Talk:Polygon

Error at mesh of polygons

Please fix the formula to: (n+1)²/2(n²) - amount of vertices/amount of triangles.

Polyhedrons and polytopes as polygons

AxelBoldt removed the statement:

Strictly speaking, every polyhedron is also a polygon as is every polytope, since they all have angles.

with the simple claim:

Polyhedra are not polygons.

Since it is obvious that polyhedron have multiple angles, and hence are polygonal, I'd like to give him a chance to explain why he removed true information.

Simple: because it's not true at all. "Polygon" is almost universally defined as a 2-dimensional figure. I don't know of any mathematics text or course that treats it as a superclass that includes 3-d polyhedra. Terms here should be used as they are commonly used in academia. --Lee Daniel Crocker

Lee,

perhaps you would like to suggest what term should be used for the class of objects that have multiple angles, regardless of the dimentionality? Then we could put in a reference to that class of objects in this article.

I had never heard the term polytope before I got involved with Wikipedia. Does the concept of the angle between two planes make sense? I, admittedly, have found very little accessible material on these topics. -- BenBaker

Maybe a phrase such as:

Even though strictly speaking, every polyhedron has multiple angles, as does every polytope, they are not considered as polygons as the angles between their faces are not two dimensional. They can be classified as 'technical-term', however.

"Polytope" is the general term, although it is typically only used to refer to 4-d and higher figures (because the 2- and 3-d figures already have names). It is, nonetheless, proper to refer to polygons and polyhedra as subclasses of polytopes. --LDC monkey

Mathematical terms are not defined etymologically. "Polygon" may mean "many angles" in Greek, but that doesn't mean that anything with many angles is called a polygon in mathematics (and yes, you can have angles between planes). Polygons are two-dimensional figures that enclose an area with straight lines. We could have links to polytopes and polyhedra I suppose. --AxelBoldt

I am not aware of any word in any context that is defined by its etymology. Words mean whatever they are defined to mean, regardless of where they happen to come from. Adding an explicit statement to that effect in this article would be silly, because that's just a case of understanding the nature of the English language and has nothing to do with polygons. This article is about polygons, which are flat. Now, if you want to add some statement to the effect that polygons are the two-dimensional instance of the more general class of polytopes, that's entirely appropriate. --LDC

"Words mean whatever they are defined to mean, regardless of where they happen to come from." -- Indeed. That's the Humpty Dumpty argument! (Through the Looking-Glass)

On the dimension issue, it might be fair to mention that a polygon is a 2D polytope, but it's not terribly interesting. The question of a "broken" polygon in higher dimensions -- ie a set of non-planar points joined by a closed, simple path -- is perhaps interesting, but completely breaks the definition of a polytope as a convex hull of point, and there's no longer any notion of area or volume. I suppose then it's merely a path. -- Tarquin

n-gons and regularity

I'd like to mention the term "n-gon" on this page, since that's the link I followed to get here. Also, the table is somewhat inconsistent: a "Triangle" may be regular or not. A "Square" is regular by definition. The other terms usually are taken to mean the regular form -- in my experience it's more common to see a phrase like "an irregular pentagon" than "a regular pentagon".

Fixed. --Damian Yerrickyuck

Proposed taxonomy

Hi. I added a proposed taxonomy, but it does have problems. There is the problem that under the definitions that I left, a complex polygon may be considered convex. Is this indeed the case? If so, the two versions of convex are surely nevetheless considered distinct, so Simple convex and complex convex are distinct classes, both denoted 'convex'? Or am I just being too hopeful in proposing a tree-based taxonomy? baby cakes

Is this polygon convex? --116.14.72.74 (talk) 02:30, 26 July 2009 (UTC)

Not by modern definitions of convexity. However Poinsot regarded it as convex: in the same paper announcing the four regular star polyhedra, he defines a "convex" polygon as having corners which are less than 180 deg (i.e. not reflex). Poinsot's definition lasted for about a century before the modern definition replaced it. -- Cheers, Steelpillow (Talk) 10:32, 26 July 2009 (UTC)

So what's the modern definition? --Euclidthegreek (talk) 05:09, 9 August 2009 (UTC), who is actually 116.14.72.74 after creating an account

Sum of inner angles

I would prefer a more simple and complete description of the sum of inner angles in an n-gon. Allthough the one used looks simple, it is based on foreknowledge of the sum of angles in a triangle. I sugest something like the following description(better phrased probably).

(For lack of a better word or expression(i can't seem to find it), i will use the term outer angle for the outside coresponding angle which is 180 - inner angle.)
The sum of the outer angles of an n-gon is a full circle.
The average size of an outer angle is therefore 360/n.
The average size of an inner angle is therefore 180 - (360/n).
The sum of inner angles is therefore n * (180 - (360/n)) <=> 180n - 360 <=> 180 * (n-2).

(exscuse the language, english is not my native tongue). Jan Pedersen 09:19 21 Jul 2003 (UTC)

I added something along these lines. - Patrick 09:48 21 Jul 2003 (UTC)

"circulated"?

its vertices, listed in order as the area is circulated in counter-clockwise fashion,

Is "circulated" the right verb here? AxelBoldt 21:48, 26 Sep 2003 (UTC)

Proof of the constructibility of the regular n-gon

Mathematicians are notoriously incompetent historians. Gauss NEVER gave a proof of the necessity of the constructibility of the regular n-gon. WANTZEL proved this in 1837. If you read otherwise, it's because mathematicians are sloppy historians. Revolver

No personal attacks. Gauß proved it. Somebody else constructed it. --116.14.72.74 (talk) 02:35, 26 July 2009 (UTC)

Gauss claimed many discoveries, in many cases without publishing any proof. After his death, his notes showed that these claims were true, although few contained formal proofs of the result. If you wish to write in an article that Gauss did indeed prove some historically contentious result, then you will need to cite a suitable reference. -- Cheers, Steelpillow (Talk) 10:24, 26 July 2009 (UTC)

Names of Polygons??

• A hectagon has 100 sides.
• A chiliagon has 1000 sides.
• A myriagon has 10,000 sides.
• A "megagon" has 1,000,000 sides.

How about 100,000 sides?? 66.245.71.11 21:53, 1 May 2004 (UTC)

Is it necessary to have "googolgon"? The word is an irregular formation and there is no evidence of its ever having been used.

I agree about removing googolgon. Have never come across it and actually sounds to me like a practicle joke that have to do with the prefix 'googol'/Google. —Preceding unsigned comment added by 78.146.179.43 (talk) 19:43, 5 April 2008 (UTC)

"Google" is famously an incorrect spelling of "googol", which is the number 10¹⁰⁰. A googolgon would have that many sides. There seems to be some doubt as to whether the term has ever been used. So yes I think it should go. -- Steelpillow (talk) 13:43, 6 April 2008 (UTC)

 ErledigtI agree, and I've simply removed it.  --Lambiam 09:48, 7 April 2008 (UTC)

Add quadrangle - "The triangle, quadrilateral, quadrangle, and nonagon are exceptions." and "quadrilateral (or quadrangle or tetragon)". 62.64.202.184 (talk) 04:02, 13 September 2008 (UTC)

Done, thanks for the nudge. Also removed googolgon again (see #Googolgon thread). -- Cheers, Steelpillow (Talk) 08:39, 13 September 2008 (UTC)

Enneagon/nonagon

The table of names of polygons specifies for a nine-sided polygon the name "enneagon" and then admonishes "(avoid 'nonagon')".

The discussion further down includes the sentence, "But beyond nonagons and decagons, professional mathematicians prefer the aforementioned numeral notation."

Personally, I think "nonagon" is in sufficiently widespread use to be acceptable even if inconsistent -- much the same way that "quadrilateral" is more widely accepted than "tetragon". So I'd prefer to see the table changed than the sentence in the discussion.

How do other people feel?

-Heath

Response

To whoever wrote the above discussion, here are some comments. First, when we sign Wikipedia articles, you don't write a word like "Heath"; you just write ~~~~ . I've studied the history of the talk page and found out that you are not a registered Wikipedian. Second, it already says just above the table that "the triangle and quadrilateral are exceptions", as well as a comment saying "(or tetragon)" meaning that quadrilateral is a more well-known word than tetragon, but that both words are equally proper. Third, the info on ennea vs. nona as the prefix for 9 is already mentioned at the bottom of the Greek numerical prefixes article, saying that "In practice, people often use Latin nona- for 9 instead of Greek ennea-." Georgia guy 21:11, 2 Feb 2005 (UTC)

Response

Thanks for the note on the Wiki etiquette -- I'm not (yet) a registered Wikipedian; still learning my way but trying to be constructive in the process. ~~~~ is a tip I hadn't yet seen.

The point that I was trying to make originally is that the table specifically says to "avoid" the word nonagon, while further down the same page the article actually uses that word. The Greek numerical prefixes article does comment further on this, but someone reading this article for the first time is not likely to have seen that other article (I sure hadn't!), and is likely to find the usage in this article contradictory to its own recommendations.

128.173.105.144 03:39, 8 Feb 2005 (UTC) (Heath)

---

• Let's get back to the subject. I feel very uncomfortable with "nonagon (or enneagon)". In practice, "nonagon" is widely used; this term, however, is etymologically dodgy as we all have been recognised by now. Also, "Enneagon" does exist in Wikipedia as an article; whereas "Nonagon" doesn't ("Nonagon" is a redirection page). We should put "enneagon" first, like "enneagon (or nonagon)". By the way, I've NEVER seen "sexagon" in my life. Is this term acceptable? --Marianne-ja 03:01, 13 July 2006 (UTC)

Names of Polygons

Currently, several Wikipedia articles on polygons between 20 and 100 sides are on Vfd and I want to see if anyone plans on creating a Names of Polygons article that is similar to the Names of large numbers article. Anyone in favor of doing so?? Georgia guy 22:56, 10 Feb 2005 (UTC)

There was a section "1.1 Naming polygons" added on 2005/04/08. I think it explains it well and there is no need to add the following 6 names to the table:

 tetracontagon 40 
 ...
 enneacontagon 90 

as 208.170.24.180 did on 2005-12-09 15:25:07. I plan to erase them. Let me know if you have a reason why not do it. --Gogino 01:31, 10 December 2005 (UTC)

Who in the world said sexagon? Is that innapropriate? —Preceding unsigned comment added by 68.42.251.79 (talk) 23:38, 17 September 2007 (UTC)

It is appropriate - it derives from the Latin for six, which is "sex". But it is not used today - we use the Greek root "hex" as in "hexagon". Whether it has ever been used, I do not know. But it does not seem to be on the page at the moment, and unless someone finds a worthwhile reference, it had best stay that way. -- Steelpillow 19:25, 18 September 2007 (UTC)

Vfd consensus

What is the Vfd consensus of the polygon articles put on Vfd about a week ago?? Has it been reached yet?? Georgia guy 20:27, 18 Feb 2005 (UTC)

"Moving around a simple n-gon"

This is only a very small problem but I didn't think I should make the edit myself just in case. When I first read this I thought it was talking about moving the whole polygon around. I quickly realised what it really meant, but thought it should be slightly clarified. Moving around on the edges of a simple n-gon There could also be a link on edges if anyone changes it. --???

I don't understand the following sentence: "There could also be a link on edges if anyone changes it." --Gogino 01:45, 10 December 2005 (UTC)

a picture of all shapes from 3 sided to 100 sided

—Preceding unsigned comment added by 116.14.95.44 (talk) 08:39, 21 July 2009 (UTC)

hexagonnaiae- or hexaconta-?

Which of these is it? 71.137.255.152 reports it as "hexagonnaiae" though all the other -conta- prefixes seems to suggest otherwise.

The edit was probably vandalism. I reverted it. Georgia guy 21:08, 6 January 2006 (UTC)

---

This is only minor non-mathematical semantics, but shouldn't the addition "-gon" to a word be a suffix, not a prefix, as used in the article?

Mully 05:38, 9 February 2006 (UTC)

I hope it's fixed now. --Gogino 05:40, 11 February 2006 (UTC)

icosa or icosi as prefix

in't the prefix for 20-side polygons icosi, as suggested on mathworld and all over the internet?

Naming 21-gon

According to your naming rules a 21-gon would be an iconikaihenagon, yet I have seen it called an icosihenagon (without the kai for "and"). I am not a greek scholar, but I wonder if there are certain times when the "kai" is not required Sorry I have no references available

I have seen icosihenagon too but icosikaihenagon is used clearly most and icosakaihenagon is not used.

--Gogino 08:12, 12 August 2006 (UTC)

Special Cases

I've added entry for squares as special cases of polygons to coordinate with triangles, quadrilaterals, rhombi, and rectangles. I also broke apart the sentence on quadrilaterals --too long without apparent need.

Under area, I clarified the term by stating its definition before giving its calculation.

--Tonea 16:39, 20 September 2006 (UTC)

-- --

User:All_am_rejects made some changes, which User:OwenX reverted. I think some of these chages are worth keeping, or at least editing into shape. Is there some special reason for reverting them, or is it a case of mistaken 'vandalism' assessment? Steelpillow 20:33, 21 March 2007 (UTC)

Simple Polygon

See also http://en.wikipedia.org/wiki/Talk:Simple_polygon

-- sumthinelse

monogon & digon

'Polygon' is a closed planar path composed of a finite number of sequential line segments. I wonder then how henagon (or monogon) & digon happens??--Praveen 18:52, 10 December 2006 (UTC)

---

This text really hurts: As mentioned above:

Mathematical terms are not defined etymologically. "Polygon" may mean "many angles" in Greek, but that doesn't mean that anything with many angles is called a polygon in mathematics (and yes, you can have angles between planes). Polygons are two-dimensional figures that enclose an area with straight lines. We could have links to polytopes and polyhedra I suppose.

Polygons are figures! Polygons are areas! Polygons are two-dimensional!

For example compare with http://mathworld.wolfram.com/Polygon.html "Polygon: A closed plane figure with n sides..."

Polygon MAY BE defined by a, what we in Germany call Polygonzug (linkage, poly-lines, i don't know the right term)

Using your definition given here, how can one call a polygon CONVEX? (see definition of convex)

Please, please, men, work out a new and consistent definition!

Author: tarantinoo, 2006/12/14

---

Apologies in advance as this is my first official edit as a new user.

I also would like some reiteration on the article on the subject of 1-gons and 2-gons.

• Digons (of which there are two possible compositions (disregarding orientation and scale).
  • Two lines where the endpoint of the second line is any place other than that of the startpoint of the first (variable line lengths or not), in which case the polygon is incomplete or 'open'. It may occupy a 2-dimensional plane, but defines no area.
  • Two lines where the startpoints and endpoints are opposite and technically enclose the system achieving one of the requirements of the definition of the polygon. It may occupy a part of the 2-dimensional plane but only defines at most part of a 1-dimension.
• A henagon is similar to the closed digon, but lacking a closed construction and being incapable of defining an area (let alone 0 area, which still implies a closure was made). It, like a polygon, can occupy a 2-dimensional plane, but is properly occupied in 1-dimension.

Assumptions will be the death of me here, but can I assume a polygon occupies a 2-dimensional plane and defines an area within it? Can a digon or henagon truely be classified as a polygon? Does this definition have enough merit to warrant refinement for submittal to the article?

This contribution is also to help immerse myself in the editing process, please offer additional responses, or suggestions personally, thank you!

--ViralStorm 07:15, 9 March 2007 (UTC)

---

Hi. I haven't been here long myself, but I do my best. I think the current state of this page is not very good. Hopefully it will get better over time. What follows will be something of a stream of consciousness. Feel free to edit into presentable form and weave it into the page - or just ignore it. Whatever.

In a broad sense, a polygon is an unbounded sequence or circuit of alternating segments (sides) and angles (corners). The modern understanding is to describe this structure as an abstract polygon which is described mathematically as a partially-ordered set (poset) of elements. The interior (body) of the polygon is another element, and (for technical reasons) so is the null polytope or nullitope.

Generally, a geometric polygon is a realization of this abstract polygon, which involves some mapping of elements from the abstract to the geometric. Such a polygon does not have to lie in a plane, or have straight sides, or enclose an area, and individual elements can overlap or even coincide. For example a spherical polygon is drawn on the surface of a sphere, and its sides are arcs of great circles. As another example, most polygons are unbounded because they close back on themselves, while apeirogons are unbounded because they go on for ever so you can never reach any bounding end point. So when we talk about "polygons" we must be careful to explain what kind we are talking about.

A digon is a closed polygon having two sides and two corners. On the sphere, we can mark two opposing points (like the North and South poles) and join them by half a great circle. Add another arc of a different great circle and you have a digon. Tile the sphere with digons and you have a polyhedron called a hosohedron. Take just one great circle instead, run it all the way round, and add just one "corner" point, and you have a monogon or henagon.

Other realizations of these polygons are possible on other surfaces - but in the Euclidean (flat) plane, their bodies cannot be sensibly realized and we think of them as degenerate.

HTH. Cheers, Steelpillow 19:26, 9 March 2007 (UTC)

Taxonomy question

I wonder if there is a term describing non-equilateral polygons whose inner angles all consist of 90° or 270°. Obviously equiangular does not fit it at 100%. (As for the case of equilateral polygons with 90° or 270°, those are the "famous" Polyominoes.)

Regards, Gulliveig 11:32, 20 December 2006 (UTC)

I recommend a "list of polygons" as a compromise

Per the recent AfD on Hectagon, which was closed as no consensus (although some questioning has occured by other users), I feel that the best course of action is to merge the stbby polygon articles into a "list of polygons" to avoid cluttering the main page and having a bunch of stubs around. Extremely significant polygons, like Triangle, will be given a main template and a brief, one paragraph explanation. If people are interested in this idea, I'd be willing to create a sandbox example so that people can see what I'm talking about. — Deckiller 01:18, 31 January 2007 (UTC)

If you think you can create a page like this (or even a section of Polygon), then by all means do so. I agree that there is probably a bit more info we could reasonably have in a list of polygons that what is in the list right now, but I'm not sure where to draw the line. Useful info would probably be name, number of vertices, approximate measure of the vertex angle (or exact, but I think that most don't have a simple representation for that).

I'm not really sure how you can make this look decent--consider that while a triangle deserves a paragraph (and its own page, of course), I don't imagine a 15-gon deserves more than a line. I'm not sure how you'd format it so that both could coexist. I'm quite interested in seeing what you come up with. --Sopoforic 02:32, 31 January 2007 (UTC)

I've started it at User:Deckiller/List of polygons (it should probably be list of named polygons). Becuase most of the lower polygons are extremely notable, most of the entries have main templates (except 1sided and 9sided, although 9sided might warrent an article). I'm still on the fence if it'll be a good idea, since I'm used to merging fictional stuff (I'm not a MathMan), but we'll see how it goes. — Deckiller 02:59, 31 January 2007 (UTC)

Well, I don't like that very much, but it did serve to make something clear: the articles on polygons need to be standardized a bit. The quadrilateral needlessly duplicates info from polygon, and several of them repeat the definition of a regular polygon, which probably isn't necessary.

Aside: I think that a polygon infobox might be in order: name, number of sides, a picture, whether it's constructible, approximate internal angle, and anything else that I've missed. From the arguments on AfD, I gather that these are the things most people are looking for.

Anyway, which polygons were you planning on including on that list? It already takes up quite a bit of space with just the first nine. Oh, and, incidentally, all polygons have names (we had a name for a 10,000-gon or 100,000-gon, or whatever we deleted a while ago), we just usually don't call the larger (more sides) ones by a name like that. So, list of polygons is fine. --Sopoforic 04:30, 31 January 2007 (UTC)

Like I said, I really didn't even bother looking at the information. I just want/wanted to try the fictional-style of merging and see if it would work. I agree that there are a lot of redundancies; perhaps a table is best. If an article is really stubby but has a bit of information, we can include it in a "notes" section. We might as well experiment with different styles and see what works the most. Ultimately, that version would include every named polygon and an "others" section to describe n-gons and whatnot. The main reason I wanted to try this format is because, like I said, some of the stubby articles have other information, like examples and history, but are too minor for articles. Each way has its pros and cons it seems. — Deckiller 04:39, 31 January 2007 (UTC)

Yeah. Maybe I'll try out a couple of different formats later and see if I hit anything I like. But as for listing 'all' of the named ones, the point I was trying to make is that they are all named. It's just easier to read 10,000-gon than decemyriagon and know what is meant. You see the use of the names a lot more in the polyhedrons than the polygons, I guess, but 'all the named ones' isn't a valid criterion. I'll think about which will be useful to include in a list later, while I'm working on my (theoretical) table layout. --Sopoforic 05:16, 31 January 2007 (UTC)

Yeah; please take over, this isn't my area of expertise. I'm struggling with algebra. — Deckiller 05:18, 31 January 2007 (UTC)

Status update on this: I can't find any nice way to present things other than the table that we already have in polygon. I'm thinking that we might want to add an 'approximate internal angle of regular n-gon' column, but otherwise I can't come up with anything that doesn't duplicate practically the whole article for less-than-notable polygons, or that isn't far too little for the more notable (i.e. writing "It's a 3-gon" for triangles, except using more words). Perhaps what is really needed is a way to call more attention to the fact that the links in the polygon table do indeed take you to articles with more information about those polygons. Maybe renaming that table/section from Polygon names to List of polygons or similar would work, but I'm not sure. Any comments will be appreciated. --Sopoforic 19:23, 7 February 2007 (UTC)

Modified talk page

I added section headers and reorganized the top a little bit to make it easier to see what was discussed, as well as to get the TOC up where it belongs. I indented some of the paragraphs to show who was replying to what (based on my reading), and I don't think I misrepresented anything. We may consider archiving a few of the old conversations that don't apply any more, just to clean this up. --Sopoforic 21:16, 9 February 2007 (UTC)

Why the names aren't proper Greek?

Although I'm Greek, I'm baffled by the names of these polygons, because they're not proper Greek. For example, Greek numbers from 13 to 19 use deca first and tri-, tetra- penta- etc second. Eg, in Greek we call a 13-gon a decatri(a)gon and a 14-gon a decatetragon, not tridecagon and tetradecagon. Now for the names between 20 and 100... The actual Ancient Greek names are as follows:

20 = icosi (pronounced ei-koh-see) 30 = triaconta 40 = tessaraconta(*) 50 = penteconta 60 = hexeconta 70 = hebdomeconta 80 = ogdoeconta 90 = eneneconta 100 = hecaton (**)

(*)Tessaraconta is the later (and standard) form of the Attic tettaraconta (since Attic double-t became double-s). The prefix tettara was usually compressed to tetra-.

(**)Hence hecatogon is proper Greek and in scientific literature the a is usually omitted so that hecato- becomes hecto-. "Hectogon" is perfectly acceptable, as is for example hectometre (=100m). --   Avg    22:01, 1 March 2007 (UTC)

Since no references are given, it is hard to say where some of these names came from. I am uncomfortable that some things on these polygon/hedron/tope pages are being invented for the Wiki, which is strongly against Wiki policy. Woodhouse's English-Greek dictionary gives thirteen as triskaideca and fourteen as tessareskaideca, so there is clearly some evidence for this construction in Classical times. As for 20, we use icosagon from long tradition, I don't know about 21, 22, etc. On 40, I note that for 4 we traditionally use tetra- rather than tessara-, so I guess the same habit spills over onto 40. Probably the main reason why some of these names may not necessarily be proper Greek is that we English can feel a need to adapt a word to something that seems more fitting to our English habits, or to established usages in related areas of mathematics. Anyway, having said all that, feel free to make corrections and additions as you see fit - this is Wikipedia! Steelpillow 22:16, 2 March 2007 (UTC)

Polygon - computer graphics area

I tried to add a brief interpretation of "polygon" as it applies in the computer graphics (image generation) area of activity. However, as you know, modifications to "polygon" are barred at the moment. I can show you my draft words if you like, where is the best place to do this? I am in the simulation business and next week I am away at the European Simulation Exhibition (ITEC) in Cologne but can make contact from there if necessary. Ian Strachan 20:16, 18 April 2007 (UTC)

Protection removed. `'mikka 21:58, 18 April 2007 (UTC)

Deleting paragraphs

Hi, User:Mikkalai. You deleted a couple of things I had put on. I have put them back (at least for now). What do you find is wrong with them? Steelpillow 17:53, 12 April 2007 (UTC)

Well, I just put the one back yet again. I've tried to rephrase it better. Any problems still? Steelpillow 21:38, 19 April 2007 (UTC)

32-sided polygon

Does anyone know the name of a 32 sided polygon? If so, please let me know ASAP. BobafettH23 21:44, 7 May 2007 (UTC)

Triacontakaidigon. Georgia guy 21:47, 7 May 2007 (UTC)

Thanks man.BobafettH23 00:25, 8 May 2007 (UTC)

Somebody check me

1. The article previously had text in it that read "(uhh... what's the dual of a hosohedron, somebody?)", which I replaced with "dihedron", because http://mathworld.wolfram.com/Dihedron.html says it is; I don't know if the resulting sentence is correct though.

Yes, that was my blank moment - and your correction does make sense, though my wording looks a little terse with hindsight. Thanks. -- Steelpillow 19:28, 9 May 2007 (UTC)

1. In the classification system, the text currently reads "A polygon is called regular if it is both cyclic and equilateral; all convex regular polygons with the same number of sides are similar to each other. A non-convex regular polygon is called a regular star polygon." The first sentences implies that all regular polygons are convex (because the definition of cyclic requires the polygon to be convex), which makes the second sentence seem somewhat strange. Presumably there's another case where a polygon can be regular that isn't described...? JulesH 18:06, 9 May 2007 (UTC)

This goes back to Poinsot's paper in which he discovered the regular star polyhedra. Consider a star pentagon - it is cyclic and equilateral, and therefore regular, but it is not convex by our modern use of the word. Poinsot considered it convex because its angles are all convex, and his notion stood for a hundred years before the modern definition came along. So the confusion is a deep one. HTH. -- Steelpillow 19:28, 9 May 2007 (UTC)

And ... help?

Does anyone know of an algorithm for calculating the area of a concave polygon (I can do convex by dividing into triangles, but some of the triangles go outside of the polygon if it's concave...)? JulesH 18:10, 9 May 2007 (UTC)

Hum, no doubt this is a well studied problem. Off the top of my head consider slicing the polygon into smaller segements. You could for example slice the polygon by cutting along the x coordinates of all the points. Between two such slices the polygon will be relatively simple, consisting of a number of convex polygons. I'm sure there are faster ways. --Salix alba (talk) 19:50, 9 May 2007 (UTC)

One way is to divide the polygon into convex regions, by drawing lines between suitable vertices. Another way, which also works for self-intersecting polygons, is quite amazing. First, decide a direction in which you will trace a circuit round the polygon and draw a little arrow next to each edge to show which way you went. Next mark any point P (easier if it's not on an edge or in line with one). Join P to each vertex of the polygon by a line. For two adjacent vertices, say A and B, the lines PA and PB together with the edge AB form a triangle. Find the area of the triangle. Note whether the little arrow on AB winds clockwise or anti-clockwise: make the area +ve for clockwise and -ve for anti-clockwise. Do the same for all triangles meeting at P, and and add their areas. The 'size' of this (ignoring any final minus sign) is is your answer. It works whether P is inside the polygon or outside, and whether the polygon is convex, concave or starry or just a tangle. Amazing! Where the polygonal circuit winds round some region more than once (say the central region of a star polygon), the area of this region counts several times in the sum - we say that it has a higher "density" than the outer parts. For a region where the circuit winds once round one way and once back round the other, you will find this region has zero area! (the triangles which cover it neatly cancel each other out). Hope you can follow all that. -- Steelpillow 17:13, 10 May 2007 (UTC)

Huh? There is a section, Polygon#Area. For partitioning in trianges, see polygon triangulation. For self-intersection polygon the notion of "area" depends on what you want. There are at least three possible interpretations. `'mikka 00:45, 11 May 2007 (UTC)

Hmm. Yes. I looked at that and dismissed it as too simple, but re-reading the description it does seem to be what I want. :) JulesH 13:49, 11 May 2007 (UTC)

"For self-intersection polygon the notion of "area" depends on what you want." Yes indeed. Somewhere I have a web page with 5 different interpretations. But if you are that advanced you won't be asking "Does anyone know of an algorithm for calculating the area of a concave polygon" 8-) -- Steelpillow 20:20, 11 May 2007 (UTC)

Classification

This taxonomy is all wrong. It mixes ideas from computer graphic with those from mathematics. Plus some obvious errors: cyclic but non-convex polygons exist (e.g. "butterfly" quadrilaterals). The type referred to in the illustration as "regular complex" is usually called a "regular star" polygon, a regular complex polygon being a figure in the unitary plane (see Coxeter's book "Regular complex polytopes"). "Simple" is used by different authors to mean what they want it to at the time. And so on. I'm deleting the ASCII art, composite graphic, and legends for the images at the top of the page. Let's rebuild something worth having, one step at a time. I've made a start on the stuff left in. -- Steelpillow 20:27, 17 June 2007 (UTC)

I'm not in a position to judge or help. Just thought I'd remind simple polygon, concave polygon, complex polygon, cyclic polygon, equilateral polygon, regular polygon articles exist and may need to be looked at too. Tom Ruen 20:32, 17 June 2007 (UTC)

Area Error?

Under Area/Simple polygons we have A=(....+ xny1 + x1yn)/2 - surely that should be A=(....+ xny1 - x1yn)/2 (sign changed)? I hesitate to make the edit, as I'm a bit tired and may possibly be wrong - could someone check this out please?

Looks like it to me too, following pattern of grouped terms (xi*yj-xj*yi) before that. I changed it. Tom Ruen 23:52, 2 August 2007 (UTC)

In the same area the following statement needs to be reversed: "The vertices must be ordered clockwise or counterclockwise, if they are ordered clockwise the area will be negative but correct in absolute value." The "if they are ordered clockwise" should be "if they are ordered counterclockwise". Simple example with a triangle going counterclockwise:

x	y	term	sum
2	1	3	3
1	2	-4	-1
3	2	-1	-2
2	1

and the same example clockwise:

x	y	term	sum
2	1	1	1
3	2	4	5
1	2	-3	2
2	1

Dugmartin 15:37, 18 October 2007 (UTC)

Mathematicians traditionally treat counter-clockwise as positive and clockwise as negative. It seems to come from the usual ordering of point (x, y) coordinates to draw a square starting with the x-axis, (0, 0) (1, 0) (1, 1) (0, 1), which is counter-clockwise. -- Steelpillow (talk) 20:14, 6 December 2007 (UTC)

Contradiction: complex polygon

At one point this article says that a complex polygon is one that lies in a complex plane (the x-y plane where x=a and y=b from the expression a+bi) it later goes on to state that a complex polygon is one which is in a way non-planar. —Preceding unsigned comment added by Mathet (talk • contribs) 04:29, 6 December 2007 (UTC)

The detail of what you say is not quite accurate, but you are right about the contradiction. In traditional geometry, the term complex polygon describes a particular kind of polygon that involves complex numbers. More recently, people have been looking for a straight forward word that means "not simple", and came up with "complex" (mathematicians call such polygons "self-intersecting"). So to these people, "complex polygon" just means a polygon which is not simple. This recent usage is common on the Internet (Google it and see), and especially so in discussions of computer graphics. I edited the article a while ago to try and clarify this point, but it looks like I could have done better. Feel free. -- Steelpillow (talk) 20:09, 6 December 2007 (UTC)

The eq. for area is wrong, because summation should include multiplication of the first and last vertices as well as given in http://mathworld.wolfram.com/PolygonArea.html . --MindaugasL (talk) 20:24, 31 May 2009 (UTC)

Tetradecagon

Sorry if this has been brought up before but why does Tetradecagon redirect to the polygons article?? Surely it would have its own article like the other shapes. JTBX (talk) 18:12, 13 March 2008 (UTC)

Presumably because nobody has created the page. Personally, I think most of those pages specific to a single type of polygon are pointless - the whole lot would be better done as a single table of values. -- Steelpillow (talk) 20:24, 13 March 2008 (UTC)

Googolgon

I added the googolgon to the chart, as googolgon redirects here, and, thus, the article should at least mention the word. As far as I can tell from comments on Talk:googolgon and Talk:googol#Adding the googolgon picture, there was an VfD (old name for WP:AfD) in 2004 (that I can't find), and the result was redirect, which is why the word does not have its own page like most of the other polygons on the chart.

I also had to add a statement that I know to be true, but cannot verify--that mathematicians will sometimes use the word + "gon" instead of number + "-gon" when the number is difficult to express numerically. I did this mainly to keep the chart from contradicting the article. Since little else is sourced on this page, I didn't bother adding the {{fact}} tag. I have also never seen 10¹⁰⁰-gon used, although it does seem practical. But I thought that was stretching the limit of what I should say.

I'm sorry that was so verbose. You can concat it if you can/want.

-- trlkly 06:51, 11 May 2008 (UTC)

While it is true, as you say, that little else is sourced on this page, most source material is scattered across a huge range of books and papers - many of whose primary concern is polyhedra rather than polygons. It would probably take a great deal of effort to put together a selected set of references that cover everything. I will add a few of the obvious ones, most of which will refer the reader on to more specialised works. On the well established subject matter, we cannot hope for more.

Returning to the googolgon, this is proving contentious, so the standards of referencing are correspondingly more strict. While the term may be used on some web sites, etc., I think we need a proper citation of a published example before such a contentious term can be accepted. I know, it's a pain, and I get annoyed when people pull policy on me in just this way. But that's policy for you. -- Steelpillow (talk) 12:21, 11 May 2008 (UTC)

The VfD discussion, which took place in 2004, can be found here. I'd say it had no clear conclusion, but a slight majority appeared in favour of merging with Googol (and not Polygon). The original Googolgon article was in fact initially merged with Googol and redirected there, but both actions were quickly undone by other editors. A second redirect to Googol was made in February 2007, but one minute later the editor changed her mind and redirected to Polygon instead.

Unless evidence can be produced that "googolgon" is a notable concept, I favour deleting the redirect page Googolgon.  --Lambiam 15:41, 16 May 2008 (UTC)

Instead of "googolagon", isn't it simpler to just say "circle"? :-) 217.171.129.73 (talk) 22:55, 30 May 2008 (UTC)

Mathematically, there is a big difference between a topological loop made of 10¹⁰⁰ points and 10¹⁰⁰ line segements, and a loop made of just one smooth curve. Also, a googolgon need not be regular, and might approximate to many other finite, closed curved - we might even consider a nonconvex zig-zag example as approximating some fractal line. -- Cheers, Steelpillow 11:57, 31 May 2008 (UTC)

This article is affirmed in large part by the article at Polygons, but, as noted on that page, does not violate copyright. Zoe17 (talk) 14:13, 22 June 2008 (UTC)

Not sure what you are getting at Zoe17. The article you link to has not yet been published and so is not visible to the casual visitor such as me. I would assume that, like many others on the New World Encyclopedia, it is an edited version of the corresponding Wikipedia article (i.e. this one), so how can it serve as a reference for Wikipedia content? Am I missing something? -- Cheers, Steelpillow (Talk) 16:17, 22 June 2008 (UTC)

Yes, the http://www.newworldencyclopedia.org page on polygons looks obviously copied from wikipedia, so that's a circular reference, a HUGE trap that often occurs from wikipedia and uncountable online reflected copies! Tom Ruen (talk) 02:29, 23 June 2008 (UTC)

P.S. I agree a googolagon is just silliness. A myriagon is at http://mathworld.wolfram.com/Myriagon.html, but without any interesting contents. Looks like the highest polygon on mathworld is a 65537-gon, http://mathworld.wolfram.com/65537-gon.html being notable (and referenced) for its constructability! Tom Ruen (talk) 02:36, 23 June 2008 (UTC)

OK, this has been done to death. I have removed it again. -- Cheers, Steelpillow (Talk) 08:39, 13 September 2008 (UTC)

Delete section?

Why on earth is there a "Things to do with polygons" section? This is an encyclopedia, not a child's activity book. Maybe there's something like that on a different website; there could be a link to that instead of having this little activity section. People might come to this article to find out more about polygons, not how to fold paper into crinkly things. 98.166.139.216 (talk) 00:36, 24 September 2008 (UTC)

For now I have renamed it Uses for polygons. Much of the content is worth preserving, but I agree it needs to be explained better. For example infinite polyhedra, described here as "crinkly things", are an important class of geometrical object. -- Cheers, Steelpillow (Talk) 18:39, 24 September 2008 (UTC)

Regardless, the terminology "crinkly things" needs to be revised. --204.111.164.69 (talk) 00:00, 10 October 2008 (UTC)

Regular Polygon

How do I Compute the Area of a polygon whose given are the coordinates? An example are the technical discreption of the land survey. Please give me details of the Formula A= n/4 S↑2cotπ/n. Thank you.

EMMANUEL ALDE, SR. Borongan City, E. Samar E-mail: [email protected] —Preceding unsigned comment added by 203.177.248.253 (talk) 06:54, 9 January 2009 (UTC)

Hi, the answer is discussed here in the article on Regular polygons. Hope this helps. -- Cheers, Steelpillow (Talk) 09:39, 9 January 2009 (UTC)

Photo Error

This is probably dumb but one of the photos identified a polygonous fruit as 'starfruit' when it is, in fact, a carambola fruit. I can't edit this page even when logged in, otherwise I would change it myself. Sorry about that. 70.250.35.158 (talk) 19:57, 9 June 2009 (UTC)Me.

It seems like starfruit and carambola are two different names for fruit of the same plant Averrhoa carambola. --Salix (talk): 21:19, 9 June 2009 (UTC)

ploygons

ploygons sheet please —Preceding unsigned comment added by 74.38.37.156 (talk) 02:32, 25 February 2009 (UTC)

Poly and vertex count

Some additional hints to expand my recent edits, in which I have stated that vertex count shall be taken with care. Take for example a vertex that is defined as {opos, onorm, col0} with opos, onorm and col0 being 3D vectors. A very minimalistic vertex processing would transform opos and pass along col0. More realistically, onorm will have to be transformed as well.

Now, suppose that per-vertex lighting is to be computed; since lighting is computed per-light, this results in a further overhead which is a function of the light count and type. In fact it was once typical to specify the vertex count and performance togheter with the number of lights used. This is a first reason for which vertex count is to be taken with some care - the same vertex does not really have to be processed in the same way on different systems.

The second reason is related to an implementative optimization exploiting temporal coherence. A cache is then used to mantain associations between the initial vertex values and the produced output. A matched vertex causes the whole processing to be short-circuited. While it's generally expected that more vertices will equate with a slower performance this is another reason for which some uncertainity should be considering when comparing similar numbers.

I hope somebody may find this interesting (I've tried to not dwelve too much in the details). Have a nice day.
MaxDZ8 talk 08:33, 14 July 2009 (UTC)

Are "self-intersecting polygons" really polygons at all?

It seems to me that self-intersecting polygons present some very fundamental problems, mostly in that they create an irreconcilable rift between a polygon's edges and the area that is bound by them. A self-intersecting quadrilateral, for example, could only be described as such if you define the polygon as a set of edges; the area cannot in any way be described as a quadrilateral, as it's really just two triangles sharing a corner. Pentagrams present another dilemma: does the pentagonal area inside the pentagram count as part of the pentagram's area, or not? If so, then the area is really that of a concave decagon, not a pentagon; if not, then the area is that of five triangles, and not a pentagon. What ever happened to the concept of degeneracy? Furthermore, in the case of the self-intersecting quadrilateral, the interior angles no longer add up to a constant amount, unless you consider that two of the angles are "inside-out", with the exteriors of the angles facing the interiors of the polygon. This sort of thing would have made Archimedes soil his toga. So at what point in he history of mathematics was it decided that the area bound by a polygon did not need to be contiguous, the sum of the interior angles of a polygon did not need to be constant, etc.? —Preceding unsigned comment added by 99.48.55.114 (talk) 09:07, 27 July 2009 (UTC)

To answer that last question first, it was decided by Pythagoras long before Archimedes. The Pythagoreans was a mystery cult founded on his teachings. and their secret sign was the pentagram. Since then Thomas Bredwardine, Kepler, Poinsot, Cauchy, Bertrand and Cayley all studied regular star figures, with Cayley finally explaining many of your questions about inside/outside by setting the idea of density on a firm mathematical footing. Modern theory, pioneered notably by Grünbaum and moved forward by Danzer, McMullen, Schulte, Johnson and others, treats real polygons as "realisations" of the abstract polygonal structure. You are right about some of the problems though, the process of "realisation" is yet to be properly formalised and various issues remain to be resolved. Meanwhile another modern theory, of convex polytopes, (and also originally set on its feet by Grünbaum) has taken his seminal work and rushed off in entirely incompatible directions; it is likely that many of your troubled feelings arise from these incompatibilities. All in all, the question as to "What is a polygon?" has yet to be answered satisfactorily. HTH -- Cheers, Steelpillow (Talk) 09:52, 27 July 2009 (UTC)

So a person would not necessarily be incorrect in stating that self-intersecting polygons are not really polygons at all, and the Kepler-Poinsot objects are not really polyhedra? This demands some serious NPOV revisions to most of the Wiki's articles on geometry... —Preceding unsigned comment added by 99.59.25.200 (talk) 05:59, 3 August 2009 (UTC)

Sorry, I should have been clearer - self-intersecting polygons have always been understood as true polygons, at least since the days of Pythagoras. There is no question about that. Most of your confusions arise where results from the theory of strictly convex polygons are taken out of context. The areas which remain to be clarified are only some of the finer theoretical concerns, which modern abstract theory has pushed into the general area of "realisation": they do not affect the status of the Kepler-Poinsot stars as true polyhedra. HTH -- Cheers, Steelpillow (Talk) 10:14, 3 August 2009 (UTC)

The Pythagoreans believed a lot of strange things. For example, they abhored the idea of irrational numbers. On what basis do modern mathematicians claim that self-intersecting polygons are still polygons, and how do they resolve the inevitable conflicts of logic that arise? —Preceding unsigned comment added by 99.48.55.129 (talk) 00:42, 5 August 2009 (UTC)

Mathworld's definition of a polygon is here; note that "Polygons can be convex, concave, or star." If you follow up the references given in the article, you will find similar definitions and remarks. -- Cheers, Steelpillow (Talk) 14:06, 5 August 2009 (UTC)