The Bring sextic of equilateral pentagons
Lyle Ramshaw

TL;DR
This paper explores the geometric and conformal structure of the moduli space of equilateral pentagons, revealing it as a Bring sextic surface with a hyperbolic tiling pattern, and provides methods for visualizing these shapes.
Contribution
It identifies the moduli space of equilateral pentagons as a Bring sextic with a specific conformal structure and develops a conformal mapping technique for visualization.
Findings
The moduli space is a genus 4 Riemann surface with 120 automorphisms.
It conformally embeds in the hyperbolic plane as a pattern of 240 triangles.
A method for plotting pentagons using isothermal coordinates and conformal maps is developed.
Abstract
Consider equilateral pentagons 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 known topologically to be a quadruple torus (genus 4). We study 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 inherits a Riemannian metric from the Grassmannian approach of Hausmann and Knutson, a metric under which has 240 isometries: an optional reflection combined with any permutation of the order in which the five edge vectors get assembled into a pentagon. Giving the conformal structure imposed by yields a compact…
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Taxonomy
TopicsMathematics and Applications · Advanced Mathematical Theories and Applications · Advanced Theoretical and Applied Studies in Material Sciences and Geometry
