  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.
