G-2019-57
Using symbolic calculations to determine largest small polygons
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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.
Paru en août 2019 , 10 pages
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