Christian Goldbach found this parametric equation for the non-trivial branch of [math]x^y=y^x[/math] in 1728 (the trivial branch is just y=x). As t goes to plus or minus infinity, the points converge to (e,e). The values where x<2 or y<2 all come from 0<t<1 and -2<t<-1. I wrote this applet to go with an article I'm writing -- just to help get a feel for the parametric equation. Incidentally, Goldbach used a slick trick to solve, setting y=ax, substituting, and then setting t=1/(a-1).
