# Talk:Polygon

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## 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
I disagree. One should not assume a polygon is regular. If one reads or hears "triangle" one should not assume it is an equilateral and equiangular triangle. The same is true if one reads "pentagon." One should not assume it's a regular pentagon.
Beth in MN —Preceding unsigned comment added by Bethhentges (talkcontribs) 21:57, 15 June 2010 (UTC)
Such terms as "triangle", "pentagon", etc should never be taken to indicate regularity unless expressly stated. However, much well-known theory derives from the study of regularity (e.g. Coxeter's "Regular polytopes"), and there is a bad habit out there of "usually" forgetting that other, non-regular examples can behave differently. As wikipedians, we should try to avoid such bad habits. — Cheers, Steelpillow (Talk) 09:49, 10 July 2010 (UTC)

## 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? Double sharp (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? Double sharp (talk) 05:09, 9 August 2009 (UTC)
See Convex set. -- Cheers, Steelpillow (Talk) 14:06, 9 August 2009 (UTC)

Usually, a polygon is not allowed to cross itself, so the pentagram you drew is not considered a polygon by many. A regular five-pointed star with just the "outside" line segments of the pentagram is non-convex because it contains two interior points whose line segment connecting them contains points in the exterior of the polygon.

Beth in MN —Preceding unsigned comment added by Bethhentges (talkcontribs) 22:01, 15 June 2010 (UTC)

Briefly, self-intersecting or coptic polygons have been recognised as true polygons since the days of Archimedes. Modern authors have taken polygon theory in many different directions, with the result that many incompatible understandings have arisen - even become commonplace. For example the star pentagon in question is neither convex nor simple, which rules it out in these specialised fields, but it is still recognised as a true polygon in any general treatment of the subject. For more about all this, see elsewhere in this talk page, e.g. Are "self-intersecting polygons" really polygons at all? and What needs to be fixed on the page because of silly errors. — Cheers, Steelpillow (Talk) 09:42, 10 July 2010 (UTC)
Actually, in the mathematical study of Euclidean Geometry, the term polygon does NOT allow edges to intersect. Doing so would cause problems to the statements of geometric results, as some would no longer be true as commonly stated.
For example, elsewhere in these discussions, there is the statement that the vertex angles in an n-gon sum to 180(n-2) degrees. This result is not true if sides are allowed to cross. One such example is a quadrilateral, which is defined as a 4-gon. If sides are allowed to cross, then the four vertex angles could sum to be less than 360 degrees.
For instance, take a square ABCD, where point A, B, C, and D are those you encounter as you move counterclockwise around the square. If we make B go to D instead of C and A go to C instead of D, then each of the four vertex angles measure 45 degrees instead of 90 degrees; thus they sum to 180 degrees instead of 360. So it would no longer be true that the vertex angles of a quadrilateral must sum to 360 degrees.
At the least for these reasons, mathematical studies of Euclidean geometry do not allow line segments to cross in a polygon. When they do, you simply have a figure that is made of multiple polygons. Dr. Belnap 04:39, 3 November 2011 (UTC)
Not all the four 45-degree angles are inside the polygon. Let E be the intersection of AC and BD (it is not a true vertex of this polygon). Now imagine that you are tracing a line along the edge of this polygon. Your line starts just outside the triangle ABE when you start tracing along AB. You then move toward D. When you pass BD, you move from outside the polygon to inside the polygon. Then you follow DC inside the polygon and then finally move back outside after passing E on the way from C back to A. Therefore you must consider the reflex angles instead of the acute ones. (BTW, I do not see any problem with the fact that star polygons do not have the same angle sum as simple polygons. All you need to do is take into account the polygon's density. It should not be too difficult to find a generalised formula that determines the sum of angles of a star polygon based on its number of sides and density.) Double sharp (talk) 08:39, 9 April 2012 (UTC)
Just a note of caution. The idea of density works well for star figures that are regular, but not for coptic figures in general. Consider this pentagon, where vertices are marked 'o' and edges are continuous across the inner two crossing points::
          o         o
/ \       / \
/   \     /   \
o-----\---/-----o
\ /
o

The idea of density does not help here*, while we can see that by pushing the bottom vertex up just above the rest of the figure we can obtain a convex figure. Seeing this one as a morph of the convex one allows us to qualify the bottom vertex as reflex. The equivalent analysis of the regular star pentagon reveals that two vertices must be reflex. But which ones? The choice is arbitrary and breaks the symmetry - the angle-sum approach is evidently not appropriate in such cases. Ultimately, neither approach can be usefully generalised for all cases.
* One might note that in the above pentagon, a density of −1 could be ascribed to the lower triangular region, and associated with the reflex nature of the tangent vertex. However, here is a certain coptic hexagon that provides a counterexample:
          o         o
/ \       / \
/   \     /   \
o-----\---/-----o
\ /
X
/ \
/   \
o-----o

To achieve the correct angle sum we must alternate adjacent vertices between interior and reflex. There is no relationship to any conceivable density values.
— Cheers, Steelpillow (Talk) 20:53, 9 April 2012 (UTC)
This is what I get for not being careful enough. I wonder if my "tracing" method above determines the inside and outside of a star polygon correctly? Double sharp (talk) 06:48, 10 April 2012 (UTC)
It depends on your definition of a "star polygon". Your tracing method assigns a consistent property to the boundary which we may call orientation. This may be used to assign a consistent set of density values to the various convex interior regions: for example a single boundary around it marked on the outside gives say a density d = 1, a boundary marked on the inside gives d = −1. Two bounding loops sum to d = +2, −2 or 0 according to their respective orientations. And so on. This is consistent with the conventional density model for regular star polygons. But for some less regular figures it can lead to ambiguities where d = 0 may indicate either two oppositely-oriented "layers" of the interior, or none at all: ask yourself whether the diamonds in this dodecagon comprise holes or overlapping regions or maybe one of each or doesn't it make sense?
                  o     o
/ \   / \
/   \ /   \
/     X     \
o     / \     o
|    /   \    |
|   o     o   |
|    \   /    |
|     \ /     |
|      X      |
|     / \     |
|    /   \    |
|   o     o   |
|    \   /    |
o     \ /     o
\     X     /
\   / \   /
\ /   \ /
o     o

If you regard such figures as "stars" then the answer is 'No', but if you require stars to "look" regular, then yes your method works. Beyond this, life gets fascinating and I must avoid posting a resumé of my unpublished study. — Cheers, Steelpillow (Talk) 19:46, 10 April 2012 (UTC)
I consider this to be a true star polygon. My tracing method ends up marking the two diamonds half on the inside and half on the outside, causing the ambiguity you mentioned. (But does it still define the density of the two diamonds well, despite causing trouble with the interpretation of their densities?) Good luck with your unpublished study! (I'd love to see it when it's published.) Double sharp (talk) 13:40, 11 April 2012 (UTC)
Yes, because the diamonds are not bounded by an oriented circuit they default to d = 0. This is more usually formalised in terms of a rotating ray, but the principle is the same. The problem comes in deciding whether they are part of the "inside" or not:  are there two overlapping regions in a diamond or aren't there? And this is where the fun really begins. We need to define "inside" with rigorous mathematical formalism. Cue much argument. — Cheers, Steelpillow (Talk) 20:41, 11 April 2012 (UTC)

I did some research into the individual (Coxeter) that Steelpillow keeps citing and I find that Steelpillow does not understand the nature of these sources. Coxeter's geometric work was in a very specialized area of mathematics and the papers he published dealt with graph theory. Graph theory uses some altered definitions from those generally used throughout the mathematical (and public) community. In graph theory, it is common for the graphical representations of paths to be comprised of edges that visually cross one another, but are assumed to be non-intersecting from a topological perspective. This leads to shapes that Steelpillow wants to call polygons, because in that very specialized study they are called that because of their close resemblance to what is broadly accepted as polygons. In the broader mathematics community, polygons do NOT allow edges to cross; that is why Coxeter, in trying to look at a broader classification of objects formed by line segments, added the adjective (simple) to polygons, so that he could distinguish the different generalizations of polygons from the more common class of polygons. We should not define polygons as broadly as Coxeter did, because his work is not indicative of the common usage of the word polygon through the mathematical community. What Steelpillow wants to call polygons is what is used in Computer Graphics and we already have a section describing those differences; there is no need to make the mathematical definitions inconsistent with their accepted uses. — Preceding unsigned comment added by Belnapj (talkcontribs) 15:57, 3 November 2011 (UTC)

Hi. Failure to understand, like polygons, comes in many forms. I would recommend you do a little more background reading. A few examples may be found in the over-simplified discussion below, I can give you plenty more if you like.
Poinsot's paper on star polygons and polyhedra might be a good place to start: L. Poinsot, Mémoire sur les polygones et les polyèdres, Journal del' École Polytechnique, 4 (1810), pp. 16-49. Cauchy, Bertrand, Cayley and Brückner built on Poinsot's ideas, while for example Grünbaum's "Poyhedra with hollow faces, Proc. of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed. T. Bisztriczky et al, Kluwer Academic (1994) pp. 43-70., "Metamorphoses of polygons". The Lighter Side of Mathematics, Proc. Eugène Strens Memorial Conference, R. K. Guy and R. E. Woodrow, eds. Math. Assoc. of America, Washington, DC 1994. Pp. 35 – 48. and "Are your polyhedra the same as my polyhedra?" Discrete and comput. Geom.: the Goodman-Pollack Festschrift, ed. B. Aronov et al, Springer (2003), pp. 461-488., trace his part in the development of a more modern form of this topological understanding, of both non-convex polygons and polyhedra, while in his seminal "Convex polytopes" he reminds us that when he makes frequent statements about 'polytopes' he is confining his remarks to the convex variety, omitting hundreds of references to convexity for the sake of readability - he points out that the clue is in the title - and overtly acknowledging the non-convex variety (A polygon is a two-dimensional example of the more general polytope in any number of dimensions).
Coxeter's equally seminal work is his "Regular polytopes", available in various editions over the years. He discusses convex figures before generalising to star polytopes. Other works of his that I happen to have read include the elementary "The beauty of geometry" and "Projective geometry", and the slightly more exotic "Regular complex polytopes". Other papers deal with the nature of hyperbolic geometry, one I have seen is devoted to its possible application to the cosmology of the day. While he was aware of - and indeed discovered and described - much deep commonality between polytope and graph theories, much else in this material has no relevance to graph theory.
The important point is that the term "polygon" is nowadays used in diverse and sometimes incompatible mathematical contexts. Since Wikipedia is an encyclopedia and not a text book, its task is to acknowledge and summarise all significant usages of the term. (locally) Euclidean convexity and graph theory are just two of them, albeit very widely used in the topological decomposition of all kinds of object, but they are far from the whole story. Several mathematicians I know have expressed the opinion to me that "a polygon can be anything you want it to be, as long as you define it at the beginning of your discussion". I disagree with the wisdom of that approach, probably as passionately as you do, but equally I have been (painfully) forced to recognise that attempting to impose a single and wholly universal mathematical definition would be unhelpful.
— Cheers, Steelpillow (Talk) 11:24, 5 November 2011 (UTC)
On the definition of a polygon, consider that the faces of polyhedra are commonly understood to be polygons, while polyhedra are said to have a bounding surface and are often described as "solids" (e.g. Coxeter's "Regular polytopes"). If a polygon is merely a circuit of segments and corners, then the polyhedron becomes more a skeleton than a solid and its boundary is no longer a surface. Hence, whatever definition the majority of mathematicians may choose to work with, the elementary logic of physical objects is best used in the introductory lead. Hence I have reverted to the form of lead that a number of editors have hammered out over the years - at the same time, trying to bring mathematics into better focus. — Cheers, Steelpillow (Talk) 12:13, 5 November 2011 (UTC)
Steelpillow, you should stop criticizing me personally. As a professional mathematician, I am full aware of the different topics that you bring up and discuss. The thing is that you are citing sources that are in highly diverse and specialized fields of mathematics. You mention, for example, projective geometry and hyperbolic geometry; these are completely separate and distinct types of geometry that have completely different assumptions (mathematical axioms). In fact, what constitutes a line in these spaces does not even closely resemble what is a line in the plane; the differences in the axioms completely change the space studied in geometry as well as the properties that result. Projective and hyperbolic geometry are very specialized mathematical fields that are not studied in elementary, middle, or high school and not even for most college students until they reach the middle of their undergraduate degrees, if that; therefore, they do not represent appropriate authoritative support for your position. When you change the geometry, you change all results. A polygon in these spaces is not the same thing as a polygon in other spaces and does not have the same properties; definitions are intricately connected to the geometric assumptions they are based upon.
Furthermore, 3-dimensional space is also different from plane geometry. Work in this space does not have simple connections to planar geometry. Furthermore, as you have mentioned, the definitions for objects in the study of 3-dimensional space are not often well-defined. For instance, definitions of polyhedron are not very well formulated in most domains.
I . We no doubt have many students who would be looking to wikipedia for information. The information we provide should be consistent with what is intended that they learn. Defining things such as poly The knowledge on wikipedia is supposed to be knowledge for the general public, not representing specialized fields. It would be greatly appreciated if you would try to understand and accurately represent the perspective of the mathematics community in your editing of the page and at the very least be courteous enough to not censor the ideas experessed herein. When I made the edits that I did, which you so dictatorily removed, I made a clear attempt to not only represent what the mathematics community accepts as a polygon, but also acknowledge that other fields defined them differently. I would appreciate your cooperation, rather than your subversion. Dr. Belnap 17:58, 24 November 2011 (UTC)
My apology if you find a personal attack in my more recent comment above. I have re-read it and can still find none. You "argue that we should make the definition of polygon consistent with what the lay public is likely to encounter and study", while I believe that "the elementary logic of physical objects is best used in the introductory lead". That looks like agreement to me. Where we seem to differ is that you regard this as calling for a "Lesson One" [corrected] mathematical definition, while I do not believe that the lay person has sufficient mathematical appreciation for that to be appropriate to the article lead.
As to the actual content, the lead grew historically as a result of several contributors thrashing out our differing approaches. What emerged should not be treated lightly, or substantially edited without prior agreement. I am happy to discuss the rest of your proposal, and no doubt others will chip in to help reach a consensus. I did restore a little of what you wrote, but in the mean time let us not play ping-pong with the page itself.
On the matter of references, Regular polytopes and Convex polytopes are widely-read works which can hardly be described as specialised: they are indeed precisely what this lay person encountered and studied. Some others which I referenced are original sources - which, out of respect to you as a professional mathematician, I would not pull away from drawing to your attention. Others do move beyond a Euclidean treatment, and were there simply to make the point which begins the subsequent paragraph.
More directly relevant is the issue that most lay people would regard a pentagram as a star polygon, recognised as such since the days of Archimedes but excluded by the definition you gave, which is more formally understood as a simple polygon. While mathematicians such as Grünbaum and yourself might choose to frequently omit descriptors such as "simple" or "convex" in treatments of these objects (Grünbaum explicitly, yourself implicitly), a general introduction to the subject can take no such luxury. The article lead should conform as far as is possible to the lay person's preconceptions, whereby polygons may be divided into those which self-intersect and those which do not.
With that proviso, if you have ideas for significant changes to the article lead, I would hope you can share them with us first. Looking back at what you wrote, I could probably second-guess some of it but would probably do little better than the current state. Yes, if we can co-operate that would be good. — Cheers, Steelpillow (Talk) 20:34, 24 November 2011 (UTC) - Updated — Cheers, Steelpillow (Talk) 20:42, 24 November 2011 (UTC)
Well, I have tried to re-create the spirit of your lead at Talk:Simple_polygon#Could_the_lead_be_better.3F. Does that help at all? — Cheers, Steelpillow (Talk) 12:00, 26 November 2011 (UTC)

## 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)

The sum of the interior angles of a convex polygon is usually given as (n-2)*180 degrees, rather than 180n-360. —Preceding unsigned comment added by Bethhentges (talkcontribs) 16:48, 15 June 2010 (UTC)

Beth in MN

Done. — Cheers, Steelpillow (Talk) 09:30, 10 July 2010 (UTC)
A correction was in order here, which I added tonight. The argument suggested about the sum of the exterior angles is incorrect, unless the polygon is convex. If the polygon is not convex, then the exterior angles do not sum to be 360 degrees, unless you establish a signed direction for "turning" at each vertex, meaning that you subtract angles where you are forced to turn in the opposite direction. Dr. Belnap 04:53, 3 November 2011 (UTC) — Preceding unsigned comment added by Belnapj (talkcontribs)

## "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

I thought Gauss proved it and somebody else constructed it? Double sharp (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??

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)
DoneI agree, and I've simply removed it.  --Lambiam 09:48, 7 April 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)

_____________________________

Everyone I know has always used nonagon for a 9-gon, and undecagon for an 11-gon, even though the Latin and Greek are mixed. After dodecagon for a 12-gon, in practice, most everyone uses the n-gon form rather than the true name. I don't have a problem with the "official" names being listed, but the common practice is nonagon, decagon, undecagon or 11-gon, dodecagon, 13-gon, 14-gon, . . .

Beth in MN —Preceding unsigned comment added by Bethhentges (talkcontribs) 22:08, 15 June 2010 (UTC)

These days quite a few of the leading experts refer to them as "enneagons", however "nonagon" is still more common. I don't think it matters as long as we give both names and remain consistent within a given article. BTW, "sexagon" should never be used, it should always be "hexagon". — Cheers, Steelpillow (Talk) 09:05, 10 July 2010 (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)
One old book defines a sexagon as a figure with six sides and six angles, but then advises readers to look at "hexagon". Double sharp (talk) 05:17, 9 August 2009 (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)

## 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)

## 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)

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)
What Jules says is correct. A polygon is regular if it is cyclic and equilateral. The start pentagon is not actually considered a polygon, but a polygram. In the present day mathematics community, polygons are defined to disallow edges to cross. Edges may pass one another and be considered to not intersect, but only under more different topologies than those studied by elementary, secondary, and early college students. This wikipedia page is predominantly aimed at the general public, which is most familiar with this traditional Euclidean geometry, using the Euclidean plane; in this study, vertices are where edges meet and only two can meet at each vertex, which precludes the possibility of intersecting edges. I think that this distinction needs to be made explicit, otherwise we are doing nothing more than to obscure peoples' understanding of geometry, particular elementary and secondary students who come here for information. Furthermore, we risk individuals assuming that certain geometric results apply to situations where they do not. — Preceding unsigned comment added by Belnapj (talkcontribs) 15:27, 3 November 2011 (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)

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 (talkcontribs) 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)

I think the formula given accounts for this, but the numbering of the vertices is different to that in the drawing shown, i.e. 0 to m-1 i.o. 1 to m. 131.180.16.252 (talk) 15:52, 25 November 2009 (UTC)

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)
It now exists. However, the polygon articles not very interesting above dodecagon, which incidentally is notable for being the highest-sided polygon that, in conjunction with other polygons, can tile the plane. By the way, if we use dodecagon above dokaidecagon, pentadecagon above pentakaidecagon, hexadecagon above hexakaidecagon...then shouldn't triskaidecagon be moved to tridecagon? Double sharp (talk) 06:02, 9 August 2009 (UTC)
Here we go...
1. Henagon is a stub that is too technical for my liking. Why isn't there a henahedron or monohedron article, especially since the reference mentions the henagonal henahedron {1, 1}?
2. Digon is quite a good article for such an obscure topic.
3. Triangle is great, but shouldn't there be a taxonomy like the next one?
4. Quadrilateral should have a definition for the area of a butterfly quadrilateral, but otherwise it's brilliant.
5. Pentagon? No comments. But I think it should be kept.
6. Hexagon should perhaps mention sexagon. And sexagon, which should probably redirect to hexagon, might have to be semi-protected because some vandals might introduce inappropriate redirecting.
7. Heptagon is probably notable for being the first unconstructible polygon. Nevertheless, I think it'll always remain like this, because not many things use heptagons.
8. Octagon has the last sentence of the "Regular octagons" section slightly inappropriate! I'm not sure about the relevance of the great dirhombicosidodecahedron to this, and perhaps the 2{4} or {$8/2$} octagram should be mentioned.
9. Enneagon - should the article be at nonagon or enneagon? But it's quite good. Maybe the {9, 3} hyperbolic tiling should be mentioned. .
10. Decagon should have an animation for its construction. I don't think the sentence "in short, a decagon is a ten-sided shape" is necessary as that was already defined in the lead section.
11. Hendecagon is a stub with hardly any useful information except that about the coins. I'd say to keep it, but only because of the use in coins.
12. Dodecagon is the last polygon article that isn't a stub...
13. Triskaidecagon has the pictures of triskaidecagrams that aren't really that relevant. We could do without this article. A stub that should be deleted (STSBD).
14. Tetradecagon? I've seen tetrakaidecagon before, which again raises questions about moving the page. Actually all articles beyond the dodecagon (except the heptadecagon) only repeat information about the polygon that should be better put in a table of polygons. Another STSBD.
17. Heptadecagon should be kept for its constructability, and maybe we could have 257-gon and 65537-gon articles for that reason.
20. Icosagon is the last STSBD we have, and thank goodness for the fact that chiliagon and myriagon have been deleted. They didn't have much content anyway, and they even lacked the polygon infobox. Double sharp (talk) 06:02, 9 August 2009 (UTC)
For extravagent fun, I linked up images to the Petrie polygon graphs of higher dimensional polytopes by their regular polygon sides. Not defendable reason to keep any above dodecagons, but pretty! Tom Ruen (talk) 10:20, 9 August 2009 (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)

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)
The content about infinite polyhedra may very well be worth preserving, but is the article on polygons really the place for it? Not every important class of geometrical object needs to be linked to every other important class of geometrical object. We could have some template "Geometrical objects" that links to all the important classes of geometrical objects. It would be a very large template, but Wikipedia has plenty of large templates already. 98.166.137.88 (talk) 19:40, 20 September 2009 (UTC)
I agree about this article. Infinite polyhedra are just a subset of polyhedra in general anyway. I have deleted it. As for large templates, I loathe the things - I always prefer a separate summary article with small links to it in other pages. -- Cheers, Steelpillow (Talk) 21:10, 20 September 2009 (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: emmanuelsr.alde@yahoo.com —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)
Hasn't it been established that Mathworld is an unreliable resource? —Preceding unsigned comment added by 99.48.55.129 (talk) 20:04, 12 August 2009 (UTC)
Then you'd better follow up the references given in the article. -- Cheers, Steelpillow (Talk) 08:27, 13 August 2009 (UTC)
Coxeter, Harold Scott Macdonald (1973). Regular polytopes. Courier Dover Publications. ISBN 9780486614809.Chapter VI, star-polyhedra defines star polygons in section 6.1, page 93. [1] His interest is for REGULAR star polygons. Tom Ruen (talk) 08:49, 13 August 2009 (UTC)

The heart of this issue lies in the fact that the study of polygons is at the intersection of several disciplines and that not everyone uses the word to represent the same class of objects. While it may be true that the ancients studied objects like these where the sides do intersect, not everyone considers such objects to be polygons. In fact, the most commonly used definition of polygon in the mathematics community at large is that the line segments that make up polygons are not allowed to intersect. — Preceding unsigned comment added by Belnapj (talkcontribs) 04:35, 3 November 2011 (UTC)

## what is a polygon?

what is the definition for a polygon? —Preceding unsigned comment added by 207.144.69.131 (talk) 22:33, 26 January 2010 (UTC)

## What needs to be fixed on the page because of silly errors

A shape with lines running though it is NOT a polygon. There is a picture at the top of the page with other examples that isn't a polygon and it is very distracting to me and my fellow colleagues that are mathematicians-in-training from Massachusetts Institute of Technology. This should be fixed immediately. I tried to myself, but I couldn't edit the page for some odd and most likely idiotic reason. Please if anyone cares about the future of math and the future of the world it should be changed.

Your favorite mathematician-in-training and future professional thinker, Bex Robington —Preceding unsigned comment added by Beeky21 (talkcontribs) 01:49, 27 January 2010 (UTC)

Hi Bex,
Bex is correct. The most commonly accepted definition of a polygon in the mathematics community is a simple, closed figure, comprised of line segments joined pairwise at the endpoints, called vertices. This definition is what is used in any study of Euclidean geometry, as the results of that study rely upon this fact. Of course, there are likely to be people in other disciplines that do not agree with this definition, but the common standard is that what you call "coptic" are not usually considered polygons. Dr. Belnap 04:46, 3 November 2011 (UTC) (Ph.D. and professor of mathematics) — Preceding unsigned comment added by Belnapj (talkcontribs)

## Clarify: what are side numbers?

The Naming polygons section states: "Exceptions exist for side numbers that are difficult to express in numerical form".

1. What are side numbers?
2. What numbers are difficult to express in numerical form?

This needs clarification/revision. hgilbert (talk) 14:53, 21 July 2010 (UTC)

Reworded. Hope that's better? — Cheers, Steelpillow (Talk) 18:54, 21 July 2010 (UTC)
Yes, much better. For a while there I thought side numbers were some exceptionally esoteric branch of topology! Sometimes the world is much simpler than we expect. hgilbert (talk) 01:14, 9 February 2011 (UTC)

## Polytoxon

Wikipedia's article Isohedral figure says:

*An isotopic 2-dimensional figure is isotoxal (edge-transitive). *An isotopic 3-dimensional figure is isohedral (face-transitive). *An isotopic 4-dimensional figure is isochoric (cell-transitive).

Given that what it says for 3 and 4 dimensions have the same roots as polyhedron and polychoron, why is this one different?? The word for a polygon is always polygon, never polytoxon. Georgia guy (talk) 13:46, 1 September 2010 (UTC)

The inconsistency originates in the term "polygon", which in Classical Greek means a "many-knee" or "many-corner" thing. This does not fit the convention of naming n-dimensional polytopes after their n−1 elements - polyhedra after their faces and polychora after their (polyhedral) cells. "Isogonal" means vertex-transitive, so when Branko Grünbaum wanted to write about edge-transitive figures he adopted the Greek word "toxon", a bow (as in bow-and-arrow), and from it coined the term "isotoxal". At least, that's what he told me, and Woodhouse's Classical Greek dictionary bears him out. "Isotoxal" seems to have caught on, while "toxon" has not. HTH. — Cheers, Steelpillow (Talk) 20:42, 1 September 2010 (UTC)

## Edit request from Pglelievre, 22 March 2011

{{edit semi-protected}}

The term "clockwise" is used but this is ambiguous unless the view is specified relative to some coordinate system axis. The following should be removed ...

"The vertices must be ordered clockwise or counterclockwise; if they are ordered clockwise, the area will be negative but correct in absolute value."

... and replaced with ...

"The vertices must be ordered clockwise or counterclockwise; if they are ordered clockwise, as viewed in the -z direction, the area will be negative but correct in absolute value."

Please confirm with others that the "-z direction" is correct there, as I believe it is, rather than "+z direction".

Pglelievre (talk) 18:17, 22 March 2011 (UTC)

But isn't it arbitrary which direction on the z-axis we call positive and which we call negative? Duoduoduo (talk) 18:47, 22 March 2011 (UTC)
Strictly, we should avoid any reference to the third dimension, and determine orientations with respect to the plane itself. That gets cumbersome and somewhat advanced, so it is conventional to assume that we are looking at the polygon as we would look at a clock on the wall in front of us - we may imagine the polygon drawn on the same wall. There is no ambiguity in this approach. It is so commonplace and, to most people, intuitively accepted, that it should not be explained at length the way I have done here. Invocation of a z-axis introduces ambiguity and so is both unwelcome and unnecessary. — Cheers, Steelpillow (Talk) 21:55, 22 March 2011 (UTC)

## Drawings of different types

User:Salix alba has added a diagram to the Classification section. I think it is unsuitable as it stands, for several reasons:

• The term "Complex polygon" usually refers to something quite different. The figures shown are "self-intersecting" or "coptic".
• "Equiangular" and "Equilateral" are spelled wrong.
• The equilateral figures need sides marking as such (two short parallel marks across each edge). Otherwise, optical illusion can make the sides appear different lengths. Ditto angle marks for the equiangular examples.
• There is no consistency of choice. Some examples such as star-shaped are omitted, while most of those shown could be classified under several types. The choice needs to be made carefully and presented in a more orderly way. I'd suggest that the types in the diagram correspond to the types in the bulleted lists. It may be that several smaller diagrams would provide a better approach.

— Cheers, Steelpillow (Talk) 11:27, 1 September 2011 (UTC)

Meanwhile, Salix alba, would you be willing to move your drawing across to the Wikimedia Commons? — Cheers, Steelpillow (Talk) 11:27, 1 September 2011 (UTC)

I'm away at the moment. Should be able to fix some of the small points, in the next couple of days. We do really need an illustration of the different types.--Salix (talk): 17:24, 6 September 2011 (UTC)
I've now updated the image, including some of your suggestions.--Salix (talk): 06:40, 14 September 2011 (UTC)
Thanks for that. What about the other suggestions? I am particularly concerned that the term "complex polygon" should not be used. — Cheers, Steelpillow (Talk) 16:07, 15 September 2011 (UTC)

## Lopshits

Was there really a guy called Lopshits or is this a very obscure vandalism attempt? — Preceding unsigned comment added by 24.2.4.28 (talk) 04:40, 3 October 2011 (UTC)

A quick web search for A.M. Lopshits should answer your question. — Cheers, Steelpillow (Talk) 19:01, 4 October 2011 (UTC)
... poor guy. 24.2.4.28 (talk) 23:40, 4 October 2011 (UTC)
Yeah, well, I know a bar on Procyon IV where saying "24.2.4.28" will get you torn limb from limb, if you have limbs. —Tamfang (talk) 21:34, 20 May 2012 (UTC)

## Polygons in nature

The description of the Lagrangian Points is flawed. It would be better to rely on a link to the corresponding Article, mentioning here only that an equilateral triangle is involved. 94.30.84.71 (talk) 19:52, 24 January 2012 (UTC)

## Constructing Higher names chart

The first chart under the "Constructing higher names" sub-section is messed up. Could someone fix it? — Preceding unsigned comment added by 50.46.122.32 (talk) 02:49, 29 January 2012 (UTC)

Thanks for the tipoff - some thoughtlessly-coded bot stripped out white space a little too enthusiastically. Let's see if my blocking tactic foils its kind in future - if not, we'll have to take the botmeisters to task. — Cheers, Steelpillow (Talk) 11:48, 29 January 2012 (UTC)

## Analogue of triangle inequality?

For triangles there are three inequalities that are necessary and sufficient for given a, b, c to be the sides of a triangle. Are there n such inequalities for an n-gon with given sides? Duoduoduo (talk) 16:16, 20 May 2012 (UTC)

There certainly are for a convex n-gon. Each inequality defines a half-plane. Given suitable inequalities, the polygon may be described as the intersection of these half-planes. This idea is very powerful, and generalises so well in higher dimensions that a convex polytope (the generalisation of a polygon to p dimensions) is sometimes defined as the intersection of n half-spaces where n is the number of (p-1)-dimensional) cells or facets. However for non-convex polytopes things get rather awkward and such treatment is less useful. — Cheers, Steelpillow (Talk) 16:54, 20 May 2012 (UTC)
If these inequalities can be expressed in a relatively succinct form for the case of an arbitrary number of sides in two dimensions, I think it would be a good idea to put it in this article. Duoduoduo (talk) 18:33, 20 May 2012 (UTC)
because this doesn't really apply to non-convex examples, I'd prefer to see it in the convex polygon article, which is currently a redirect to a rather messy and trivial page. Alternatively, might it be enough that the convex polytope article already discusses the general case? — Cheers, Steelpillow (Talk) 18:47, 20 May 2012 (UTC)
I assume Duoduoduo meant the rule that no side-length of a triangle can exceed the sum of the other two side-lengths; what this has to do with half-planes is beyond me. —Tamfang (talk) 19:06, 20 May 2012 (UTC)
No, that is a single inequality. If I am not mistaken, those that Duoduo refers to are inequalities which define a "half-plane" - the region on one side of a line - and sometimes the bounding line as well. For example the equation y = 2x + 3 defines a line, while the inequality y < 2x + 3 defines an open half-plane (excluding the line) and y ≤ 2x + 3 defines a closed half-plane (i.e. including the line). By defining the n half-planes which lie on the inside of the edge lines of an n-gon, we can define the polygon as the intersection of these half-planes (i.e. the area common to all of them). Some treatments include the bounding lines, others exclude them. For a more general mathematical description, see Convex polytope#Intersection of half-spaces. — Cheers, Steelpillow (Talk) 20:20, 20 May 2012 (UTC)
Read the OP again: "for given a, b, c to be the sides of a triangle." I took a, b, c to mean lengths, but they could be lines. For N given lines to be sides of a convex polygon, a necessary condition (not sufficient for N>3) is that no three concur in a single point. For N=3 this reduces to one inequality, that the pair x,y which solves linear equations a and b does not also solve c; for N>3 I can't say offhand how many such conditions suffice. (For a non-convex polygon they're not all necessary: a vertex may happen to be collinear with a non-incident edge.)
Can the triple condition a+b>c, b+c>a, c+a>b really be reduced to a single inequality? —Tamfang (talk) 20:52, 20 May 2012 (UTC)
Sorry I wasn't clear! Tamfang's interpretation is what I had in mind -- each side needs to be shorter than the sum of the others. Now that I think about it, I guess this set of n conditions -- each side needs to be shorter than the sum of the others -- is necessary and sufficient for any set of n sides a. b, c, ... to be the sides of an n-gon. Sorry for the confusion and the (I now assume) triviality of the question. Duoduoduo (talk) 22:57, 20 May 2012 (UTC)
You ask, "Can the triple condition a+b>c, b+c>a, c+a>b really be reduced to a single inequality?" Well, if a, b and c are lengths and we know that one edge is at least as long as either of the others, then yes. But I suppose that strictly, one should include those two other inequalities as well. — Cheers, Steelpillow (Talk) 20:51, 13 September 2012 (UTC)

## What? No traps?

I can hardly believe that this article has been rated at 4.3 on completeness. One of the polygons taught in grammar school (at least when I was that young) is the trapezoid. Why isn't it mentioned?Kdammers (talk) 12:47, 13 September 2012 (UTC)

There are so many kinds of polygon. There is not space to mention them all here. But if the trapezoid has an important place in the general picture, then by all means explain its importance and inclide a link to trapezoid. I have to say, my own understanding is that it can be a useful tool for teaching elementary geometry but has no real encyclopedic significance. — Cheers, Steelpillow (Talk) 20:09, 13 September 2012 (UTC)

I came here wondering whether "polygon" refers to just the closed chain of segments or whether it also includes the interior. Unfortunately, the first two sentences of this article say first one and then the other:

In geometry a polygon /ˈpɒlɪɡɒn/ is a flat shape consisting of straight lines that are joined to form a closed chain or circuit.
A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments (i.e., by a closed polygonal chain).

I notice that there are a couple of talk page threads above that argue about which definition is right, and perhaps the present wording was intended as a compromise. But to me it just comes across as self-contradictory. I suggest the following wording instead of the above two sentences:

In geometry a polygon /ˈpɒlɪɡɒn/ is a plane figure consisting of a closed finite sequence of straight line segments (i.e., a closed polygonal chain) and, in some contexts, all of the points that are enclosed by that chain of line segments.

Would there be any objection to my putting in this revised wording? Duoduoduo (talk) 22:23, 13 June 2013 (UTC)

Have to agree with your criticism. So I just did a makeover of the whole lead section. The original and traditional meaning refers to the whole thing. When Euclid constructed polygons he just constructed the chains, because of course the interior then becomes evident and need not be constructed as such. Then he famously used his polygons to construct polyhedra which he - and everyone since - happily refers to as "solids". Only in the 19th Century did Poinsot first ignore the fact that you cannot make a solid from daisy-chains, and modern mathematicians have since run away with edge chains, vertex sets, convex intersections of half-spaces, skew polygons and other goodly but inconsistent specializations. — Cheers, Steelpillow (Talk) 10:37, 14 June 2013 (UTC)

## Degeneracy and realization

A recent edit has changed the description of the monagon and digon in the Euclidean plane from "degenerate" to "not realizable". I have a couple of issues with this:

• It is technically incorrect. These polygons cannot be faithfully realized with straight sides but they can be realized after a fashion, with curved sides. [update 13:14, 12 November 2013 (UTC) - or with zero-length sides].
• That fashion is most usefully described as degenerate.
• I am not sure if the monagon is even a valid abstract polytope. If not, then it must have a geometric definition and the idea of realization is not appropriate anyway. It remains degenerate in the Euclidean plane, in any event.

— Cheers, Steelpillow (Talk) 09:32, 12 November 2013 (UTC)

What say the sources? - DVdm (talk) 09:54, 12 November 2013 (UTC)
A good general reference is Grünbaum's Are your polyhedra the same as my polyhedra? (2003) where he remarks that "every abstract polyhedron has realizations" and notes that in some the affine hull may be sub-dimensional. I note that by analogy this would apply to the digon, while the case when side length = 0, as would apply to the monogon, is described by him as "trivial". However his definition explicitly demands at least three sides for a polygon and further the monogon and digon are explicitly excluded. McMullen and Schulte are the main authorities on abstract theory. I have to hand only their 1997 paper Regular polytopes in ordinary space. In it they define "faithful" with respect to realization, in such a way that "a (faithfully realized) finite regular polygon P can only be either planar, and thus a pure polygon {p}, or skew...." The case where edges have length = 0 and vertices coincide is again dismissed as a "trivial" realization. Their definition of the minimal case is less arbitrary than Grünbaum's: "Finally, if F and G are a (j − 1)-face and a (j + 1) face with F < G, then there are exactly two [their italics] j-faces H such that F < H < G." I note that this rules out the monogon but allows the digon. Cromwell's Polyhedra discusses "degeneracy" in the context of the geometric form failing to match our expectations as raised by Euler's formula, it does not address abstraction, digons or monogons. Coxeter's Regular polytopes explicitly allows the digon with curved sides but ignores the monogon, and uses "degenerate" in the context of plane tilings as polyhedra. "Degenerate" is not a fully-defined technical term but in my opinion it is a useful descriptive word for the article writer - and the book author - where a figure falls short of the defined model in some way. For the digon one could alternatively write "unfaithful" but that would not appear to be applicable to the monogon which is not merely geometrically but also abstractly degenerate, in a sense close to that used in Cromwell's discussion of non-Eulerian forms. — Cheers, Steelpillow (Talk) 13:14, 12 November 2013 (UTC)
Looks good. Go ahead and make the changes you fancy, but make sure you add the source. - DVdm (talk) 13:55, 12 November 2013 (UTC)

FWIW, I made those changes because suddenly the article was talking about "realizations" with curves, in an article that is about polygons, which are supposed to have straight sides. Yes, people have considered curvilinear generalizations of polygons, but that concept needs to be introduced before talking about realizations of monogons, and it needs to be made clear that any realization is necessarily curvilinear. Currently, our article has nothing about allowing the sides to be curved, and talking about monogons is not the way to introduce that subject. —David Eppstein (talk) 17:16, 14 November 2013 (UTC)

## The top two pictures

I am not sure if the use of binary filling for the coptic hexagon at the top of the article is representative: I think that, just as we mix boundary and filling types, we should also show a coptic polygon filled in using the density method (e.g. filling the centre of a regular pentagram).

Also, I'm not too keen on the use of "regular complex" in the second picture to describe the category including the regular pentagram: the word "complex" makes me think of complex polytopes instead. Perhaps "self-intersecting" would be a better alternative. ("Coptic" is more concise, but outside Grünbaum it doesn't appear to have generated a large following.) Double sharp (talk) 14:16, 17 April 2014 (UTC)