Talk:Einstein problem

Unclear statement: other sets

"Chaim Goodman-Strauss suggested that a tiling be considered strongly aperiodic if it admits no infinite cyclic group of Euclidean motions as symmetries, and that only tile sets which enforce strong aperiodicity be called strongly aperiodic, while other sets are to be called weakly aperiodic."

This should make clear exactly what kind of sets "other sets" refers to.

New Research

Read https://arxiv.org/pdf/2303.10798.pdf 103.216.190.117 (talk) 03:33, 21 March 2023 (UTC)

Added. ArXiv preprints are essentially self-published sources but I think both Kaplan and Goodman-Strauss meet the "recognized expert" clause of WP:SPS. —David Eppstein (talk) 06:15, 21 March 2023 (UTC)

3D Socolar-Taylor tiling is weakly aperiodic?

The article currently reads:

"but this tile allows tilings that are periodic in one direction, and so it is only weakly aperiodic"

in reference to the 3D Socolar-Taylor tiling. However this is a contradiction, if it permits periodic tilings then it is not a weakly aperiodic set of prototiles. I don't know what construction is being discussed here, since there's no in-line reference, but I think it is likely one of the two claims is a mistake. The naive construction of a 3D Socolar-Taylor tile does indeed permit periodic tilings in one direction, but there may be a construction that permits only screw symmetries, that I am not aware of. AquitaneHungerForce (talk) 18:27, 24 March 2023 (UTC)

It is not a mistake. It is a misunderstanding by you of the terminology in this area. "Periodic", for a $d$-dimensional tiling, means that the symmetries of the tiling include translations in a $d$-dimensional group $\mathbb {Z} ^{d}$. "Weakly aperiodic" means that the symmetries are nonexistent or not full-dimensional. "Aperiodic", without qualification, still allows some symmetries, but only a finite group. A tile or prototile is aperiodic if it only allows aperiodic tilings, and weakly aperiodic if it only allows weakly aperiodic tilings. For instance the sun version of the Penrose tiling has a dihedral group of ten symmetries, but this is finite, so the Penrose tiling is still aperiodic despite having some symmetry. The 3d ST tiling has translational symmetry in only one of the three dimensions, so it is weakly aperiodic. —David Eppstein (talk) 18:36, 24 March 2023 (UTC)

Ah thanks. That resolves it. I had taken weakly aperiodic to mean that there was no translational symmetry. The lead in to the section certainly seem to imply that to be the definition. When I have the time I'll try to supply the article with a source for an actual definition. AquitaneHungerForce (talk) 18:54, 24 March 2023 (UTC)

Unsolved problem

@AquitaneHungerForce: You reverted my edit removing the paragraph saying "The existence of a strongly aperiodic tile set for the Euclidean plane consisting of one connected tile without matching rules is an unsolved problem". I believe this is a mistake, as the "hat" tile described in the preceding paragraph is the (purported) solution to this problem. colt_browning (talk) 14:44, 30 March 2023 (UTC)

No, not a mistake. The solution is still an unpublished pre-print. It still needs to go through peer review, and since the problem has had various statements rather than a single precise definition, it is to be seen whether this is accepted as a solution to the problem, or whether like previous "solutions" the problem simply narrows in scope. AquitaneHungerForce (talk) 14:55, 30 March 2023 (UTC)

@AquitaneHungerForce:thank you for the reply, I see your points, but I still think I'm right. First, whenever there is a preprint with a proposed solution to a problem co-authored by a well-known expert, they normally don't just say that the problem is open; instead, in such cases, it is normally said that there is a proposed solution, but it is yet to be checked etc.; and this is precisely what is already made clear in the preceding paragraph. Second, even though "the einstein problem" indeed can be stated in various ways, "The existence of a strongly aperiodic tile set for the Euclidean plane consisting of one connected tile without matching rules" is "a single precise definition" which the "hat" (purportedly) satisfies. colt_browning (talk) 16:29, 30 March 2023 (UTC)

I'm not against rewording the sentence to more clearly reflect the status of the problem. Mathematics is slow, and I don't think we should rush to declare anything over. I will say that that definition is not precise, and the hat arguably doesn't meet that definition. In order to tile the plane you need both the hat and its mirror image, whether that is a "one tile" solution is up to interpretation. The SCD prototile holds the opposite assumption, it is only weakly-aperiodic if consider it without its mirror pair, so its clear that there's some ambiguity in what is counts. AquitaneHungerForce (talk) 17:28, 30 March 2023 (UTC)

Seems like someone else has made this rewording in question. Here is a tweet from Craig S. Kaplan (one of the authors of the new paper) that supports the statement, but given that it's a tweet and that Kaplan doesn't have a verified Twitter account, it might not necessarily be suitable as a citation. Edderiofer (talk) 17:24, 4 April 2023 (UTC)

The easiest would be to wait for the published peer-reviewed paper, then cite that. –jacobolus (t) 17:36, 4 April 2023 (UTC)

Voderbergs tiling

Hello, I am a french Wikipedianer, bad english speaking, please apologize ! The article says "The existence of an "einstein" prototile such that the tiling does not involve mirror images is an unsolved problem".

Why is the Voderberg tiling not a solution for this problem ? Is it not un full planar tiling?
Is it not aperiodic ? Tank You for your responses, fliendly.--Jacques Mrtzsn (talk) 08:19, 9 May 2023 (UTC)

The tricky part is not finding a tile that can tile the plane non-periodically: a rectangle can accomplish that. The tricky part is finding a tile which can tile the plane but (provably) cannot do so periodically. –jacobolus (t) 08:48, 9 May 2023 (UTC)

Thank you, @Jacobolus Jacques Mrtzsn (talk) 11:11, 9 May 2023 (UTC)

Wiki page for David Smith

Hi. I'm wondering if a page should be made for David Smith. It's clear his continued effort on envisioning new shape variants (besides his initial discovery) was very valuable. Thoughts? Electricmaster (talk) 12:14, 6 September 2023 (UTC)

Evidently someone agreed: David Smith (hobbyist) was created September 11. —Tamfang (talk) 02:49, 28 September 2023 (UTC)

Unclear statement

"but only in a nonperiodic way"

So, in order to prove that hat/spectre fits the requirements, just an example of nonperiodic tiling would not be enough. We also need to show that there are no periodic tilings using the hat/spectre. Or did I get it wrong? 2A02:AB88:C8F:7400:493A:7A4B:A260:59E8 (talk) 21:56, 23 January 2024 (UTC)

That is correct, and that is what has been shown. —David Eppstein (talk) 23:12, 23 January 2024 (UTC)