Talk:Apeirogon

Symmetry groups

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)

Teragons

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)

Large-scale changes and removal of OR/unsourced content

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. — MarkH₂₁^talk 10:14, 19 December 2019 (UTC)

Definition images

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)

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)

Re the second paragraph of the introduction, A regular apeirogon can be defined equivalently as a partition of the Euclidean line E1 into infinitely many equal-length segments, an infinite polygon inscribed in a horocycle, or an infinite polygon inscribed in the absolute circle of the hyperbolic plane.[2]: how are these supposed to be equivalent definitions? —David Eppstein (talk) 21:57, 6 June 2020 (UTC)

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. — MarkH₂₁^talk 22:18, 6 June 2020 (UTC)

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)

This sentence from the article is directly from the cited reference:

A regular apeirogon (∞) is either a partition of the Euclidean line E¹ into infinitely many equal-length segments or an infinite polygon inscribed in a horocycle or in the absolute circle of the hyperbolic plane.
— N.W. Johnson, Geometries and Transformations

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 equivalently. But they're certainly viable alternate definitions for what an "apeirogon" is. — MarkH₂₁^talk 23:00, 6 June 2020 (UTC)

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)

(edit conflict)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. — MarkH₂₁^talk 00:45, 7 June 2020 (UTC)

This convex subset of the hyperbolic plane, derived from a partition of the ideal circle into arcs, has 10 one-sided limit points, unlike the horocycle example which has a single two-sided limit point. Is this an apeirogon? I can't tell from our article and I think that my inability to tell is a serious problem with the article. —David Eppstein (talk) 01:36, 7 June 2020 (UTC)

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)

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 absolute circle. 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 usual definitions of the word "partition" deviate from this? — MarkH₂₁^talk 01:10, 7 June 2020 (UTC)

(edit conflict)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. — MarkH₂₁^talk 00:57, 7 June 2020 (UTC)

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)

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. — MarkH₂₁^talk 01:29, 7 June 2020 (UTC)

Re Nobody said that a partition of the Euclidean line is a polygon...?: really? Because our article says that the partition of the Euclidean line into regular intervals is an apeirogon and that an apeirogon is a kind of polygon. Also, another example: is the Euclidean convex hull of the points $\{(x,x^{2})|x\in \mathbb {Z} \}$ an apeirogon? —David Eppstein (talk) 01:36, 7 June 2020 (UTC)

(edit conflict)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 Classification of polygons may be confusing. — MarkH₂₁^talk 01:43, 7 June 2020 (UTC)

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)

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 infinite 2-polytope. 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. — MarkH₂₁^talk 01:58, 7 June 2020 (UTC)

There's an actual rigorous definition in McMullen, Peter (1994), "Realizations of regular apeirotopes", Aequationes Mathematicae, 47 (2–3): 223–239, doi:10.1007/BF01832961, MR 1268033! McMullen defines an apeirogon as a two-dimensional abstract polytope (a partially ordered system of the empty set, vertices, edges, and whole apeirogon) with the properties you would expect: each vertex is below exactly two edges in the partial order, each edge is above exactly two vertices, the empty set is below all the vertices, the whole apeirogon is above all the edges, and the collection of edges and vertices have the combinatorial structure of a single connected doubly-infinite path. He defines a realization of an apeirogon to be a mapping of its vertices into a finite-dimensional geometric space (he considers only Euclidean spaces but one could also immediately apply the same definition to the hyperbolic plane) such that every symmetry of the abstract polytope (infinite dihedral symmetry) corresponds to an isometry of its geometric image.

With this definition, unlike with our article in its present state, I can immediately answer all my questions above about what is or isn't an apeirogon.

• The subdivision of the Euclidean line into unit segments is not an apeirogon. It is, however, a realization of an apeirogon.
• The convex hull of equally-spaced points on a horocycle is also a realization of an apeirogon, if one allows hyperbolic as well as Euclidean realizations.
• The Euclidean infinite zigzag (equally spaced points with equal angles that turn in alternating directions) is also a realization of an apeirogon, because every symmetry of the abstract apeirogon is an isometry of this set.
• Equally spaced points on a three-dimensional helix are yet another realization of the apeirogon.
• The hyperbolic shape with 10 one-sided limit points is not a realization of an apeirogon, because it has the wrong group of symmetries.
• Unevenly spaced points on a line or a horocycle are not realizations of apeirogons, because they don't have enough symmetry.
• The convex hull of integer points on a parabola is not a realization of an apeirogon, again because it has too little symmetry.
• The "infinite polygon inscribed in the absolute circle" is not a realization of an apeirogon, because it is not a mapping from the vertices of the abstract apeirogon to points in a geometric space.
• The whole set of points on a Euclidean circle is not a realization because it has too many points to be the image of a mapping from the vertices of the abstract apeirogon.
• Countably infinitely many points on a Euclidean circle, with the spacing between consecutive points being an equal irrational multiple of 2π, are a realization of an apeirogon (one explicitly mentioned by McMullen). It does not matter that these points do not form a partition of the circle into arcs, nor that they form a dense subset of the circle, nor that their cyclic ordering is very different from the consecutive ordering along the apeirogon. What matters is that the symmetries of the abstract polytope are reflected in symmetries of this set of points, and they are.
• The obvious mapping from the apeirogon onto any regular polygon is also a realization. The mapping from the apeirogon onto a single point is also a realization. McMullen doesn't require realizations to be injective, only surjective. (He uses the group-theoretic terminology "faithful" for the injective ones.)

There is nothing in these definitions about convexity or about partitions of curves. I would be much happier with our article if it followed McMullen's viewpoint. We can still lead with the intuitive version but it should be backed by mathematical rigor. —David Eppstein (talk) 04:15, 7 June 2020 (UTC)

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). — MarkH₂₁^talk 05:32, 7 June 2020 (UTC)

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)

Original research

The section Classification of Euclidean apeirogons appears to be original research or at the very least a poor synthesis. It has extremely poor sourcing and uses language that sounds like a strange synthesis of different sources. McMullen & Schulte offer a classification of symmetric realizations of polygons (not just finite ones), but uses very different language. Discussions of antiprism polygons is given by Coxeter, but not in relation to this, I have to infer what is meant here is non-faithful symmetric realizations on Coxeter's antiprism polygons. Helices and zigzags are discussed by Grünbaum in the linked source, but none of these other classes. Lastly this classification has two mistakes:

• It seems "infinite skew polygons" is intended to mean zigzag polygons, however this category are called "infinite skew polygons" which should arguably include helices and others.
• This classification completely neglects faithful dense polygons, e.g. {π} (2D) or {π}#{} (3D). Polygons like {π}#{∞} are ostensibly covered by the helix category, but it's not clear. Based on this I'd think we are talking about discrete realizations, but non-faithful realizations are discussed which are not discrete. Something's wrong here.

Maybe these errors are because this section is using different definitions, but without sources it's impossible to know. And for me to "fix" these errors would itself be WP:OR. This should be sourced so that the errors can be fixed. Otherwise I'm going to delete it and replace it will a McMullen & Schulte's classification of polygons. AquitaneHungerForce (talk) 23:23, 27 February 2024 (UTC)

I think this was based on the 7 groups of Grünbaum's article (pp. 2-3). David Eppstein may be of help since he wrote some of it here.

I don’t know what you mean by Helices and zigzags are discussed by Grünbaum in the linked source, but none of these other classes since Grünbaum's Group 1 is "Convex n-gons", Group 2 is "Star n-gons", Group 3 is "Apeirogons", and Group 5 is "Anti-prismatic n-gons". — MarkH₂₁^talk 01:31, 28 February 2024 (UTC)

Ech. I seem to have checked through it to hastily. Grübaum has a later discussion of zigzags and helices in isolation. Checking back a second time I see Grünbaum's classification. It makes sense how this came about from that source. There's still a problem with the synthesis here, as Grünbaum is discussing faithful symmetric discrete polygons, and the article is not. AquitaneHungerForce (talk) 01:44, 28 February 2024 (UTC)

I think the faithful and dense subsets of a circle is supposed to be covered by the star polygon case. In any case I'd be happy for the entire section to be removed. My edit was more an attempt to do the same thing as you are doing: puzzle out what is really meant by vague language and vague references to sources. It is mostly User:Steelpillow who has been pushing here the idea of "realizations" of abstract polytopes that map their vertices to points of a geometry; perhaps you might ask him what was intended and how it can be better sourced. —David Eppstein (talk) 01:52, 28 February 2024 (UTC)

Realizations are covered extensively by McMullen & Schulte in Abstract Regular Polytopes, the article's discussion in the other sections seems very based on that source. That source offers a much less convoluted classification of regular polygons in Euclidean space of any dimension, although I don't recall if it focuses on apeirogon realizations specifically. AquitaneHungerForce (talk) 01:58, 28 February 2024 (UTC)

I don't think I have edited this article much, at least not for a very long time. Polytopes as realizations of some abstract construct have been discussed by several prominent researchers, including McMullen, Schulte, Grünbaum and Johnson. All evolved their ideas and terminology as time went by, so while the approach has verifiable significance, any individual assertion is less hard to justify. For example everybody has their own understanding of what makes a realization unfaithful. Similarly, there are many attempts to classify apeirogons, but as far as I know the phrase "Euclidean apeirogon" has not been much used. "Aperiogon in Euclidean space" is more precise, but then the article tangles itself up in that faithfulness issue again. I'd be in favour of deleting anything that smells of OR, and of reorganising the article along more conventional lines.

On the matter of dense apeirogons (or apeirohedra for that matter), typically within a finite region, I have seen little worth noting: most authors are obsessed with rational symmetries, and treat dense figures as degenerate. I'd love to discover a more neutral-minded treatment cited here!

— Cheers, Steelpillow (Talk) 21:57, 28 February 2024 (UTC)

I've rewritten the section attempting to account for the feedback here. There might be more to contribute or improve here still. AquitaneHungerForce (talk) 23:14, 29 February 2024 (UTC)

Looks like a definite improvement in both clarity and sourceability to me. —David Eppstein (talk) 07:13, 1 March 2024 (UTC)

I have now made a bold edit, to further reorganized it and remove the emphasis on abstraction and realization as the guiding principle. Still plenty to be done. — Cheers, Steelpillow (Talk) 10:01, 1 March 2024 (UTC)

Hyperbolic pseudogon

Does anyone know what Norman Johnson is talking about when he describes "the hyperbolic line" [1] and regular pseudogons on it [2]? One-dimensional hyperbolic geometry is the same as one-dimensional Euclidean geometry — they only differ in higher dimensions. And why does this part of our article, supposedly on hyperbolic geometry, not discuss apeirogons inscribed on horocycles (or hypercycles), as seen in the figure in the "symmetries" section? —David Eppstein (talk) 18:24, 1 March 2024 (UTC)

"One-dimensional hyperbolic geometry is the same as one-dimensional Euclidean geometry" – while this is locally/intrinsically true, and I'm sure there are plenty of sources saying so, I have found it personally helpful to conceptually distinguish them, thinking of the Euclidean line as represented by points on a "straight line" but thinking of the "hyperbolic line" as represented by points on (one or both branches of) a (unit) hyperbola in the Lorentzian plane, i.e. the points $p$ whose coordinates satisfy $p_{x}^{2}-p_{y}^{2}=1$ in a context where squared distance is defined by $(q-p)_{x}^{2}-(q-p)_{y}^{2}$. I wish I knew of more sources clearly discussing this in detail though, and trying to work carefully through it; most sources focus on higher-dimensional cases without much thought for the 1-dimensional case. Norman doesn't elaborate much.

Most times you come across hyperbolic functions, the (hyperbolic) angle measure that is their argument can be thought of as representing the magnitude of a hyperbolic rotation or the geodesic distance between two points along the hyperbolic line; it is in my opinion substantially conceptually misleading to think of this as a Euclidean translation or distance. The metrical geometry of higher-dimensional hyperbolic space still holds for the linear case (sometimes in a degenerate way), and geodesics in higher-dimensional hyperbolic spaces are in my opinion best thought of as hyperbolic lines. By contrast, a horocycle in the hyperbolic plane is in some sense Euclidean or "parabolic", straddling the boundary between hyperbolic "hypercycles" and elliptic "circles". It's somewhat an accident of history that all of these objects were first studied and named based on experience with Euclidean 2–3 dimensional space, so that there's some conceptual friction involved in the current abstractions and tools used.

As one example difference between Euclidean and hyperbolic lines, points in the hyperbolic line have an "antipodal" or diametrically opposite point on the other branch of the hyperbola. If the hyperbolic line is represented under stereographic projection as the Euclidean interval with single coordinate in $(-1,1)$ ("the Poincaré segment model", if you like), this antipodal point on the other branch has the reciprocal as its coordinate. If the hyperbolic line is represented under projection through an ideal point (which seems to me to be also a kind of stereographic projection) to the Euclidean half-line with single coordinate in $(0,\infty )$, then the antipodal point on the other branch has the additive inverse as its coordinate.

While we can't continuously get from a point to its antipodal point, and we can't represent all of both branches of the hyperbola as real-valued distances from a specific origin, we can look at isometries of the hyperbola which exchange the branches, and we can do various kinds of arithmetic between the points on the two branches (and the ideal points). In particular in the "Poincaré segment", whose coordinates are the hyperbolic half-tangents of hyperbolic angle measures, the change of variables $\psi \mapsto \tanh {\tfrac {1}{2}}\psi$ the natural operation is "hyperbolic tangent addition", with coordinates adding like $a\boxplus b:={\tfrac {a+b}{1+ab}}.$

One practical place the distinction comes up is that relative velocities in spacetime can be thought of as hyperbolic; if we take a slice that chops out two spatial dimensions, the remaining velocities ("Lorentz boosts") can be represented as points on a hyperbolic line. These are usually represented by the quotient between space and time differences, a "Gnomonic projection" of the hyperbola, resulting in what might be called the "Beltrami–Klein segment model" of the hyperbolic line. But can also be stereographically projected. In either case there's an ideal point, the "speed of light", bounding the velocity. If we instead use a Galilean model of spacetime, velocities would be representable by points on a Euclidean line instead, without any such speed of light. –jacobolus (t) 20:03, 1 March 2024 (UTC)

I've generally seen "psuedogon" used to refer to apeirogons on a hypercycle (i.e. hypercompact). And Johnson's Schläfli symbol would match this definition based on the circumradius calculation. Why Johnson would call a hypercycle a "hyperbolic line" is beyond me, and I can't be sure of what is meant. AquitaneHungerForce (talk) 20:08, 1 March 2024 (UTC)