# Talk:Pick's theorem

I submit to thee that the statement provided below is not an explanation, but merely a set of instructions. I won't make any changes myself due to the popularity of rote learning. A more intuitive explanation should be shown at the top of the page.
>A simple way to explain to middle school kids in 4 simple steps. 1) count the dots on the perimeter of the figure. 2) divide the number by 2. 3) subtract 1. 4) add the interior dots. The example above would be 14 / 2 = 7 − 1 = 6 + 39 = 45.

How do you compute the Euler characteristic?  —Preceding unsigned comment added by 79.37.207.183 (talk) 20:06, 27 February 2009 (UTC) 

I've rewritten the article so that you don't need to. —David Eppstein (talk) 21:50, 27 February 2009 (UTC)

References are all from the web, it's pretty sad as this seems like it surely made it into a book somewhere. I'll try to find one. Cliff (talk) 05:48, 22 March 2011 (UTC)
The proof described on this page can be found in Chapter 2 of [Beck, Matthias; Robins, Sinai (2007), Computing the Continuous Discretely, Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, New York: Springer-Verlag, ISBN 978-0-387-29139-0, MR 2271992]. --Mattbeck (talk) 07:55, 17 June 2013 (UTC)

This review is transcluded from Talk:Pick's theorem/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Horsesizedduck (talk · contribs) 14:57, 3 July 2021 (UTC)

Greetings! I'm here to review this article for GA status! It caught my attention, and I find it more interesting the more I look at it! Let's have a look! Horsesizedduck (talk) 14:57, 3 July 2021 (UTC)

Images: spot on;
Sources: reliable, more investigation to follow;
No OR spotted;
Neutral point of view: nearly certain;
Stability: stable as it gets, nominator is also greatest recent contributor;
Will continue in a few hours, after consulting bibliography adequately and dealing with other circumstances.Horsesizedduck (talk) 16:14, 3 July 2021 (UTC)

Save for some technical article concerns I would like to discuss with the nominator, the article seems close to passing the review. Noting the nominator has mentioned being busy today, I will take some more time before leaving instructions. Horsesizedduck (talk) 17:49, 3 July 2021 (UTC)
Actually, after more careful examination, my plans have changed: the prose of the article has notable readability issues. Aside from being very dense, some sentences run rather long; the ability to discern meaning ("1a") may be compromised for the average reader, and correction in that case would involve copyediting work - not just the nominator executing suggestions by the reviewer.
Recognizing this new setback, to salvage a possibility of passing the article, I am contemplating two options, aside from (perhaps more wisely) failing it:

Putting up some copyedit request;
Calling for a second opinion, from a more experienced reviewer;
At this point, I will put the article on hold and wait for input from @David Eppstein: the nominator. Standing instructions are to improve the proofs and generalization, discussing here or in the talk page. Horsesizedduck (talk) 22:24, 3 July 2021 (UTC)

@Horsesizedduck: This is the paradigm of a non-constructive review: Your article is bad but I'm not going to tell you anything useful about where to find the bad parts so you can fix them, and I'm not going to give you any detailed feedback on any point other than this one thing I thought was bad. Additionally, the review appears not to have understood a very important piece of wording in the GA criteria: the article should be "understandable to an appropriately broad audience". That does not mean the same thing as understandable to all audiences. It is not reasonable or within the criteria to demand, for instance, that a mathematical proof be readable to someone who does not understand mathematical proofs or who shies away from them as being too scary to read. The review's invocation of "the average reader" shows that the reviewer has missed this point; this part of the article is not aimed at "the average reader", and should not be.
It is also crucial to understand that different parts of the article are aimed at different audiences, per WP:TECHNICAL, with the more basic material earlier and the more advanced later. I think that the opening part of the article, the introduction and "formula" section, should be readable by high school students (if it isn't, it could use improvement) but that the later parts of the article are and should be more advanced.
That said, I have gone over the entire proof section, adding another illustration and splitting it into more smaller sentences. I'm not entirely convinced that it's more readable that way, but it's what you asked for. I have not done the same to the "Other proofs" section; this section is aimed at more advanced readers who might already have some idea what a Voronoi diagram or a doubly periodic function might be, rather than the target audience of the "Via Euler's formula" section (readers who are comfortable with mathematical proof in general but without much more assumption about their background knowledge).
I also (more or less unrelatedly — I was thinking of doing this anyway before seeing your review) expanded the see-also section into article prose, adding sources for it.
Please, take another look, both at the parts of WP:TECHNICAL about what an "appropriate audience" is and is not, and at the updated version of the article. —David Eppstein (talk) 01:31, 4 July 2021 (UTC)
@David Eppstein:, you are entirely correct.
I apologise for any distress my misunderstandings might have caused. I can only defend that my immediate thoughts were to involve someone with more insight to come assist the review.
Allow me to explain my thought process. As you noticed, WP:TECHNICAL was a part of what made me hesitate. Finding myself slightly adrift, I believed the article might not fit the criteria, and found it difficult to give proper feedback. Indeed, saying "rewrite these sections" would be absurd, and in trying to avoid saying it, I think I actually sent precisely that message across.
In response, you went above and beyond, and essentially did just that.
My feedback now is:

"This is the value given by Pick's formula for this triangle, but it would be circular reasoning to use Pick's formula in its own proof": Perhaps this could be turned into a footnote, with a few adjustments to the following sentence;
" ... contains twice as many triangles (in the limit as the scale factor goes to infinity) as the number of grid points it contains." Could perhaps be turned into "... contains twice as many triangles as grid points."
"Any other polygon can be subdivided by the repeated addition of line segments within the polygon between pairs of grid points, not crossing any other added line segment, until no more line segments can be added." Perhaps this could be "... can be subdivided by adding line segments between pairs of grid points within it, without crossing, until no more can be added" (Unless it's too close to the cited material)
For now this will have to do for me, as in my time-zone it is quite late. I will have more suggestions tomorrow, within 8 hours of this edit, and am confident it will be at the GA status when we're done. I am genuinely grateful for your willingness to put up with errors on my part. Wikipedia is doubtless better for it.Horsesizedduck (talk) 02:57, 4 July 2021 (UTC)

Re 'Could perhaps be turned into "... contains twice as many triangles as grid points."': unfortunately no. It is not exactly twice, because of boundary effects, but those effects become smaller as the scale factor gets larger. The other two changes look possible, though. —David Eppstein (talk) 06:26, 4 July 2021 (UTC)

As it stands, the article definitely meets all criteria, and the nominator has put in a considerable amount of work (this is an understatement).
Accordingly, I will pass the article. @David Eppstein: are there any more changes you wish to make?

I'm happy with the article as it stands, for now. —David Eppstein (talk) 16:24, 4 July 2021 (UTC)
I will give it GA status immediately. Thank you once more for an amazing contribution. Horsesizedduck (talk) 16:39, 4 July 2021 (UTC)

This is my first time posting on a talk page, as far as I can remember, so I apologize if I'm doing this wrong! Please let me know if so.
The proof in the article that makes use of Voronoi diagrams isn't correct. It seems likely that there's a valid proof that works by using the Voronoi diagram of the integer lattice, but the version laid out in the article isn't right. It's fairly easy to construct counterexamples to both of these assertions:
- "For each interior grid point of the polygon, the entire Voronoi cell is covered by the polygon."
- "Grid points on an edge of the polygon have half of their Voronoi cell covered."
Does anyone see how to fix this proof while keeping it reasonably concise? I'm not seeing it at the moment. Hmkgx (talk) 21:34, 13 December 2021 (UTC)

Removed. I agree that it has problems, and it isn't clearly formulated as a proof in the source. It would be nice if something like that could be made to work, but trying to fix it here is the wrong way to go; we should only include it if we have a source that formulates it properly. —David Eppstein (talk) 23:29, 13 December 2021 (UTC)