Hyperbolic Structure of the Equilateral Pentagon

TL;DR

This paper explores the hyperbolic geometric structure underlying the realization space of equilateral pentagon linkages, providing a conformal parameterization that captures all configurations equally.

Contribution

It introduces a full conformal parameterization of the linkage space using hyperbolic geometry, symmetry, and the Riemann mapping theorem, extending previous combinatorial correspondences.

Findings

Realization space corresponds to a hyperbolic tiling by right-angled pentagons.

The parameterization is fully conformal and treats all points equally.

The approach combines symmetry, Riemann mapping, and normalization techniques.

Abstract

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.