I couldn't express in radicals the (decic) Hexagon coordinates (http://mathworld.wolfram.com/GrahamsBiggestLittleHexagon.html) so I N A I V E L Y sought the Octagon. As far as I can Google, the univariate polynomials have never been written. A few hours with Resultant and 8G suggests why. If the bottommost vertex lies at (0,-1), the equation for the next-bottommost is 0= an (apparently irreducible) polynomial of degree 1872 with leading coefficient ~ 2.500474035501180*10^4244. The relevant root is ~-0.8946150293614985078063489766550846192654082886696687277707235119368\ 3381052661981830070725316844917984914222814666702729402156616431103191\ 5113334527424812898682528354633020185161200054393827649877953027688718\ 9821519807938387261042335825894072479219978185428285819499151742726948\ 1253539986988090644767767756895648857009098846656474169735583498317364\ 0417156963302378587405533205650717901440893242162746162522154127519400\ 68014831008223709670737308174873863117110982157364646771271039885075... This polynomial is so huge that it's not even possible to guess it by PSLQing the numerical solution to a multivariate partial solution. Apparently, it was not even clear until last year which equations to solve, i.e. which chords are unit length. --rwg