JJ, Concerning the ring of Eisenstein integers: There is no room for argument! They are the complex numbers of the form a + b omega, where omega = e^(2 Pi i / 3) = -1/2 + i sqrt(3)/2 and a and b are ordinary integers. I'll use w for omega from here on The norm of a+b w is (a + b w)(a + b w^2) = a^2 -ab + b^2. w satisfies w^3 = 1. You should not use a sixth root of unity. (Yes, -w is in the ring, but so what) Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Mon, Dec 21, 2015 at 10:00 AM, Joerg Arndt <arndt@jjj.de> 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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