# Talk:Binary tiling

Does anyone have a view of this tiling in the Poincaré disk model? I think this would illustrate better the fact that the tiles are not polygons. — Preceding unsigned comment added by TheKing44 (talk • contribs) 21:53, 28 January 2018 (UTC)

There's one in the Penrose article, but we can't just copy it directly and I don't think it does what you want it to do. (The horocycle edges are drawn as circular arcs, just as straight lines in the disk model would be, but that's not different from what we have now where the horocycles are horizontal lines. Anyone who knows enough about the hyperbolic plane to know that the circular arcs of the disk model are the wrong direction to be straight lines would also know that the horizontal lines in the halfplane model are the wrong direction to be straight lines.) —David Eppstein (talk) 22:32, 28 January 2018 (UTC)
@David Eppstein: Well, maybe a view in which the horocycles are replaced by straight edges would be useful (to demonstrate what the difference looks like). (In particular, the edges would form a bunch of regular aperigons.)
Ok, done. —David Eppstein (talk) 01:55, 29 January 2018 (UTC)
Nice! It would still be informative to see the shape of a single tile drawn "centered" on a Poincaré disk model. Then it would be more clear that the bottom pair edges are the same length as the top one. Tom Ruen (talk) 01:58, 29 January 2018 (UTC)
It's not a regular polygon, so I'm not sure what "centered" means here. —David Eppstein (talk) 04:26, 29 January 2018 (UTC)
Ha, that's why I put in quotes! Call it a centroid, or somethig close, as you like. Tom Ruen (talk) 05:59, 29 January 2018 (UTC)
@Tomruen: Or perhaps displayed in the band model, centered on a line going through a bunch of tiles. The band model preserves area near a line, so that would show the fact that the "parent" tiles are the same size as the "child" tiles. (For reference here is what the {5,4} looks like in the band model: http://bulatov.org/math/1001/band/tiling_425_band_00.png although I was thinking our band model would go through the centers of the tiles.) TheKing44 (talk) 17:52, 29 January 2018 (UTC)

The binary tiling is related to the one in figure 3 of this paper about a strongly aperiodic tiling of the hyperbolic plane (in particular, it also uses rectangles in the half-plane model, and they have a picture of the Binary tiling in Figure 1). Should we talk about in the article somehow?
(Maybe we could move this article to Böröczky tilings, and describe all such tilings (including higher dimensional versions).) TheKing44 (talk) 17:50, 29 January 2018 (UTC)

Theorem 3.12 of https://www.math.uni-bielefeld.de/baake/frettloe/papers/hyp-art-final.pdf describes the symmetry group of this tiling. (Just though y'all would like to know.) TheKing44 (talk) 20:50, 29 January 2018 (UTC)

This review is transcluded from Talk:Binary tiling/GA1. The edit link for this section can be used to add comments to the review.
Nominator: David Eppstein (talk · contribs) 05:56, 24 September 2024 (UTC)
Reviewer: DoctorWhoFan91 (talk · contribs) 06:44, 24 October 2024 (UTC)
I'll take this one. Expect initial comments in 24-48 hours. DoctorWhoFan91 (talk) 06:44, 24 October 2024 (UTC)

Thanks! —David Eppstein (talk) 07:08, 24 October 2024 (UTC)
I'll go section by section. DoctorWhoFan91 (talk) 14:24, 24 October 2024 (UTC)


### Lead
, meaning: which means(I think that'll sound better)
 Done —David Eppstein (talk) 07:13, 27 October 2024 (UTC)


### Tiles
An example is the familiar tiling...: maybe an example where such tiling is used for easier visualization?
 Done —David Eppstein (talk) 07:13, 27 October 2024 (UTC)
* Wikilink asymptote?(in the caption of the top pic too)

I don't think we have a good article to link to for the asymptotic point of a hyperbolic line. The article you link to is for the asymptotic line of a Euclidean curve, a different concept. The meaning intended here is glossed in the first paragraph of this section; that's why it's in italic in that paragraph. The closest link we have to the correct concept is ideal point which is already linked. —David Eppstein (talk) 07:13, 27 October 2024 (UTC)
Suggestion revoked, my brain incorrectly assumed its must be the same definition as for Euclidean geometry
Start a new paragraph from Two common models..., maybe
 Done —David Eppstein (talk) 07:13, 27 October 2024 (UTC)


### Enumeration and aperiodicity
I am a bit confused with what the first paragraph and the whole section is trying to say, am I understanding it correctly - the tiling is not actually symmetric in the hyperbolic plane, but in modeling to the Euclidean plane they are? If so  Done

Mention that it is in the Hyperbolic plane
There are uncountably many different tilings of the hyperbolic plane by these tiles, even when they are modified by adding protrusions and indentations to force them to meet edge-to-edge.: There are uncountably many different binary tilings of the hyperbolic plane, even ones which are modified by adding protrusions and indentations to force them to meet edge-to-edge.
The tiles are not symmetric to each other, period. It doesn't matter whether you consider them in the hyperbolic plane or via its models in the Euclidean plane. Even though every two tiles have the same shape there are always pairs that are not positioned in the same way with respect to the other tiles that surround them. Anyway, I edited this part, I hope to clarify these matters. —David Eppstein (talk) 07:46, 27 October 2024 (UTC)
no tiling has an infinite group of symmetries.: add to the end , as it is possible for the one-dimensional group
I split the sentence in a different way. —David Eppstein (talk) 07:46, 27 October 2024 (UTC)
Thank you, I understand it better now, marked all three points as done

As a tile: Maybe split the first para here
 Done —David Eppstein (talk) 07:46, 27 October 2024 (UTC)
* The first corona is the set of tiles touching a single central tile.: Remove this, as the wikilink before this explains coronas

In general, I think it is better per WP:TECHNICAL (part of the GA criteria) to provide a brief gloss of technical terms, when possible, instead of relying on readers to follow wikilinks and then return to where they were before following them. —David Eppstein (talk) 07:46, 27 October 2024 (UTC)
Oh, okay, I usually read the wikilink lead to undertstand unknown concepts, didn't think of WP:TECHNICAL
Corresponding to the fact: Also, corresponding to the fact (because it seems to following the previous paragraph's line of thought)  Done
Copyedited this whole paragraph to put the description of the dual tilings before their properties. —David Eppstein (talk) 18:49, 27 October 2024 (UTC)
need a clarification- so the dual tilings aren't also binary tilings, bcs the pre-edit paragraph made me think so? DoctorWhoFan91 (talk) 19:07, 27 October 2024 (UTC)
They are not binary tilings. They have triangles and quadrilaterals as their tiles. I added another sentence explaining this. —David Eppstein (talk) 20:08, 27 October 2024 (UTC)
A suggestion, not required for GA: another sections have images, can this have an image, if possible?  Done
I added two new images. —David Eppstein (talk) 03:42, 28 October 2024 (UTC)


### Images
"Is the average number of red points per tile 1/3 (left) or 2/3 (right)?" : Are questions as captions appropriate? I'm not familiar if that is correct per MOS or not?
I don't see anything in the MOS against it. I couldn't think of a way to make the same point as concisely as a statement rather than a question. I don't want to say that the left has 1 point per 3 tiles and the right has 2 per 3, because the point is actually that the left and the right have the same points and that trying to define an average number of points per tile doesn't work. But just saying "the left and the right have the same points and that trying to define an average number of points per tile doesn't work" doesn't work because without an explanation of 1 point per 3 tiles or 2 points per 3 tiles, it might not be obvious to the reader why it doesn't work. —David Eppstein (talk) 18:57, 27 October 2024 (UTC)
Suggestion revoked
* This isn't GA criteria, just an additional suggestion, to make the article look better- the images do not fit well into the layout of the lead and first section- so if anything could be done about that?

I removed some unnecessary detail from the lead caption and moved the next section's images up to improve the layout. —David Eppstein (talk) 18:54, 27 October 2024 (UTC)
I feel like the information in the lead caption wasn't unnecessary- it helped visualize. Also,  the images to the side of the corresponding paragraph also helped better. Perhaps you can revert the change, better to have the correct content, then trying to make the layout better. DoctorWhoFan91 (talk) 19:05, 27 October 2024 (UTC)
Ok, longer caption and image locations restored. —David Eppstein (talk) 03:46, 28 October 2024 (UTC)


### Related patterns
three right triangles.: three isoceles right triangles.  Done
When interpreted as Euclidean shapes rather than hyperbolically, the tiles are squares and the subdivided triangles are isosceles right triangles.: Remove, redundant bcs that has already being mentioned in the article.  Done
by part of a binary tiling, the tiling of a horoball: confusing, is the tiling of a horoball made of binary tilings?  Done
Ok, I think I've handled all these. The tiling of a horoball is like what you get from the half-plane model binary tiling by keeping only the part above one of the horizontal lines. —David Eppstein (talk) 04:00, 28 October 2024 (UTC)


### Applications
I'm confused here too-
*If the tiling is extends infinitely, can't it be done by using calculus, or is it saying that that calculus in hyperbolic geometry can't be modeled to Euclidean geometry or something?

I'm not sure what you're asking here. If you try to define average number of points per tile using a limit (of #points/#tiles in large regions) instead of by counting points and tiles in finite repeating patterns, you run into the same problems for the same reasons. —David Eppstein (talk) 04:24, 28 October 2024 (UTC)
I must be misunderstanding the geometry, suggestion revoked
* The area of the tiles are actually the same, right, bcs its measured differently in this non-Euclidean geometry. If yes, can it be mentioned, it's not easy to remember?

As it says in the second sentence of the article, the tiles are congruent. That means among other things that they have the same area. —David Eppstein (talk) 04:14, 28 October 2024 (UTC)
Yeah, sorry, I missed the obvious effect of the tiles being congruent
* The tiles of a binary tiling ...: I feel like the paradoxical issues should be explained in a different paragraph

The paradoxical issues are the application. You think one application should have more than one paragraph? I would think that would be confusing. —David Eppstein (talk) 04:14, 28 October 2024 (UTC)
Hmm, suggestion revoked
Adjusting the distance: I thought it was a continuation of the previous paragraph, can you add "also" or something to make it obvious it's explaining other applications.  Done
Copyedited including a sentence stating that this is a different application. —David Eppstein (talk) 04:14, 28 October 2024 (UTC)
* Just to make sure, are these the only applications?

Um. I would say "the only ones I know about" but that's not actually true; see the application to proving that origami-folding is hard in section 6 of https://arxiv.org/abs/2410.07666 . I didn't want to add that one because of the COI and because it hasn't been reliably published yet. —David Eppstein (talk) 04:02, 28 October 2024 (UTC)
Suggestion revoked (also, nice paper)


### Spot-check
Checking every 5th ref in general (the parts I can understand, atleast)

Ref-1: In particular, it is shown ... there are uncountably many tilings with a fixed prototile
Ref-6: Hyperbolic length=Euclidean length/y
Ref-11: due to Boroczky
Ref-17: Escher created a few ... patterns in hyperbolic geometry
Ref-22: "diagram of an infinite binary tree"
Ref-26: On the Hyperbolicity of Small-World and Tree-Like Random Graphs


### Overall
Two sections reviewed. As an aside, you write well, and this is so easy to understand. Thank you! Will definitely read and review more articles created by you when I get the time. DoctorWhoFan91 (talk) 16:30, 25 October 2024 (UTC)
Another section reviewed. Most of the changes suggested are probably bcs I do not understand the topic enough. DoctorWhoFan91 (talk) 08:13, 26 October 2024 (UTC)
Just one section remains to be reviewed. DoctorWhoFan91 (talk) 07:05, 27 October 2024 (UTC)
Completed review, putting on hold. Your responses to the review have been great and your changes to the article even more clearer than I thought could be made. Thank you for writing such a great and clear article. DoctorWhoFan91 (talk) 12:26, 27 October 2024 (UTC)
That was quick, and your changes were better than my actual suggestions. I need to write down the spot-check properly, so I'll do that and pass it, probably in a few hours, a day at the most. DoctorWhoFan91 (talk) 06:49, 28 October 2024 (UTC)
Did the source check, and everything seems fine (article is even better written than I thought, very understandable to a layman, great job). Thank you for such an informative, clear and well-written article. Congratulations, David Eppstein, passing the article to GA! Keep up the good work, helping even those with less technical knowledge to understand complex mathematical concepts. Thank you again. DoctorWhoFan91 (talk) 14:25, 28 October 2024 (UTC)