Talk:Keller's conjecture/GA1

GA Review
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Reviewer: Pbrks (talk · contribs) 06:37, 30 March 2021 (UTC)

Happy to review this article. Pbrks (talk) 06:37, 30 March 2021 (UTC)

Intro

 * In reference to edge to edge and face to face, It should be more clear what these mean, since there are not widely accepted definitions for these terms. I see that this is defined in the body of the article, but a mention of what it means in the intro would be helpful (e.g. "... that meet face-to-face (that is, they share an entire face)."
 * When referring to faces, it should be more clear that you are referring to k-dimensional faces, since the next sentence talks about a two-dimensional example, where face could have a different meaning. Perhaps even just mentioning that a "face" in two-dimensions in this context refers to the edges.
 * For instance, as shown in the illustration, in any tiling of the plane by identical squares, some two squares must meet edge to edge. This implies that the image proves it's true for any 2D tiling. Perhaps something like "For instance, the illustration to the right shows an example of a tiling of the plane by identical squares, and necessarily, there exist two squares that have an entire edge in common."
 * ... showed that it is false in high dimensions... Any reason to leave this unspecific?
 * Ok, rewrote lead. Moved the phrase "as shown in the illustration" to avoid the implication that the illustration is in any way a proof. Replaced "high dimensions" by "ten or more dimensions". And removed the face-to-face and edge-to-edge parts entirely, instead describing them as sharing an (n-1)-dimensional face. I couldn't find a clear way of phrasing this without using formulas, though, so this may have made the lead a little more WP:TECHNICAL. —David Eppstein (talk) 01:00, 31 March 2021 (UTC)

Definitions

 * ... the tetrastix structure formed by three perpendicular sets of square prisms can be partitioned into a cube tiling... in which one fourth of the cubes are surrounded by twelve other cubes without meeting any of them face-to-face. This entire paragraph is eluding me. The tetrastix tiling leaves gaps in space. If we partition the prisms to cubes and fill in the gaps, then every cube would meet face-to-face on all six of its faces. Also, how would one fourth of the cubes be surrounded by 12 other cubes? I don't seen anything in the reference that answers these questions.
 * The gaps are cubes. The sticks are square prisms that can be partitioned into cubes. Putting the partitioned sticks and gaps together produces a cubical tiling, one in which the gap-cubes are not face-to-face with anything else. Copyedited to try to clarify this. —David Eppstein (talk) 01:04, 31 March 2021 (UTC)
 * The final two paragraphs here seem misplaced since they are not defining terms. Perhaps include the final sentence in paragraph two as well as the final two paragraphs in a separate section titled "Statement of the conjecture".
 * The whole section is about the statement of the conjecture. The intent of these two paragraphs are to clarify variations in the statement: it doesn't change the conjecture to require entire columns of cubes to be face-to-face rather than a single pair (penultimate paragraph) but it does change it to require all cubes to meet face-to-face with another cube (last paragraph). Renamed the section to "Statement" rather than "Definitions", and added an explanatory sentence to the start of the last paragraph of the section, to try to make this more clear. —David Eppstein (talk) 01:08, 31 March 2021 (UTC)

Group-theoretic reformulation

 * $0, g_{i}, 2g_{i}, 3g_{i}, ..., (q_{i} &minus; 1)g_{i}$ What are the gs and qs here? Also, the subscripts need consistent markup.
 * Explained g's, rewrote to avoid q's, and took more care with italic subscripts. —David Eppstein (talk) 01:18, 31 March 2021 (UTC)

Keller graphs

 * It would be a bit clearer if it was explained how the Keller graph edge conditions relate to cube tiling before going on Corrádi and Szabó's result.
 * ... by cubes of side two whose... Side-length two?
 * Together with the fact that they don't overlap Isn't this redundant to two three sentences ago?
 * Striking these out. On reading the section again, I feel like it is indeed clear as it is. Pbrks (talk) 04:26, 31 March 2021 (UTC)

Related questions

 * ... when n is at least eight, then this maximum dimension is at most n − 2. Lagarias & Shor (1994) showed more strongly that it is at most n − √n/3. Lagarias & Shor's result is not stronger for n=8,9, unless I am missing something?
 * Replaced by "stronger for ten or more dimensions". —David Eppstein (talk) 01:21, 31 March 2021 (UTC)

Comments
Overall, it is well-written. Per MOS:INTRO, the intro should be able to stand as its own concise version of the article, so I would like to see the intro be a bit more clear. I fixed quite a few comma splices and general punctuation issues. I'll place this article on hold to let you respond and address these issues. Pbrks (talk) 06:37, 30 March 2021 (UTC)
 * Items have been taken care of. Enjoyed reviewing this article -- passing Pbrks (talk) 04:26, 31 March 2021 (UTC)
 * Thanks! —David Eppstein (talk) 05:14, 31 March 2021 (UTC)