Ranking Small Regular Polygons by Area and by Perimeter

Asymptotic Behaviour of Optimal Spectral Planar Domains with Fixed Perimeter

We consider the problem of minimizing the kth Dirichlet eigenvalue of planar domains with fixed perimeter and show that, as k goes to infinity, the optimal domain converges to the ball with the same perimeter. We also consider this problem within restricted classes of domains such as n−polygons and tiling domains, for which we show that the optimal asymptotic domain is that which maximises the ...

Measuring regularity of convex polygons: experimental results

We aim to evaluate to which extent the shape of a given convex polygon is close to be regular, focusing on diverse characteristics of regularity: optimal ratio area-perimeter, equality of angles and edge lengths, regular fitting, angular and areal symmetry. We have designed and implemented algorithms to compute the resulting measures and we provide and discuss experimental results on a large se...

On the Symplectic Volume of the Moduli Space of Spherical and Euclidean Polygons

In this paper, we study the symplectic volume of the moduli space of polygons by using Witten’s formula. We propose to use this volume as a measure for the flexibility of a polygon with fixed side-lengths. The main result of our is that among all the polygons with fixed perimeter in S or E the regular one is the most flexible and that among all the spherical polygons the regular one with side-l...

Scaling function for self-avoiding polygons

Exactly solvable models of planar polygons, weighted by perimeter and area, have deepened our understanding of the critical behaviour of polygon models in recent years. Based on these results, we derive a conjecture for the exact form of the critical scaling function for planar self-avoiding polygons. The validity of this conjecture was recently tested numerically using exact enumeration data f...

Ranking efficient units by regular polygon area in DEA