[math-fun] Ever seen a spacefilled pentagon?

Yesterday I told tutor Julian an idea for spacefilling a pentagon, and he said I could just conformally map Peano's filled square onto whatever, but then he got into it and helped make www.tweedledum.com/rwg/pentagonfill.pdf . It self-contacts, but the median curve doesn't. Note the density variations, which become unbounded in the limit. I don't have a fix, but I can turn a slight variant of this one into one of those exact continuous maps of the rational unit interval onto some dense set of algebraic points. This would permit sampling at various frequencies and phases, which has proven interesting in previous cases (www.tweedledum.com/rwg/sampeano.htm, www.tweedledum.com/rwg/sam306090.htm ). --rwg