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

## TL;DR

This paper derives explicit formulas for counting convex tilings of the sphere by triangles, squares, or hexagons, using advanced lattice and modular form techniques to encode and compute these counts.

## Contribution

It extends Thurston's work on convex triangulations to squares and hexagons, explicitly computes the associated lattice, and derives formulas via modular forms.

## Key findings

- Explicit formulas for weighted counts of convex tilings
- Extension of Thurston's lattice correspondence to squares and hexagons
- Use of modular forms to encode tiling counts

## 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 result is 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 $\Lambda\subset \mathbb{C}^{1,9}$. 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.