Finding the Sequence of Largest Small n-Polygons by Numerical Optimization

TL;DR

This paper develops high-precision numerical methods to determine the largest small polygons with n vertices, especially for even n, providing new estimates and models that improve upon previous results.

Contribution

The authors introduce a revised optimization model and apply numerical methods to accurately estimate the maximum area of LSP(n) for even n up to 1000, surpassing prior results.

Findings

Numerical estimates for A(n) for even n up to 1000.

Close agreement with or improvement over previous results.

Regression models for the entire sequence of A(n).

Abstract

LSP(n), the largest small polygon with n vertices, is the polygon of unit diameter that has maximal area A(n). It is known that for all odd values $n \geq 3$, LSP(n) is the regular n-polygon; however, this statement is not valid for even values of n. Finding the polygon LSP(n) and A(n) for even values $n \geq 6$ has been a long-standing challenge. In this work, we develop high-precision numerical solution estimates of A(n) for even values $n \geq 4$, using the Mathematica model development environment and the IPOPT local nonlinear optimization solver engine. First, we present a revised (tightened) LSP model that greatly assists the efficient solution of the model-class considered. This is followed by numerical results for an illustrative sequence of even values of n, up to $n \leq 1000$. Our results are in close agreement with, or surpass, the best results reported in all earlier studies. Most of these earlier works addressed special cases up to $n \leq 20$, while others obtained numerical optimization results for a range of values from $6 \leq n \leq 100$. For completeness, we also calculate numerically optimized results for a selection of odd values of n, up to $n \leq 999$: these results can be compared to the corresponding theoretical (exact) values. The results obtained are used to provide regression model-based estimates of the optimal area sequence {A(n)}, for all even and odd values n of interest, thereby essentially solving the entire LSP model-class numerically, with demonstrably high precision.