The Bring sextic of equilateral pentagons

Consider equilateral pentagons $V_1,\ldots,V_5$ in the Euclidean plane. When we identify pentagons that differ by translation, rotation, and magnification, the moduli space of possible shapes that we get is an oft-studied polygon space: a 2-manifold $E_5$ known topologically to be a quadruple torus (genus 4). We study $E_5$ geometrically, our goal being a conformal map of that terrain of possible shapes. The differential geometry that we use is all due to Gauss, though much of it is named after his student Riemann. The manifold $E_5$ inherits a Riemannian metric from the Grassmannian approach of Hausmann and Knutson, a metric $e_5$ under which $E_5$ has 240 isometries: an optional reflection combined with any permutation of the order in which the five edge vectors $V_{k+1}-V_k$ get assembled into a pentagon. Giving $E_5$ the conformal structure imposed by $e_5$ yields a compact Riemann surface of genus 4 with 120 automorphisms: the 120 isometries that preserve orientation. But there is only one Riemann surface with those properties: the Bring sextic. So $(E_5, e_5)$ conformally embeds in the hyperbolic plane, like the Bring sextic, as a repeating pattern of 240 triangles, each with vertex angles of $\frac{\pi}{2}$, $\frac{\pi}{4}$, and $\frac{\pi}{5}$. That conformal map realizes our goal. To plot pentagons on our map, we compute an initial pair of isothermal coordinates for $E_5$ by solving the Beltrami equation \`a la Gauss. We then use a conformal mapping to convert one of those isothermal triangular regions into a Poincar\'e projection of a $(\frac{\pi}{2},\frac{\pi}{4},\frac{\pi}{5})$ hyperbolic triangle.