Two renderings of Gosper's island, one traverses all points of the (3.4.6.4)-tiling once, the other (was awfully hard to obtain and) traverses all edges once: http://jjj.de/tmp-xmas/ The edge version is to my best knowledge the first such curve for (3.4.6.4), versions for (3^6) and (4^4) are known for quite some time. Ventrella gave the first for (3.6.3.6) in 2012 in his book "Brain-Filling Curves". All other uniform tilings have odd incidences on some points, hence no such curve exists on them. The curve I use is not Gosper's (I need a curve that turns by 120 degrees after every edge). Enjoy! Now that is my chance to ask about terminology (again). Do the following appear OK? I call... ... the arrangement of points and edges of some tiling a "grid", as in the "square grid" for what is (4^4), the (uniform) tiling into unit squares in Gruenbaum and Shephard. ... specifically, the grids for (3^6) the "triangular grid", for (6^3) the "hexagonal grid", for (4^4) the "square grid" (as said), and for (3.6.3.6) the "tri-hexagonal grid". ... curves that traverse all points once "point-covering" (and could in analogy call those that traverse all edges once "edge-covering", but indeed call them "grid-filling", should I prefer "edge-covering"?). The (pdf) images above are examples of each. These are two corner cases of "plane-filling" on a grid. Also I define Eisenstein integers as numbers of the form x + \omega_6 * y while the rest of the world seems to use x + \omega_3 * y (my norm is x^2 + x*y + y^2, the other is x^2 - x*y + y^2). Does anybody want to kill me for that? Yes, I am writing something up and would like to avoid annoying the readers with bad terminology (and nobody in my personal reach could possibly answer the questions above). Best regards, jj