Talk:Heilbronn triangle problem

message: :If we are posing the question of the worst case for the existence of small triangles, it will happen with all the chosen points on the boundary.
Might be true (in some sense, asymptotically) but I no longer find this at all evident. Another way to pose the question: for P, Q points from the configuration X, the line segment PG having length L, we want to draw strips of width inversely proportional to L and look whether any other point R lies in them (because the area of the triangle PQR is L times the perpendicular distance from R to the line).
Charles Matthews 09:35, 25 Sep 2004 (UTC)
thread_title: Comments

message: Hi there, I'm pleased to inform you that I've begun reviewing the article "Heilbronn triangle problem" you nominated for GA-status according to the criteria.  This process may take up to 7 days. Feel free to contact me with any questions or comments you might have during this period.  Eluike (talk) 19:43, 8 April 2022 (UTC)
replies: text: Note: reviewer was blocked indefinitely; review has been deleted and nomination returned to pool of those awaiting a review with no loss of seniority. BlueMoonset (talk) 05:13, 9 April 2022 (UTC)
thread_title:  Your [[WP:Good articles|GA]] nomination of ''Heilbronn triangle problem'' 

message: I'm making this a new section rather than appending it to the one above. I consider most of the stuff discussed in the above section as resolved, though I'm open to suggestions for changes in the diagrams. But there's one aspect which still worries me.
When  suggested "highlight a single one of the minimal area triangles", my views followed this trajectory:
#There'll only be a few minimal triangles for each diagram, I'll colour them all.
#Oh dear, there's a lot more minimal triangles than I expected – 28 for the 11-point case. I can't highlight them them all. But most of the diagrams have some symmetry. The minimal triangles fall into sets, with with symmetries of the diagram mapping members of a set to one another. I'll just highlight one member of each such set. 
#I can reduce the number of highlighted triangles further, because
##When a diagram has rotational symmetry, there are lots of parallelograms. Specifically, if it has 2-fold rotational symmetry and 2n points, there are at least n(n-1)/2 parallelograms.
##The vertices of a parallelogram define four triangles all of equal area.
#:so there are plenty of cases where two triangles in a symmetric diagram necessarily have the same minimal area, even though there's no symmetry operation that maps one to the other. I need not highlight triangles that are "duplicates" in this way.
#:The current state of the diagrams reflects is intended to reflect this logic.
#In the diagram for the 11-point case there are four parallelograms whose existence is not a consequence of the overall symmetry of the diagram, and which are small (thin) enough that halving them along a diagonal gives minimal triangles. Using this I could reduce the number of "independently" minimal triangles to highlight by (I think) five. This would make that diagram look less messy.
I'm not going to do anything about this for a while. Maybe my thoughts will be different in a few days. Maybe someone will offer some good advice.
I've long stopped pretending to myself that I'm avoiding WP:OR. Maybe WP:OR doesn't apply to diagrams? If I add a picture of a dandelion to Taraxacum, no-one complains about WP:OR.   Maproom (talk) 19:35, 8 May 2022 (UTC)
replies: text: It's definitely possible for WP:OR to apply to diagrams. But in this case the coordinates are sourced, and the question of which triangles are smallest falls under WP:CALC. So the only OR issue is your choice of which triangles to highlight; that seems insufficiently significant to be a problem to me. —David Eppstein (talk) 20:39, 8 May 2022 (UTC)
replies: text: Agreed with David that the smallest triangles, and the number of them, are roughly within WP:CALC. In any case, your choice of highlighting isn't a big deal. The footnote [d] is perhaps pushing it, but... meh, bigger fish to fry imho. Indeed, great work on these diagrams! :) For the 11 point diagram, are the horizontal-line-shaded triangle and upper vertical-line-shaded triangle not congruent? Ovinus (talk) 03:16, 9 May 2022 (UTC)
replies: text: Thank you, Ovinus, for your improvement to the footnote. Yes, those two triangles are congruent. But that's so because the horizontal edge joining their non-common vertices is midway between the two horizontal edges that run from the left side of the square to the right side – it's not a consequence of the symmetry of the overall diagram (a vertical mirror). I'll probably be removing the shading from one of those two, but I want another day or two to clear my head first. Maproom (talk) 08:42, 9 May 2022 (UTC)
thread_title:  Highlighted triangles in the gallery of optimal solutions