Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $\pi/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}} = \mathbb{PC}^{10}_+/\Gamma$. Here $\mathbb{C}^{10}_+ = \{ v \in \mathbb{C}^{10} \vert \; Q(v) > 0\}$ for some quadratic form $Q$ of signature $(1,9)$ on $\mathbb{C}^{10}$, $\mathbb{PC}^{10}_+$ is its projectivization, and $\Gamma$ is a certain discrete group of linear transformations of $\mathbb{C}^{10}$ preserving both $Q$ and the lattice $\mathbb{E}^{10} \subset \mathbb{C}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $\pi/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.

This paper establishes a generalized relationship between the arc length of sinusoidal spirals \(r^n=\cos(n\theta)\) and the area of generalized Lamé curves defined by \(x^{2n}+y^{2n}=1\). Building on our previous work connecting the lemniscate to the squircle, we prove an integral identity relating these two curves for any positive integer $n$, which we further generalize to arbitrary positive real exponents and general superellipses. We further extend this correspondence to a geometric relationship between radial sectors of the Lamé curve and arc lengths of the spiral, providing a physical interpretation where keplerian motion on the Lamé curve corresponds to uniform motion on the spiral. Additionally, we derive an explicit central force law for keplerian motion along the Lamé curve. Finally, we introduce policles--a new class of curves generalizing the squircle--and demonstrate a direct geometric mapping between their sectors and the arc lengths of sinusoidal spirals.