### G-2019-57

# 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.

Published **August 2019**
,
10 pages