Talk:Degenerate polygon/Archive 1

Sourcing the definition

Coxeter's Regular polytopes does not appear to mention "Improper regular polygons". It discusses the digon as a spherical polygon not an improper one, and when it incorporated digons into a tessellation (of the Euclidean plane), it describes the tessellations as improper, not the digons. As such, I cannot see this work as a suitable reference for the article lead. — Cheers, Steelpillow (Talk) 12:44, 31 January 2015 (UTC)

Coxeter was talking about "improper" tessellations of the sphere, not the Euclidean plane. He was expressing a limit on spheres, while the same limit exists on circles. The only other application I see is Regular_map_(graph_theory), but it also is focused on spherical tessellations rather than just circular ones, even if they are a subcase. Tom Ruen (talk) 12:56, 31 January 2015 (UTC)

It's irrelevant really what surface he was discussing. The point is, what we think is the truth plays second fiddle to what the sources say. And whatever we might think Coxeter meant, what he actually says does not support the lead. — Cheers, Steelpillow (Talk) 13:46, 31 January 2015 (UTC)

I added the term, based on a slight extrapolation of Coxeter (but unfortunately still OR), so that we didn't have to put the monogon under an article called "digon" when obviously a monogon cannot be a digon. Since there does not seem to be a simple term that covers both monogons and digons perhaps they ought to be split back up into separate articles. (After all, while {2} is reasonably well-behaved as a spherical polygon, {1} doesn't seem to be as nice.) Double sharp (talk) 13:50, 31 January 2015 (UTC)

OK, unfortunately I do not think that was a good choice, at least for the digon. It is not improper by most modern definitions, it just happens to be degenerate under certain admittedly commonplace conditions. The monogon I am less sure about: in most formulations of polytope theory it is certainly improper, but ISTR you said elsewhere that the term is used in graph theory. Is it treated there as proper or improper? — Cheers, Steelpillow (Talk) 11:05, 4 February 2015 (UTC)

I don't think I mentioned that the name "monogon" was used in graph theory: I did mention at Talk:Quasiregular polyhedron that the regular maps {2,2n}/2 and {2n,2}/2 (which are essentially {1,2n} and {2n,1}) are certainly referred to often, but when they get called anything it appears to be "hemi-dihedron" and "hemi-hosohedron". Analogously one could call {1} a "hemi-digon", which would not be that inaccurate a description (but of course not on WP because I haven't seen that anywhere). These would all be tessellations of the elliptic plane. (Coxeter mentions {2,2n}/2 and {2n,2}/2, but without naming them, although he does name {1} the "monogon": see a few sections above.)

The article cites Coxeter as saying "improper tessellations for p = 2": if so then "improper" would refer to only {2} and not {1} (which, as you say, is slightly weird as {2} is certainly not improper, just usually degenerate.)

(Currently a bit skeptical about including the monogonal polytopes without comment in a list, BTW: at most Coxeter verifies only {1,2} and {2,1} as legitimate spherical polytopes, and even {2,1} is weird – its single digonal face has coinciding edges, but its area is the whole sphere! {1,1} doesn't appear to be in any reliable source, and I'm not sure I really understand it: if it has a monogonal face, where is the edge of that face?) Double sharp (talk) 13:50, 4 February 2015 (UTC)

When Coxeter writes of improper tessellations, he is referring to their degeneracy as tessellations, and not specifically to the digon as a general geometric object: that is a very different thing. {2,1} is just an arc bounded by polar points, it does not subdivide the sphere. It is basically a line segment, a 1-polytope, only it is said to "enclose" the sphere. Clearly, as a boundary it is not polygonal but is of one dimension less. Dualising this criticism must likewise cast doubt on {1,2} (a great circle with a point on it), though the ideas involved are more subtle: in essence it has no 1-polytopes as elements, because a 1-polytope is bounded by two end points (a 0-sphere) and this has only one. I am dubious whether {1,1} has been correctly identified in the discussions above here. It is not simply a dot on the sphere, as that would be {0,1} (0-sided-gon, 1 round each vertex). {1,1} describes a 1-sided-gon, 1 at each vertex. This does not exist on the sphere but does exist on the projective plane as a line through a point. It is effectively a hemi {2,2}. Neither it nor {2,1} and {1,2} are polygons in most usages, even those that allow {2,2}, and I would want to see more than the odd cheeky remark by Coxeter to establish their verifiability (supporting policy/guideline links for this stance can be provided for the asking, but I do hope you folks can research Wikipedia's policies and guidelines for yourselves). — Cheers, Steelpillow (Talk) 16:43, 4 February 2015 (UTC)

(Is {0,1} even a legitimate polytope in the first place? How does its dual {1,0} even make sense? In fact, what in the world is {0}?)

It might be significant that Coxeter calls the dihedra, hosohedra, and Platonics (presumably not including {1,2} and {2,1}) as regular maps and regular tessellations, but only calls {1,2} and {2,1} regular maps (which they still are). Given the weirdness of the monogonal cases you bring up, I am getting more and more convinced that they should not be covered as standard regular polytopes and incorporated into such lists without comment. Monogon could contain a discussion of these problems, provided that they can be reliably sourced – which I am not sure about. If not, all we could mention is that while the monogonal cases are certainly regular maps, their status as regular polyhedra is shaky (for various reasons which we unfortunately wouldn't be able to give). Double sharp (talk) 13:40, 5 February 2015 (UTC)

P.S. Chinese Wikipedia has a markedly silly article subsection on the 0-sided polygon, that consists only of repetitious statements of how {0} doesn't exist and can't be constructed and how all the usual formulas are meaningless on it. Is there anything in reliable sources at all about 0-gons? Double sharp (talk) 09:02, 6 February 2015 (UTC)

Most of these figures are not usually accepted as polytopes. Schläfli symbols are typically interpreted as construction recipes, e.g. {3,4} means take some 3-gons and place 4 around each vertex. So {0,1} is a 0-sided "polygon". Pushing the idea, one might suggest that a polygonal boundary divides the sphere into two polygons - a small one on the inside and a large one on the outside, which is the rest of the sphere. If you morph the boundary out to a great circle and then beyond, the outside region eventually becomes the inside and vice versa. So a 0-gon on a sphere must have a boundary but it can have no sides. A single point kind of fulfils this condition. But a point is a 0-tope not a 2-tope so to distinguish our {0,1} from it we include an interior surface, which can only be the rest of the sphere. The dual {1,0} must be a set of monogons, with none at each vertex. The only solution appears to be an empty set of monogons, i.e. a bald sphere with no map or graph on it at all. Despite the apparent rather neat duality between everything and nothing, all this of course defies standard definitions of a polytope, in varying degrees as we push our OR onward. I would be astonished if any RS were to waste space on the idea, save only perhaps to cheekily poke the student in an exercise at the back.

Regular maps use regularity in a purely combinatorial sense, much like that used for abstract polytopes. Here, the simplest admissible polygon is the digon. This highlights a useful distinction to bear in mind. In modern polytope theory we say that a geometric figure is a "realization" of the abstract one. A digon is permissible but its realizations are often degenerate, while a monogon is not even a valid abstract construct. Wikipedia does not document belly-button fluff, not even Coxeter's, so I would expect to see use of the term "monogon" verifiable as mainstream before I would accept its notability as anything other than a brief comment on its impropriety. For example does the term appear in graph theory as anything other than throwaway jargon in a handful of obscure papers? Otherwise, one might find something in an intuitive introduction to topological analysis of piecewise decompositions such as CW-complexes, which is close to graph theory in many ways. — Cheers, Steelpillow (Talk) 11:08, 6 February 2015 (UTC)

Is this relevant? Double sharp (talk) 12:04, 6 February 2015 (UTC)

Yes indeed, though not in decompositions as such. The book is published by Dover, so nobody can call it fringe. He uses the word monogon in a theorem that he attributes to Cohn-Vossen, a central figure in the development of topology. So clearly, there is value to be had in seeking out the proper usage of the term "monogon" - in this case the study of closed geodesics on a manifold. But it should not be linked to the idea of polytopes, especially regular polytopes, unless we find comparably reliable sources.— Cheers, Steelpillow (Talk) 13:11, 6 February 2015 (UTC)

I don't understand DoublSharp's statement: {2,2n}/2 and {2n,2}/2 (which are essentially {1,2n} and {2n,1}). The purpose of Coxeter /2 is to imply something on the real projective plane, and if you remove that, you're going back into the spherical tilings. Tom Ruen (talk) 18:23, 5 February 2015 (UTC)

Things like {1,2n} and {2n,1} (n > 1) are probably not realizable on the sphere, but {1,2n} has a simple construction on the projective plane, which is just n lines through a point ({2,2n}/2). These lines are monogons and 2n meet at the same vertex. {2n,1} is dually a 2n-gon around a line subdivided into n equal segments ({2n,2}/2): there is one 2n-gonal face and there is only one such face around each vertex.

However, I'm not sure about Steelpillow's statement that {1,1} = {2,2}/2, being a line through a point on the projective plane. Wouldn't that be {1,2}? In which case {1,1} is not realizable either as a spherical tiling or a tiling of the projective plane. In fact, I think the same would have to be true for {1,n} and {n,1} for odd n. Can they even exist? More and more I am thinking that anything with 1 in its Schläfli symbol is suspect as a standard polytope – {1} isn't even a proper abstract polytope. Double sharp (talk) 09:02, 6 February 2015 (UTC)

Where does Coxeter describe and define his {p,q}/2 notation? To my knowledge it is not widely used and would represent something of a historical backwater. Perhaps my writing of {{2,2}/2 as {1,1} is a misinterpretation. I don't think it matters as this whole topic is degenerate in every sense of the word. The simplest polytopes are: the null polytope or −1-tope (abstractly the empty set), the 0-tope or point, the 1-tope or bounded line segment (including its two end points) and the digon. Of these, only the digon {2} has a widely-used Schläfli symbol, and anything less would need exceptional verification. For example I have seen { } suggested for the line segment but this is also sometimes used for the empty set, so again the odd mention by some obscure small fry would establish a minor PoV but Wikipedia does not usually document such minor PoVs. — Cheers, Steelpillow (Talk) 11:08, 6 February 2015 (UTC)

According to list of regular polytopes and compounds, the Schläfli symbol {} for the line segment was used by Coxeter himself in Regular Polytopes (3rd ed., pp.294–6), as well as McMullen and Schulte's Abstract Regular Polytopes (p.30). {1} for the monogon was also used by Coxeter in Introduction to geometry, as were {1,2} and {2,1} for the monogonal hosohedron and dihedron (in an exercise, however, so YMMV on that): so the symbols should be OK as simple references ({} is almost certainly OK to mention; {1} is slightly less sure). The "/2" I'm not sure about: McMullen and Schulte use a notation based on the Petrie polygons (pp.164–5), so the hemi-dodecahedron is {5,3)₅, the hemi-24-cell is {3,4,3}₆, and the hemi-600-cell is {3,3,5}₁₅. I guess Coxeter's usage would be {5,3}/2 (this is attested in Coxeter), {3,4,3}/2, and {3,3,5}/2 (I'm not sure if these two are, but it would be a very logical extension). Nonetheless I do not know where Coxeter defines his "/2" notation, although it fits nicely with the use of the prefix "hemi-". Double sharp (talk) 11:55, 6 February 2015 (UTC)

That list is pretty awful in lots of ways, it needs a good spring-cleaning. Coxeter mentions the line segment (on a different page) but I can find no symbol for it, certainly not on the pages cited. I do not have a copy of McMullen and Schulte, but if that cite is genuine, how do they distinguish it from the point and the empty set? What was Coxeter's context in his Introduction to Geometry? The fact that /2 "fits nicely" in a cursory dabble does not mean it is reliably so. Still, we have some progress. — Cheers, Steelpillow (Talk) 13:11, 6 February 2015 (UTC)

Hmm, that citation seems to be in error: it is on p.129, where he writes "α₁ = { }...δ₂ = {∞}" (α_n being of course the n-simplex; β_n is the n-orthoplex, γ_n the n-cube, and δ_n the n-hypercubic honeycomb). Double sharp (talk) 14:05, 4 March 2015 (UTC)

Here's a rather nice reputable source for "hemi-hosohedron" and "hemi-dihedron" as regular maps, courtesy of Maproom (talk · contribs): Carlo Séquin; "Symmetrical immersions of low-genus non-orientable regular maps", Symmetry: Culture and Science, 2013. Note his avoidance of Schläfli symbols. — Cheers, Steelpillow (Talk) 13:56, 6 February 2015 (UTC)

"can not"

In the first sentence of the article "a polygon is degenerate if it either does not or can not conform to the standard definitions", what does "can not" mean? I can't tell if it's a typo for "cannot", or a confusing way of writing "need not". Either way, it could be clearer. Maproom (talk) 15:27, 6 September 2015 (UTC)

"Can not" is the original written form of "cannot" or "can't" and is still common in at least some regional variants. It is also common in verbal form where the speaker wishes to emphasise the not. Check the regional variation used for the rest of the article and then check whether either "can not" or "cannot" is preferred there. "Can't" should be avoided. — Cheers, Steelpillow (Talk) 16:19, 6 September 2015 (UTC)

To follow that up a little more. There doesn't seem to be much regional variation these days. Other things being equal, it is indeed more common to use "cannot". But "can not" has its place, see e.g. here. The phrase "does not and can not" is comparing "does" with "can" and therefore splits off the "can" for emphasis, much as those who seek to highlight the "not" also split the written form. The split in order to emphasise the "can" is in itself perfectly correct and cannot (sic) be argued against on grounds of unfamiliarity by one editor. The argument can only be, is it justified here? — Cheers, Steelpillow (Talk) 16:42, 6 September 2015 (UTC)

If it was just a matter of regional variation, I wouldn't have bothered commenting. The sentence in the article is ambiguous, and need not be. I guess from the responses that "can not conform" is intended to mean "cannot conform" rather than "may fail to conform", but I'm still not sure. Maproom (talk) 18:05, 6 September 2015 (UTC)

Ah, I get the ambiguity you mean. In this context, "can not..." meaning "need not..." or "may not..." would be an ugly use of slang - so ugly it took me a while to catch on to. The interpretation as a synonym for "cannot" is the more correct one. Condensing it to "cannot" would lose an intended nuance but would also lose an unintended ambiguity. If you change it, I will not argue the point further. — Cheers, Steelpillow (Talk) 21:21, 6 September 2015 (UTC)

Thank you. As a player of "Euro" style boardgames, I am painfully aware of this particular ambiguity. All too often, I have seen German "muss nicht" translated as "must not", when it should be "need not". Maproom (talk) 23:05, 6 September 2015 (UTC)

I was surprised at the change because, in my (British) English, "can not" is clearer. I accept that there might be varieties of English which use words in a non-standard way. Perhaps we should make it clearer by stating why it "cannot" or "can not" ... Dbfirs 07:23, 7 September 2015 (UTC)

History split

The history of Degenerate polygon before Double sharp's move back in January should be moved back to Digon. To do this, follow these steps:

1. Delete Degenerate polygon
2. Restore the revisions of Degenerate polygon before the January move
3. Delete Digon
4. Move Degenerate polygon to Digon without leaving a redirect
5. Restore the deleted revisions of Digon, as well as the revisions of Degenerate polygon after the January move.

GeoffreyT2000 (talk) 02:24, 7 September 2015 (UTC)

•  Done Anthony Appleyard (talk) 06:22, 25 February 2016 (UTC)

Confusing

It seems confusing to have the article called degenerate polygon because flat nontessellation expressions are degenerate, and then use this as a reason to remove the nondegenerate digon cases. The entire monogon section is about nondegenerate usages in tilings on the circle and sphere. Are you planning on moving the monogon section to its own article?! You previously reverted this [1]. Tom Ruen (talk) 13:10, 11 December 2015 (UTC)

Resolved

After discussion by multiple editors, consensus was reached to reinstate the monogon and digon articles in their original form, and to make degenerate polygon into a redirect to degeneracy (mathematics). -- The Anome (talk) 12:09, 27 December 2015 (UTC)