Abstract: A triangulation of $S^2$ is combinatorially non-negatively curved if each vertex is shared by no more than six triangles. Thurston showed that non-negatively curved triangulations of $S^2$ correspond to orbits of vectors of positive norm in a lattice $\Lambda \subset \mathbb{C}^{1,9}$ under the action of a group of isometries. Using this description, we show that an appropriately weighted count of triangulations of $S^2$ with $2n$ triangles are the coefficients of a modular form, and specifically that the number is
$$\frac{809}{2612138803200} \sigma_9(n)$$
where $\sigma_9(n) = \sum_{d|n}d^9$. Similar formulas hold for tessellations by squares and by hexagons. This is joint work with Philip Engel.