A small polygon is a polygon of unit diameter.
The question of finding the largest area of small
has been answered for some values of
\(n-\)gons are optimal when
\(n\) is odd and
kites with unit length diagonals are optimal when
\(n=6\), 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.
\(n=6\), we propose a new formulation which involves the factorization of a polynomial with integer coefficients of
degree 14 rather than 40.
\(n=8\), 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