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

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 ≥ 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 ≥ 6 has been a long-standing challenge. In this work, we develop high-precision numerical solution estimates of A(n) for even values n ≥ 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 modelclass considered. This is followed by numerical results for an illustrative sequence of even values of n, up to n ≤ 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 ≤ 20, while others obtained numerical optimization results for a range of values from 6 ≤ n ≤ 100. For completeness, we also calculate numerically optimized results for a selection of odd values of n, up to n ≤ 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 modelclass numerically, with demonstrably high precision.