Geometry Seminar


Circle Packings and the Weierstrass P-function


Dr. Edward Crane

University of Bristol, UK & IMR Junior Fellow, HKU




In the 1930s Koebe proved that any triangulation of the sphere can be represented by a finite collection of geometric discs in the standard 2-sphere, one for each vertex, such that adjacent vertices correspond to externally tangent discs. Moreover the resulting circle packing is unique up to Mobius transformations. The theorem was rediscovered in the context of polyhedra in hyperbolic 3-space by Andreev, and then reinterpreted by Thurston, who viewed it as a discretization of the conformal structure on the Riemann sphere. He also gave an algorithm for computing the packing. Thurston's idea has been developed into an interesting theory of "discrete analytic functions", which captures the geometry but not the arithmetic of holomorphic functions. There are interesting connections with 3-dimensional hyperbolic geometry, integrable systems, and discrete minimal surfaces.

I will describe the main results of this area and also some open problems, then demonstrate a new construction of discrete versions of Weierstrass P-functions.



June 10, 2008 (Tuesday)


3:00 – 4:00pm


Room 517, Meng Wah Complex, HKU




All are welcome