Bug on Notes of Thurston

by Jeffrey Brock and Emily Dumas

This ray-traced image shows approximately 18000 overlapping coplanar solid glass spheres pictured in front of a page from the lecture notes of William Thurston's 1978 course The Geometry and Topology of Three-Manifolds. The intersection of the spheres with the plane containing their centers is a domain of discontinuity of a quasi-Fuchsian group of genus 2 created by a shear-bend deformation of a Fuchsian group.

This image was created in January 2006 for a poster announcing the conference Geometry and Dynamics in Surfaces and 3-Manifolds (held at Brown University, May 3-6, 2007).

For another image in a similar spirit, see “Thurston's Jewel”.

Computation and rendering

The following procedure was used to create the image:

1. Compute generators of the quasi-Fuchsian group from the shear-bend parameters using Mathematica
2. Generate the limit set as a postscript file, using Curt McMullen's program lim
3. Convert the limit set to a list of (center,radius) tuples for the spheres, using lim2pov
4. Scan the background from a mimeographed copy of Thurston's lecture notes (page 8.36 in the original numbering)
5. Integrate the background image and spheres (with glass texture, refraction, reflection, etc.) in a POV-Ray scene file.
6. Render horizontal slices of the scene in parallel on several computers (using a total of 726 CPU-hours)
7. Combine the slices using assembleppm, and convert to a reasonable image format (PNG)

Downloads

• Full-resolution version (2100x1975 PNG, 2.4M)
• Source code (POV-Ray scene files, sphere data, and background image)

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