Jörg, these recursive arcs are beautiful and ingenious, but in the limit, they all describe the same area-filling function, where in this case, the area filled is 1/3 of a "France Flake" island. All such area-fills map closed intervals onto closed sets, hitting *all* the points at least once, uncountably many at least twice, and at least countably many at least thrice. Your island/3 looks to be self-similarly dissectible. Could you render a multicolored one to show how many pieces? --rwg On 2015-12-21 07:00, Joerg Arndt wrote:
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
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