Talk:Apeirogon

message: Can someone add some explicit information about the symmetry groups of the regular apeirogon and the irregular apeirogons.
It seems to me that the symmetry groups for all of them would be isomorphic to the dihedral group of infinite order, but for the irregular apeirogons, the "rotations" and "reflections" would be twice as far apart when compared to the regular apeirogon. 
Hopefully someone who is more familiar with this than me can add this information to the article. --AndreRD (talk) 14:19, 28 June 2019 (UTC)
thread_title:  Symmetry groups 

message: Are these defined as synonymous to apeirogons, or must they be self-similar like the examples given there are? Double sharp (talk) 05:58, 27 September 2019 (UTC)
thread_title:  [[Teragon]]s 

message: The article was just reorganized and rewritten due to a large amount of unsourced (or unclearly sourced) content which may have been original research. The discussion regarding this is here in WikiProject Mathematics. — MarkH21talk 10:14, 19 December 2019 (UTC)
thread_title:  Large-scale changes and removal of OR/unsourced content 

message: I don't think the images in the definition section are useful on this page as they depict pseudogons rather than apeirogons, which are distinctly different. It is also not helping that, in the first of the two images clearly says that the pseudogon is bounded by a hypercycle, which is very different to how the apeirogon is bounded by a horocycle. Alexhughes123456787 (talk) 16:39, 12 May 2020 (UTC)
replies: text: This tripped me up as well. I decided to be bold and replaced it to (hopefully) reduce confusion. Maybe Pseudogon can be recreated as a separate article at some point. -pinkgothic (talk) 21:24, 6 June 2020 (UTC)
replies: text: Re the second paragraph of the introduction, : how are these supposed to be equivalent definitions? —David Eppstein (talk) 21:57, 6 June 2020 (UTC)
replies: text: I did some minor tweaking, but they’re effectively the same thing. You can view a Euclidean horocycle as the Euclidean line since it’s the limit of circles with a fixed tangent as the radius tend towards infinity. I’m not sure what “absolute circle of the hyperbolic plane” means in the Johnson book; it appears to mean the boundary of the Poincaré disk model, in which case it’s the same. — MarkH21talk 22:18, 6 June 2020 (UTC)
replies: text: Geometrically they're totally different: one has its points and edges on a Euclidean line, one has points on a line (horocycle) and edges outside it, one has no points (they are not points in the hyperbolic plane) and only edges. Combinatorially they're different: the first two have a sequence of vertices and edges order-equivalent to the integers, the third may be some much nastier cyclic order-type. Conflating them and saying they're identical is just confusing. We need to decide what this article is about and make it be about that definition, not pretend that different things are really the same. —David Eppstein (talk) 22:31, 6 June 2020 (UTC)
replies: text: This sentence from the article is directly from the cited reference:  They're the same in the sense that a polygon can be defined as a finite partition of a circle, a collection of finitely many line segments (and the same number of vertices) embedded in the Euclidean plane, or even a collection of finitely many line segments embedded in another space. They're different objects (a collection of points on a circle, a collection of line segments in a space X, a 2-simplex = the face, a simplicial 2-complex = the face + edges + vertices, etc.), but they're all polygons.Also all three of the apeirogon definitions require that there are countably infinitely many sides; there's no issue there.One's definition of "the same" can be different from the next reader based on what characteristics we focus on, so I already removed the . But they're certainly viable alternate definitions for what an "apeirogon" is. — MarkH21talk 23:00, 6 June 2020 (UTC)
replies: text: Having a reference doesn't automatically imply that the reference is actually mathematically rigorous, in this case it appears not to be, and trying to rationalize it by making up explanations for how it might mean something beyond what it actually says is original research. Another question: why does the Euclidean line version require equal-length segments when the other two make no such requirement? And is an apeirogon always (weakly) convex, as these definitions all imply, or does an infinite zigzag in the Euclidean plane count as a non-convex apeirogon? Also, "they're all polygons" is dubious, unless you have some definition of polygon in mind different than the ones I've commonly seen (which generally require that the vertices and edges form a cycle of finitely many objects). Is a circle a polygon? What definition are you using that would allow an apeirogon but exclude it? As for "they're certainly all viable explanations": certainly the first two are definitions of something. I'm not convinced that the third one is even that. I mean, I can imagine that "infinite polygon inscribed in the absolute circle" means that we take some subset of ideal points and of hyperbolic lines between the chosen points. But which subsets of the ideal points are allowed as vertices? Must it be a closed set? How many limit points can it have, as a topological subset of the circle? (It must have at least one, but can it have more? Infinitely many?) Must it be countable? Do we take all hyperbolic lines that bound maximal halfspaces empty of chosen ideal points, or what? —David Eppstein (talk) 00:41, 7 June 2020 (UTC)
replies: text: I can take a look, but I don’t think there’s much besides Coxeter’s old works.The equal-length requirement isn’t to ensure that the polygon or apeirogon is regular; it does seem that this wording was implied but omitted for the other two definitions stated by Johnson.A finite partition of a circle uniquely determines a finite set of points on the circle, which uniquely determines the inscribed polygon with vertices at the given points. A polygon also uniquely determines a partition of the circumscribed circle. So giving a partition on a circle is equivalent to giving a Euclidean polygon. A circle is not a non-trivial partition of itself, and is not polygon nor apeirogon. — MarkH21talk 00:45, 7 June 2020 (UTC)
replies: text: When you say that "equal length" was implied for the version in which the vertices are ideal points and the sides are entire hyperbolic lines, what do you mean? It cannot be equal length in the hyperbolic plane, because all lines have infinite length, and it cannot be equal angles of the circle, for then there would be only finitely many points. Even if you are somehow inferring that the definition involves a covering of the circle by interior-disjoint circular arcs (what I think you mean by "partition", since your usage does not match the usual definitions of the word "partition") there is still the question of how many limit points there may be: the horocycle has only one, but arbitrary coverings of the circle by interior-disjoint circular arcs may have many or even infinitely many. —David Eppstein (talk) 00:53, 7 June 2020 (UTC)
replies: text: To be honest, I hadn’t really seen the second or third definitions before seeing it in this article, and I was confused by the non-standard term . I just guessed that equal length was implied for the hyperbolic definition with line segments. Maybe regularity is being sloppily dropped in the sentence by Johnson for the third definition. There’s no question that the quoted sentence from Johnson isn’t as precise as one would like.I use the word partition in the same way that one partitions the unit interval or partitions a set, which coincides with covering with disjoint arcs or picking finitely many (and at least two) points on the circle in this case. Which  deviate from this? — MarkH21talk 01:10, 7 June 2020 (UTC)

text: Since the Coxeter definition of an apeirogon is generalizing the polygon via partitions of a Euclidean line, there is no formal convexity or non-convexity. The whole “infinite polygon” is to provide an intuitive description, but non-recreational mathematical works just use the partitions language. — MarkH21talk 00:57, 7 June 2020 (UTC)
replies: text: A partition of an interval describes a finite subset of interior-disjoint intervals whose union is the whole interval. The intervals overlap at their endpoints. A partition of a set describes a collection of disjoint subsets whose union is the whole set; there is no geometric structure implied to these sets, and they cannot overlap. Neither of them matches what is being used here, exactly. As for convexity: we have three definitions, one in a line but two describing (somehow) convex subsets of the hyperbolic plane. Are non-convex subsets allowed? Why not, when quadrilaterals, pentagons, hexagons, etc are all generally allowed to be non-convex? And if we are supposed to think of the partition of the Euclidean line as somehow being the "real" definition, why is it reasonable to even call this a polygon, when we would certainly not call a partition of the interval into five subintervals a pentagon? —David Eppstein (talk) 01:16, 7 June 2020 (UTC)
replies: text: A partition of an interval is just the sequence of numbers themselves. A partition of the unit interval uniquely determines (and is uniquely determined by) a (yes, finite) partition of the set [0,1] via {[0, x1), ...}. They’re not inherently different besides partitions of an interval necessarily being finite. Similarly, a partition of a circle is a collection of points on the circle which is uniquely determines (and uniquely determines by) a set of disjoint arcs whose union is the circle.Nobody said that a partition of the Euclidean line is a polygon...? The Coxeter definition of an apeirogon via partitions of the Euclidean line excludes (finite) polygons. Rather confusingly though, some authors use “polygon” to include both finite polygons and apeirogons.The definitions probably only allow for convex apeirogons, since these definitions are supposed to be related by identifying the Euclidean line with the horocycle or “absolute circle”. Similarly, the partition of a circle only uniquely determines (and is uniquely determined by) a convex polygon. — MarkH21talk 01:29, 7 June 2020 (UTC)

text: Where does it say that an apeirogon is a polygon? A polygon is usually defined to be finite, whereas apeirogons are countable infinite. There are authors who use “polygons” to collectively refer to both finite polygons and countably infinite apeirogons (e.g. the existing reference from Branko Grünbaum). Perhaps a note is needed in the article, as the section title  may be confusing. — MarkH21talk 01:43, 7 June 2020 (UTC)
replies: text: In the first sentence it says that an apeirogon is a generalized polygon. You could define a "generalized polygon" to be either a finite polygon or one of the zebras in the San Diego Zoo, but people would still think you meant that a zebra was some kind of polygon and wonder how you meant it. It also says they are 2-polytopes without even the word "generalized" as a fig leaf. Later in the same first paragraph we also say that apeirogons "can be considered as limits of an n-sided polygon as n approaches infinity". Is there any mathematical sense in which the partition of the Euclidean line into unit intervals can be obtained as a limit of polygons? Of regular polygons? This whole article is vague mathematics-sounding wording with nothing behind the curtain when you try to figure out what it actually means. —David Eppstein (talk) 01:48, 7 June 2020 (UTC)
replies: text: Yeah I think that’s one of the confusing uses of “generalization” that doesn’t technically include the original concept on its own. The polytope wording is another example of mixed terminology; infinite polytopes are “generalized polytope” that doesn’t include classical polytopes, which are assumed to be finite like classical polygons. Here, the article says . The article certainly does need more cleaning up though.Re limits, it’s only slightly more indirect than that. There are many ways to obtain the Euclidean line as the limit of circles. For instance, curves of constant positive curvature are circles and curves of constant zero curvature is a line. As the radius of a circle tend to infinity, the curvature tends to 0. So one could say that the Euclidean line is a limit of circles, and by extension via partitions, an apeirogon is a limit of polygons.When it comes down to rigorous mathematical works that discuss apeirogons, it’s generally restricted to the Coxeter definition without these semi-rigorous intuitive descriptions. We could just focus the article on the formal definitions and properties. — MarkH21talk 01:58, 7 June 2020 (UTC)

text: Ah yes, McMullen's definition is a more workable modern definition that we can use as the basis for the article, and it's the same as the one in the McMullen-Schulte book (ref 14 in the paper you linked) that I added to the article several months ago whose existence I have completely forgotten about. Sorry, I didn't see some of the questions about those examples earlier in the edit conflicts; some of those questions can actually be answered using the old-fashioned Coxeter definition with the same justification (e.g. that they don't share the same cardinality or symmetries of the countable partitioning of the Euclidean line) I'll work on reworking these into the article.By the way, I don't see McMullen mentioning infinitely many points on a Euclidean circle as being an example of a realization of an apeirogon. The example that he gives on pages 231-232 has automorphisms that are products of nonzero translations and rotations (and possibly a reflection). — MarkH21talk 05:32, 7 June 2020 (UTC)
replies: text: It was the line near the top of p.232 "each irrational angle \vartheta_j corresponds an apeirogon, in the latter case bounded and non-discrete" that I was referring to. But perhaps I'm misreading the intention there. —David Eppstein (talk) 06:54, 7 June 2020 (UTC)
thread_title:  Definition images