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Finding Largest small polygons via symbolic computations

A small polygon is a convex polygon (in a plane) of unit diameter. The problem of determining the largest area of small n-gons was already studied by Reinhardt in 1922. He showed that for n odd the regular n-gon is optimal. For even n this is not the case. For n = 6 the largest area F6, a plane hexagon of unit area can have, satisfies a 10th degree irreducible equation wit integer coefficients. This is the famous Graham’s largest little hexagon (1975). R.L. Graham (with S.C. Johnson) needed factoring a 40-degree polynomial with up to 25-digit coefficients. Graham introduced the diameter graphs by joining the vertices at maximal distance. For n=6 (resp. 8) there are 10 (resp. 31) possible diameter graphs. The case n = 8 was attacked by C. Audet, P. Hanson, F. Messine via global optimization (10 variables and 20 constraints) which produced (an approximate) famous Hansen’s little octagon. In this talk we report on reduction for F6 of the auxiliary polynomial to degree 14 (instead of 40) by rational substitutions (a “missed opportunity” in Graham and Johnson’s approach). Also for the first time, under axial symmetry conjecture, we obtained explicit equations for F8 (resp. F10) of degree 42 (resp. 152) via intriguing symbolic iterated discriminants computations (sometimes involving 2800 digit numbers).

small polygons , maximal area, symbolic computations , diameter graphs,

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