2004/01/01 by Philip L. Bowers, Kenneth Stephenson · 73 citations
Computer Science · Engineering · Mathematics · #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #3D Modeling in Geospatial Applications #Equilateral triangle #Mathematics #Menagerie #Circle packing #Algebra over a field #Pure mathematics #Combinatorics #Geometry #Art #Art history
paper · doi:10.1090/memo/0805
published in Memoirs of the American Mathematical Society 170(805), 0 (American Mathematical Society)
openalex publication_date 2004/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Grothendieck's theory of Dessins d'Enfants involves combinatorially determined affine, reflective, and conformal structures on compact surfaces. In this paper the authors establish the first general method for uniformizing these dessin surfaces and for approximating their associated Belyi meromorphic functions. The paper begins by developing a discrete theory of dessins based on circle packing. This theory is surprisingly faithful, even at its coarsest stages, to the geometry of the classical theory, and it displays some new sources of richness; in particular, algrebraic number fields enter the theory in a new way. The paper goes on to show that the discrete dessin structures converge to their classical counterparts under a hexagonal refinement scheme. In addition, since the discrete objects are computable, circle packing provides opportunities both for routine experimentation and for large scale explicit computation. A range of examples up to genus 4 is given in the paper, and an a...