David Dumas and François Guéritaud
In this project, we study the space PML(S) of projective measured laminations on a surface S of negative Euler characteristic. Our goal is to make meaningful pictures of this space and its hierarchical structure using William Thurston's embedding of PML(S) into a cotangent space of Teichmüller space. In these visualizations, the points in PML(S) representing simple closed hyperbolic geodesics are marked by spheres; larger spheres correspond to shorter geodesics. In this way, the intricate structure seen in these images reflects both the topology of the surface and its hyperbolic geometry.
Two views of the space of projective measured lamination space of the five-punctured sphere, shown in stereographic projection from the 3-sphere to flat 3-dimensional space. The positions of 200,000 simple closed geodesics are marked by spheres. The lower panel highlights geodesics that lie in certain four-punctured sphere subsurfaces.
This animation shows a mathematical object rotating in 4-dimensional space, flatted into 3-dimensions using stereographic projection. The object depicted is the space of projective measured laminations on the 5-punctured sphere, which is a sort of generalized polyhedron with an infinite number of faces. In this animation, only the "corners" of the polyhedron are shown. Each blue sphere represents a closed geodesic on a hyperbolic 5-punctured sphere, with the radius of the sphere inversely proportional to the hyperbolic length of the associated geodesic.
The projective measured lamination space of the three-punctured projective plane, realized as the boundary of a convex body in 3-dimensional space using Thurston's embedding. Larger dots represent one- and two-sided simple closed geodesics, and smaller dots represent multicurves. While each one-sided simple geodesic is isolated among simple geodesics, the two-sided geodesics are dense in an apollonian gasket (as shown by M. Scharlemann in 1982). Production assistance by Bathsheba Grossman.
This animation shows the projective measured lamination space PML(N_{1,3}) of the non-orientable surface obtained by removing three points from the real projective plane. This space is shown in Thurston's d(log(length)) embedding for a hyperbolic structure on N_{1,3}.
Colored spheres represent simple curves, either one-sided (blue) or two-sided (red). Larger spheres represent shorter curves. The transparent gray spheres represent multicurves.