The Equilateral Pentagon at Zero Angular Momentum: Maximal Rotation Through Optimal Deformation

TL;DR

This paper investigates the maximum rotation of an equilateral pentagon with zero angular momentum by analyzing its shape space, identifying optimal shape changes, and quantifying the geometric phase accumulated during these deformations.

Contribution

It characterizes the shape space topology, derives optimal deformation loops for maximal rotation, and quantifies the geometric phase for equilateral pentagons.

Findings

Maximum rotation of approximately 45 degrees around a regular pentagram shape.

Optimal shape change corresponds to zero contours of a specific function B.

Restricting to convex shapes yields a maximum rotation of about 19 degrees.

Abstract

A pentagon in the plane with fixed side-lengths has a two-dimensional shape space. Considering the pentagon as a mechanical system with point masses at the corners we answer the question of how much the pentagon can rotate with zero angular momentum. We show that the shape space of the equilateral pentagon has genus 4 and find a fundamental region by discrete symmetry reduction with respect to symmetry group D_5. The amount of rotation \Delta \theta for a loop in shape space at zero angular momentum is interpreted as a geometric phase and is obtained as an integral of a function B over the region of shape space enclosed by the loop. With a simple variational argument we determine locally optimal loops as the zero contours of the function B. The resulting shape change is represented as a Fourier series, and the global maximum of \Delta \theta \approx 45\degree is found for a loop around the regular pentagram. We also show that restricting allowed shapes to convex pentagons the optimal loop is the boundary of the convex region and gives \Delta \theta \approx 19\degree.