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 model-class 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 model-class numerically, with demonstrably high precision.


Working Paper. Lazaridis School of Business and Economics, Wilfrid Laurier University, Waterloo, ON, Canada.



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