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

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 D5. The amount of rotation ∆θ 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 ∆θ ≈ 45◦ 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 ∆θ ≈ 19◦.