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I enjoy math, but I am a novice with it. I was wondering today if there is a proof that a simple polygon can always be made by connecting any set of dots on a plane, so I went online to try to find out. Is it possible that this sort of info could be provided on this page?
Thank you.
2Plus2Is4 ( talk) 01:19, 27 June 2013 (UTC)
Our section currently states "If a closed polygonal chain embedded in the plane divides it into two regions one of which is topologically equivalent to a disk, then the chain is called a weakly simple polygon." But this cannot be correct: if the chain is embedded, it cannot touch itself (that's what the definition of "embedding" is), so it is simple (not just weakly simple) and by the Jordan curve theorem always divides the plane into two parts. And the associated illustration, which is supposed to match this "definition", does not show an embedded chain, but one that has self-intersections. So what is the correct form of the definition that is intended here? — David Eppstein ( talk) 04:55, 12 December 2015 (UTC)
It may be worth adding a discussion at the end of polygons on the sphere. These are practically important for GIS etc. have some interesting differences from planar polygons; for example, there is no a priori reason to declare one side or the other as the "interior". In particular things get tricky when a polygon does not fit within a hemisphere, e.g. such polygons don't have an easily defined convex hull. – jacobolus (t) 03:25, 18 July 2023 (UTC)
At some point a while back I looked up a whole bunch of analysis textbooks to compare their definitions/usage of the terms "domain" and "region", as evidence for figuring out what the top couple sections of domain should say. (The "historical notes" section there is not really right; Caratheodory definitely did not introduce this concept.) From everything I can tell, these terms are largely used interchangeably across mathematical subdisciplines to mean "connected open set in a topological space", which is a precise formalization of a casual term meaning something like "the inside or outside of a shape". (Some sources, rather confusingly in my opinion, specifically define the two to mean subtly different things.)
The reason I think it's helpful to wikilink terms like this, even if the link might sometimes fly over the head of some of the intended audience, is that many readers don't actually know that this is what "region" informally means in mathematics. The article circle (whose text I didn't write), says:
Just yesterday someone tried to change it to say that the circle divides the plane into 3 regions: the interior, the circle itself, and the exterior. To someone who is not familiar with the customary techical meaning of the jargon word "region", it's not inherently obvious that a circle (or a simple polygon) can't be a "region". Obviously the word "region" originally comes by analogy with the geographical / cartographical term region, and you'd think people would roughly have a sense of the distinction between a region – say an administrative district – and its border or boundary, but when a word is transplanted from one context to another like that, it's helpful to be able to point to a definition to avoid confusion and give readers something to look into if they are curious to learn further details.
The better solution in my opinion would be to add some material at the top of domain which is more accessible to non-technical readers, along with a picture or two and some examples. – jacobolus (t) 08:01, 9 September 2023 (UTC)
For a simple polygon, the turning number or " density" is always equal to 1. Perhaps this should be worked into the paragraph about being a Jordan curve, or into the paragraph about internal and external angles. – jacobolus (t) 17:21, 17 September 2023 (UTC)
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Reviewer: Dedhert.Jr ( talk · contribs) 04:36, 2 January 2024 (UTC)
I'll be honest. I'm trying to avoid reviewing this article, but I have no choice but to do so. If I'm not actively reviewing this article due to up to my neck in real life, I may need a second opinion to assist me. Anyway, at least let me try.
I see that @ David Eppstein and @ Jacobolus are the main writers of this article, so I think this could be counted as two users nominating the article. I'll begin reviewing this article now.
I also wonder if you can put most of the see-also links in every section of the article, which may be helpful for expansion, or whatever it is. The article has met six criteria good article: It is neutral (GACR4) and stable, although we have one [3], maybe? (GACR5) Weirdly, the copyvio mentions it has the similarity of words comparing the 86% in group google, and over 8% in AMS. [4] Anyway, if you have any questions, or if you have completed them, please let me know. Good luck! Dedhert.Jr ( talk) 07:21, 2 January 2024 (UTC)
Do you have an illustration for the ear and mouth of a simple polygon,– scroll up a bit, this is one of the parts illustrated in the figure in the "Definitions" section. – jacobolus (t) 16:35, 2 January 2024 (UTC)
why do you need the plural word for "vertex"– this seems pretty helpful to me, since the words vertex and vertices appear all over this article, and it's a non-standard pluralization. – jacobolus (t) 16:38, 2 January 2024 (UTC)
This is the
talk page for discussing improvements to the
Simple polygon article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Archives: 1Auto-archiving period: 2 years |
Simple polygon has been listed as one of the Mathematics good articles under the good article criteria. If you can improve it further, please do so. If it no longer meets these criteria, you can reassess it. | ||||||||||
|
This article is rated GA-class on Wikipedia's
content assessment scale. It is of interest to multiple WikiProjects. | |||||||||||
|
I enjoy math, but I am a novice with it. I was wondering today if there is a proof that a simple polygon can always be made by connecting any set of dots on a plane, so I went online to try to find out. Is it possible that this sort of info could be provided on this page?
Thank you.
2Plus2Is4 ( talk) 01:19, 27 June 2013 (UTC)
Our section currently states "If a closed polygonal chain embedded in the plane divides it into two regions one of which is topologically equivalent to a disk, then the chain is called a weakly simple polygon." But this cannot be correct: if the chain is embedded, it cannot touch itself (that's what the definition of "embedding" is), so it is simple (not just weakly simple) and by the Jordan curve theorem always divides the plane into two parts. And the associated illustration, which is supposed to match this "definition", does not show an embedded chain, but one that has self-intersections. So what is the correct form of the definition that is intended here? — David Eppstein ( talk) 04:55, 12 December 2015 (UTC)
It may be worth adding a discussion at the end of polygons on the sphere. These are practically important for GIS etc. have some interesting differences from planar polygons; for example, there is no a priori reason to declare one side or the other as the "interior". In particular things get tricky when a polygon does not fit within a hemisphere, e.g. such polygons don't have an easily defined convex hull. – jacobolus (t) 03:25, 18 July 2023 (UTC)
At some point a while back I looked up a whole bunch of analysis textbooks to compare their definitions/usage of the terms "domain" and "region", as evidence for figuring out what the top couple sections of domain should say. (The "historical notes" section there is not really right; Caratheodory definitely did not introduce this concept.) From everything I can tell, these terms are largely used interchangeably across mathematical subdisciplines to mean "connected open set in a topological space", which is a precise formalization of a casual term meaning something like "the inside or outside of a shape". (Some sources, rather confusingly in my opinion, specifically define the two to mean subtly different things.)
The reason I think it's helpful to wikilink terms like this, even if the link might sometimes fly over the head of some of the intended audience, is that many readers don't actually know that this is what "region" informally means in mathematics. The article circle (whose text I didn't write), says:
Just yesterday someone tried to change it to say that the circle divides the plane into 3 regions: the interior, the circle itself, and the exterior. To someone who is not familiar with the customary techical meaning of the jargon word "region", it's not inherently obvious that a circle (or a simple polygon) can't be a "region". Obviously the word "region" originally comes by analogy with the geographical / cartographical term region, and you'd think people would roughly have a sense of the distinction between a region – say an administrative district – and its border or boundary, but when a word is transplanted from one context to another like that, it's helpful to be able to point to a definition to avoid confusion and give readers something to look into if they are curious to learn further details.
The better solution in my opinion would be to add some material at the top of domain which is more accessible to non-technical readers, along with a picture or two and some examples. – jacobolus (t) 08:01, 9 September 2023 (UTC)
For a simple polygon, the turning number or " density" is always equal to 1. Perhaps this should be worked into the paragraph about being a Jordan curve, or into the paragraph about internal and external angles. – jacobolus (t) 17:21, 17 September 2023 (UTC)
GA toolbox |
---|
Reviewing |
Reviewer: Dedhert.Jr ( talk · contribs) 04:36, 2 January 2024 (UTC)
I'll be honest. I'm trying to avoid reviewing this article, but I have no choice but to do so. If I'm not actively reviewing this article due to up to my neck in real life, I may need a second opinion to assist me. Anyway, at least let me try.
I see that @ David Eppstein and @ Jacobolus are the main writers of this article, so I think this could be counted as two users nominating the article. I'll begin reviewing this article now.
I also wonder if you can put most of the see-also links in every section of the article, which may be helpful for expansion, or whatever it is. The article has met six criteria good article: It is neutral (GACR4) and stable, although we have one [3], maybe? (GACR5) Weirdly, the copyvio mentions it has the similarity of words comparing the 86% in group google, and over 8% in AMS. [4] Anyway, if you have any questions, or if you have completed them, please let me know. Good luck! Dedhert.Jr ( talk) 07:21, 2 January 2024 (UTC)
Do you have an illustration for the ear and mouth of a simple polygon,– scroll up a bit, this is one of the parts illustrated in the figure in the "Definitions" section. – jacobolus (t) 16:35, 2 January 2024 (UTC)
why do you need the plural word for "vertex"– this seems pretty helpful to me, since the words vertex and vertices appear all over this article, and it's a non-standard pluralization. – jacobolus (t) 16:38, 2 January 2024 (UTC)