Gallery of Mathematical Images

Introduction

This gallery contains an assortment of mathematical images I have created over the years. While some are related to limit sets and CP1 structures, these images don't quite fit into my specialized galleries of Bers slices (including images, animations, and 3D slices), limit sets, and ray-traced convex hulls.

These images are © 2006-2008 David Dumas. Contact me at ddumas@math.uic.edu if you want to use them elsewhere.

Images

Each thumbnail links to a page with the full image and a brief explanation.
The "converse" of McShane's identity.  In a one-parameter family of punctured torus groups, McShane's sum appears to converge only when the group is quasifuchsian--approximately the open interval (0,0.235) in this case. Bers fire: The hex torus Bers slice colored according to the generalized McShane's sum (which is constant for quasifuchsian groups) Bers fire: The hex torus Bers slice colored according to the generalized McShane's sum (which is constant for quasifuchsian groups) Bers galaxy: Small section of the hex torus Bers slice colored according to the generalized McShane's sum.  Parabolic words produce bright "stars" that fill out the indiscrete region. A bug in my Bers slice program Bear resulted in this strange picture.  It is the discreteness locus in a non-holomorphic slice whose graph in trace coordinates looks like an egg carton.
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Another distorted Bers slice image resulting from a bug in Bear.  This is the discreteness locus in a very strange (non-holomorphic) slice of the representation variety. The lift of a maximal geodesic lamination (red) on a hyperbolic punctured torus to the universal cover. (Produced with lim.) Horocycle orbit for a Kleinian group produced by a cuspidal connected sum (or "plumbing") construction. (Produced with lim.) Horocycle orbit for a Kleinian group produced by a cuspidal connected sum (or "plumbing") construction, drawn in inverted coordinate system.  (Produced with lim.) Horocycle orbit for a Kleinian group.  This image reminds me of threading die.  (Produced with lim.)
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Overlap of cuspy curves:  These roughly self-similar curves are shaped like a cardioid at a dense set of points.  Curves like this form the boundary of the discreteness locus in a slice of the punctured torus representation variety. Trajectories of two different strebel differentials on a genus two surface obtained by doubling the torus along a slit.  These differentials define Teichmüller geodesics that are asymptotic in one direction and divergent in the other (as explained in my paper Grafting, Pruning, and the Antipodal Map on Measured Laminations). Fuchsian centers in a Bers slice without symmetries.  The main theorem in my paper The Schwarzian derviative and measured laminations on Riemann surfaces implies that these lie within bounded distance of a straight line. The square lattice (left) and its image under f(z)=z2 (right).  The images of primitive vectors (filled dots) provide a rough model for the locations of the Fuchsian centers in the square torus Bers slice. The limit set (left) and convex hull (right) for a Kleinian group obtained by bending a Fuchsian group.  (Produced with lim and POV-Ray.)
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Found in my archive of figures; from July 2003.
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(This gallery was created with igal, a static HTML image gallery generator.)