sets created with the open-source program
POV-Ray. It is a more informal complement to my
gallery of 2-dimensional limit set pictures, and like that gallery the basic tool used to created the limit sets is Curt McMullen's program
lim.
I'm not the first person to make pictures of limit sets and their 3-dimensional convex hulls using ray-tracing; for other images of this type, see:
Software and Hardware
The images and animations below were created using POV-Ray 3.5 running on GNU/Linux (Debian AMD64 unstable). Most were rendered serially during the summer of 2004 on a machine with two AMD Opteron 250 processors and 2Gb of RAM, with per-frame render time on the order of several minutes to one hour. The POV-Ray scripts 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.
Animations
Note: large file sizes!
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.
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.)
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.
