Talk:Voderberg tiling

"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)

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)

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)

@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)

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)

Now that you mention it, I recall seeing this same tiling printed in black-and-white in an old (1980s) issue of Scientific American. 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)

@SebastianHelm: Aha! I found it. I was mistaken, it was the late 1970s. Someone maintains a set of photographed pages here: https://web.ma.utexas.edu/users/radin/gardner.html - the black-and-white Voderberg tiling is shown on page 2. ~Anachronist (talk) 16:48, 22 January 2021 (UTC)

Thank you. That's interesting; I'll read the rest of the article later. So far, I noticed the tiling on the top right, which seems to have the same structure as the Voderberg tiling, except that the tiles, being isosceles triangles, are much simpler. What do the irregular nonagons give us that the triangles don't? (Forgive me for bothering you with the question if it's already covered in the article.) ◅ Sebastian 09:40, 25 January 2021 (UTC)

@SebastianHelm: Six extra sides? :) Actually I wondered the same thing, because it is possible to create a spiral tiling with just triangles that resemble a 1/12 pie slice. I added an external link to the article - https://docplayer.net/154111976-Waldman-voderberg-deconstructed-9-september-voderberg-deconstructed-triangle-substitution-tiling-cye-h-waldman.html - which goes into interesting variations of the Vorderberg tiling and includes an illustration of doing it with just triangles. That article also explains how to create the Vorderberg nonagon (the author refers to it as an ennagon) by deforming a triangle.

One thing in that article that surprised me is that Vorderberg created his tiling in 1936, while the triangular spiral tiling, created simply by taking a circular tiling of triangles and shifting half of the circle over by one triangle unit, was discovered much later, and seems trivial by comparison to Vorderberg's complicated way of doing it. ~Anachronist (talk) 16:56, 25 January 2021 (UTC)

Yup, it's good that Voderberg (only two “r”s, btw!) didn't live to see that! But isn't that often the case in math that we first treat something in a complicated way, until we find an easier way? I remember reading (in Scientific American, too) that the first number system in Mesopotamia used different signs and even different bases for different items and positions, and then (somewhere else) that the first logarithm table was based on something like 0.99999. ◅ Sebastian 17:11, 25 January 2021 (UTC)

Oops, my brain keeps reading it with that extra 'r'. And yes, it's nice to see an easier way of doing something after a complicated way has been discovered. It's a pity that was never the case with Fermat's last theorem! ~Anachronist (talk) 18:55, 25 January 2021 (UTC)

You know German too well!

So I read the article, and it's certainly interesting, although most of it is about Penrose tiling. On the first page, Gardner agrees with us: “Goldberg's method of obtaining [the Voderberg tiling] makes it almost trivial.” ◅ Sebastian 12:42, 29 January 2021 (UTC)

@Anachronist:I added some words in the article to explain why Voderberg created such a complicated spiral. The spiral structure was more or less a side-effect of a more important problem he was able to solve. Namely, can a polygon be fully surrounded by two of its copies? He even solved a more advanced problem: In the tiling's center there are two tiles (blue and yellow) which are surrounded by two others (red and purple). Vorderberg's adviser thought that this might be impossible with congruent shapes. So, it is often stated that Voderberg constructed the first spiral tiling. However, his task for his thesis was a different and more sophisticated one. --Nessaalk (talk) 17:10, 13 September 2021 (UTC)

Good addition. ~Anachronist (talk) 16:15, 14 September 2021 (UTC)

No.

The article had a sentence that read:

"... it consists of only one shape, tessellated with congruent copies of itself."

No. The shape was not "tessellated". The shape tessellates the plane. It is the plane that is tessellated. (Now fixed.) 2601:200:C000:1A0:2DC4:5FF4:5924:1238 (talk) 04:48, 1 April 2021 (UTC)