Julian & I (again mostly Julian) found exactly the two continuous functions schematized by this old Macsyma plot <http://gosper.org/hemis.gif> of a closed and unclosed curve. So now we can sample the functions at arbitrary rationals. E.g., triple frequency <http://gosper.org/semihexes3x.png>. But actually the median curve <http://gosper.org/semihexesmedian.png> most resembles the Macsyma plot. Since Julian recently found a sampling of Heighway's Dragon equal to its median curve, it's likely that there are samplings of these semihexes coincident with their median curves. This will be easy to test when we manage to invert the semihex functions. Which will also let us directly exhibit the sextuple points. Failing that, it should be possible to tediously guess and check a few. (Actually, since the Dragon truly spacefills, there is obviously some way to hit all the vertices on the median curve. But "luckily" not crazy and disordered.) Heretofore, the only spacefill dense with sextuple points I'd seen was Julian's huge Fourier triangle (@http://howwords.com/triangles/artcircle, which seems to be down), made from 25 copies of itself. Unfortunately, sextuple points are visited very furtively, so the Fourier plot is unconvincing, never (in six tries) coming very close to the actual sextuple points. For the exact definition, see my 21 Aug 2016 math-fun, which gives the six preimages of In[381]:= untrifil25[1/2 + I Sqrt[3]/10] Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} (In arithmetic progression. Always?) —rwg