Volume 22, issue 5 (2018)

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

Philip Engel and Peter Smillie

Geometry & Topology 22 (2018) 2839–2864
Abstract

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 $\Lambda \subset {ℂ}^{1,9}$. 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.

Keywords
triangulations, nonnegative curvature, sphere, tiling, modular form, Thurston, polyhedra, shapes of polyhedra
Mathematical Subject Classification 2010
Primary: 05C30, 32G15, 53C45
Secondary: 11F27