Group for Research in Decision Analysis

# Using symbolic calculations to determine largest small polygons

## Charles Audet, Pierre Hansen, and 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