Abstract
A tiling of the sphere by triangles, squares, or hexagons is convex if every vertex has at most , , or 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 . First, we extend this result to convex square and hexagon tilings. Then, we explicitly compute the relevant lattice . Next, we integrate the Siegel theta function for 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.
Citation
Philip Engel. Peter Smillie. "The number of convex tilings of the sphere by triangles, squares, or hexagons." Geom. Topol. 22 (5) 2839 - 2864, 2018. https://doi.org/10.2140/gt.2018.22.2839
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