# Talk:Three-gap theorem

This review is transcluded from Talk:Three-gap theorem/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Olivaw-Daneel (talk · contribs) 20:31, 6 January 2022 (UTC)

The theorem is stated in the lead but not in the body; I believe this is required per MOS:LEAD.
It was stated in the body, actually. The statement was buried in the history section. I moved it up to a new section. —David Eppstein (talk) 01:10, 22 January 2022 (UTC)


### Applications
First sentence of Plant growth: The source (Adam) says phyllotaxis is the arrangement of leaves, not the theory of plant growth. (A biology paper doi:10.1016/j.cub.2017.05.069 calls it a phenomenon, but I think it agrees with Adam.)
Ok, changed to better match the definition. —David Eppstein (talk) 02:28, 18 January 2022 (UTC)
The phenomenon of having three types of distinct gaps depends only on fact that the growth pattern uses a constant rotation angle, and not on the relation of this angle to the golden ratio; the same phenomenon would happen for any other rotation angle, and not just for the golden angle. – I'm not seeing this in the source.
I found another source (Akiyama) that talks about the three-gap theorem both in general for arbitrary angles and specifically for the golden angle, in order to prove that the golden angle leads to additional properties (point placements that form a Delone set and bounded ratios between the three gaps). Although Akiyama does not outright say explicitly "The phenomenon of having three types of distinct gaps depends only on fact that the growth pattern uses a constant rotation angle", I think the implication is clear; what he does say would make no sense unless this were true. —David Eppstein (talk) 08:07, 18 January 2022 (UTC)
The first sentence in Music theory is a bit hard to follow; suggest splitting it into smaller sentences.
This sentence only made sense, I think, for people who already know what it means. I didn't see how to split it without significantly expanding this whole section to provide a gloss of all the terms it uses, for readers who might not already be familiar with this aspect of music theory. So now instead of a single paragraph it is three paragraphs. —David Eppstein (talk) 03:21, 18 January 2022 (UTC)
Same comment for the first sentence of Mirrored reflection.
This one was easier to just split without adding much new material. —David Eppstein (talk) 07:23, 18 January 2022 (UTC)
Mirrored reflection: It would be helpful to add the page range cited in Lothaire, which has 58 pages. I believe it's pp. 72–73, which talk about three distinct frequencies; but I wasn't able to verify this sentence: The proof involves partitioning the y-intercepts of the starting lines (modulo 1) into n + 1 subintervals within which the initial n elements of the sequence are the same.
Pages added. Our citation templates do not actually provide a way to cite both the page range of a chapter within a book, and the precise location of the material cited within the chapter, so I added it as text in the footnote after the citation template. I think the material about partitioning the y-intercepts of starting lines is more or less equivalent to the partition of the circle into intervals discussed in the first full paragraph of page 73, that these intervals are generated by the rotation angle discussed in the same paragraph, and that the frequencies are just the lengths of these intervals. But despite that equivalence this source proves the three-frequency property directly rather than pointing to the three-gap theorem. Maybe I can find a source that makes these connections more clearly. —David Eppstein (talk) 08:31, 18 January 2022 (UTC)
I found and added another source, Alessandrini and Berthé, which does describe the connection between the three-frequency theorem and the three-gap theorem more explicitly. —David Eppstein (talk) 01:07, 22 January 2022 (UTC)


### History and proof
Allouche (cited in the lead) and Shiu (used for the proof) state an algebraic formulation of the theorem; in fact they don't mention the geometric statement in the lead. Suggest adding their version as well since it seems to be widely used. (Also, it wasn't obvious to me that the two versions are equivalent until I looked at Mayero.)
You mean the formulation in terms of fractional parts of real numbers instead of angles on a circle, right? Added, and moved from history to a new earlier section. —David Eppstein (talk) 00:33, 22 January 2022 (UTC)
Liang's proof is not specifically for this theorem, but for a generalization called the "3⁠
  
    
      
        d
      
    
    {\displaystyle d}
  
⁠ gaps theorem" (source: Shiu p. 266); may be worth mentioning.
Added this to the short paragraph about other things that Liang proves. —David Eppstein (talk) 06:39, 18 January 2022 (UTC)
It would be helpful to have a citation for the fact that Sos' (and others) were the first proofs.
Another source added explicitly stating that they were first. —David Eppstein (talk) 07:18, 18 January 2022 (UTC)
This isn't really a GA criterion; just a suggestion. The list of 5 citations after Several later proofs have also been published seems a bit ugly. You could perhaps discard the less notable ones, or group them in a manner similar to Marklof & Strömbergsson: Various new proofs have appeared since then, with connections to continued fractions [5, 10], Riemannian geometry [1] and elementary topology [4, App. A], as well as higher-dimensional generalisations [2, 3, 11].
Grouped following Marklof & Strömbergsson, with a citation to them for the grouping. —David Eppstein (talk) 01:54, 22 January 2022 (UTC)
Placing it on hold; please ping if you have questions/once done. Thanks, Olivaw-Daneel (talk) 07:09, 16 January 2022 (UTC)

@Olivaw-Daneel: Ok, I think I have addressed all of your comments here. Please take another look. —David Eppstein (talk) 01:54, 22 January 2022 (UTC)
Looks great; promoting to GA. I wasn't expecting such a substantial addition to the Music section; I can appreciate that it's a difficult topic to gloss. Olivaw-Daneel (talk) 08:18, 23 January 2022 (UTC)