A small polygon is a convex polygon in the euclidean 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 F_6, a plane hexagon of unit diameter can have, satisfies a 10th degree irreducible equation with 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 F_6 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 an explicit equation for F_8 (resp F_10) of degree 42 (resp 152) via intriguing symbolic iterated discriminants computations (sometimes involving almost 3000 digit numbers).
As a second topic we shall explain a direct approach to computing the area- (or the circumradius-) equations for cyclic polygons in particular heptagons and octagon. The results for the circumradius is cyclic heptagons or octagons are obtained in extremely compact form, but for area we still need more powerful computers (probably with more than one TB of RAM) than we have so far at our disposal.
It i s a part of an ongoing collaboration with Charles Audet and Pierre Hansen.
Welcome to everyone!