The number of convex tilings of the sphere by triangles, squares, or hexagons

摘要

A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most 6, 4, or 3 polygons adjacent to it, respectively. Assigning an appropriate weight to any tiling, our main results are explicit formulas for the weighted number of convex tilings with a given number of tiles. To prove these formulas, we build on work of Thurston, who showed that the convex triangulations correspond to orbits of vectors of positive norm in a Hermitian lattice Lambda subset of C-1,C-9. First, we extend this result to convex square and hexagon tilings. Then, we explicitly compute the relevant lattice Lambda. Next, we integrate the Siegel theta function for Lambda to produce a modular form whose Fourier coefficients encode the weighted number of tilings. Finally, we determine the formulas using finite-dimensionality of spaces of modular forms.