  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.
