Talk:Voderberg tiling

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"It exhibits an obvious repeating pattern" - really?

This is a fascinating tiling; I still have a hard time believing that it works. Given that the curvature of the spiral has to keep changing, I wonder how it keeps doing that.

If we look at the tiles from the center, they can lie in 4 different rotations (roughly, that is, neglecting the wiggles of the order 10°), which I'll call J,L,7, and P, depending on whether the sharp hook points to the lower left, lower right, upper left, or upper right, respectively. Then the blue and red tiles form the following pattern, beginning at the center:

JJJJJJJJJJJJJJJJPJJPJJ ...

Here I lost count, but when they reach the left side, they switch to

PJPJJ...

How long will this pattern go on? Until it reaches the right side, where the yellow/purple tiles switch from PJJ to PJPJJ? What does it change to then? And how will the next layer react to that change?

This is far from obvious to me. Can we cut this sentence, or change it to something like "The Voderberg tiling is [obviously] non-periodic."? (I'm leaving out the "Because it has no translational symmetries" part, since it doesn't have any other symmetries, except for C2 inversion symmetry in the center, which is no big deal for a spiral.) — Sebastian 19:41, 19 August 2015 (UTC)[reply]

After studying it some more, I'm beginning to see how it works. One key element is that pattern #3 in the table at https://www.uwgb.edu/dutchs/symmetry/radspir1.htm (L77 in my nomenclature, read from bottom to top, or PJJ, if mirrored) shows a straight line on the right end for the two blue tiles combined. Another one is that this pattern can be extended in such a way that the right end remains a straight line. A similar straight line can be formed at the left end, albeit somewhat offset. Of course, this is my original research; is there anything we can quote for that? — Sebastian 20:14, 19 August 2015 (UTC)[reply]

Coloring

The question has been raised about the meaning of the coloring. It seems to me this is just an application of the 4-color theorem in that it is an easy (or the only possible?) way to color all tiles with 4 colors. Maybe that should be mentioned in the article? — Sebastian 19:41, 19 August 2015 (UTC)[reply]

@SebastianHelm: I think the coloring has nothing to do with the 4-color theorem, but rather the coloring is simply a visual aid for telling the tiles apart. For this tiling, an individual tile is a rather convoluted shape with small narrow details, and the colors give a visual indication of the extent and shape of each tile. Because each tile shares edges with exactly four other tiles (a checkerboard does this also), it may be possible to use less than four colors. ~Anachronist (talk) 02:31, 22 January 2021 (UTC)[reply]

Indeed, Anachronist, you're right: It would be possible to paint all red tiles yellow and all blue ones purple. ◅ Sebastian 14:49, 22 January 2021 (UTC)[reply]

Now that you mention it, I recall seeing this same tiling printed in black-and-white in an old (1980s) issue of Scientific American. So yes I remember it was an article about Penrose tiling that showed some other examples of non-periodic tiling such as Voderberg. ~Anachronist (talk) 16:40, 22 January 2021 (UTC)[reply]