Hyperbolic Structure of the Equilateral Pentagon

The combinatorial structure of the realization space of the euqilateral pentagon linkage is closely related to a tiling of the hyperbolic plane by right-angled pentagons. In this correspondence lower dimensional faces of the tiling correspond to degenerate realizations of the linkage. We extend this combinatorial correspondence to a full conformal parameterization of the space of all such linkaged controlled by one point in the hyperbolic plane. To do so we exploit the symmetry of the realization space, combine it with the Riemann mapping theorem and a normalization procedure introduced by Springborn. The resulting parameterization is “democratic” in the sense of Yoshida Masaaki: All points are treated exactly equal.