Groupe d’études et de recherche en analyse des décisions

# Using symbolic calculations to determine largest small polygons

## Charles Audet, Pierre Hansen et Dragutin Svrtan

A small polygon is a polygon of unit diameter. The question of finding the largest area of small `\(n-\)`gons has been answered for some values of `\(n\)`. Regular `\(n-\)`gons are optimal when `\(n\)` is odd and kites with unit length diagonals are optimal when `\(n=4\)`. For `\(n=6\)`, the largest area is a root of a degree 10 polynomial with integer coefficient having 4 to 6 digits. This polynomial was obtained through factorizations of a degree \$40\$ polynomial with integer coefficients.

The present paper analyses the hexagonal and octogonal cases. For `\(n=6\)`, we propose a new formulation which involves the factorization of a polynomial with integer coefficients of degree 14 rather than 40. And for `\(n=8\)`, under an axial symmetry conjecture, we propose a methodology that leads to a polynomial of degree 344 with integer coefficients that factorizes into a polynomial of degree 42 with integer coefficients having 21 to 32 digits. A root of this last polynomial corresponds to the area of the largest small axially symmetrical octagon.

, 10 pages