Talk:Euclidean tilings by convex regular polygons

Title

Shouldn't this be "Tilings by regular polygons" or even "Plane tilings by regular polygons"? Melchoir 01:18, 9 October 2005 (UTC)

• I agree. Feel free to move/rename this title - I don't know how! (besides copying contents and putting old under delete) Tom Ruen 01:46, 9 October 2005 (UTC)

• Done. Oh, for future reference, check out Wikipedia:Merging and moving pages. Melchoir 02:20, 9 October 2005 (UTC)

Cleanup and topics not covered

I cleaned up the article and removed the cleanup tag. There are still a number of topics discussed in Grünbaum and Shephard which could perhaps be discussed on this page but aren't yet:

• More on $k$-uniform (and $k$-isohedral, $k$-isotoxal) tilings, with additional examples and something on the Krötenheerdt tilings; equitransitive tilings.
• Non-edge-to-edge tilings: equitransitive unilateral tilings by squares; the problem of tiling the plane with exactly one square of each integer edge length.
• Star polygons, both in the style of Kepler (Grünbaum and Shephard section 2.5, the lists there being incomplete) and as hollow self-intersecting polygons (section 12.3); I shouldn't make the call as to notability of the former, having published regarding them.
• The duals of the uniform tilings (Laves tilings).
• Archimedean and uniform colourings of tilings.

Joseph Myers 00:41, 6 October 2005 (UTC)

This is wrong, but I'm not sure how to fix it

This part:

In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same size. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd.

is correct, but the tilings it makes impossible are uniform (semiregular) tilings.

However, immediately below is a section which says things like "cannot appear in *any* tiling of regular polygons". (Not just uniform tilings).

There is no justification for the claim that it can't appear in *any* tiling. I can't figure out if this is a true claim with no justification, or a false claim that happened because someone confused "any tiling" and "any uniform tiling". Ken Arromdee 18:31, 22 November 2006 (UTC)

I'm pretty sure it can all be justified. E.g. for a tiling involving 3.7.42: a triangle with a vertex of this type would have to have another vertex of this type at the other endpoint of the edge shared between the 3 and the 42; consequently the third vertex of the 3 would have to be 3.7.7, which doesn't work. The same reasoning shows 3.8.24, 3.9.18, and 3.10.15 impossible. 4.5.20 is the only vertex type involving 20-5 and 4-5, so if a vertex of this type existed, you'd have to have 4's and 20's alternating around the 5, impossible by the same reasoning as you've blockquoted above, and the same reasoning works for 5.5.10.

Changed it to read "cannot appear in any uniform tiling of regular polygons". Double sharp (talk) 12:20, 27 April 2014 (UTC)

I changed it back to read "cannot appear in any tiling of regular polygons". That statement is true, and is much stronger. Technically, what's true is that those vertex types cannot appear in any edge-to-edge tiling of regular polygons, so I added the phrase "edge-to-edge" at the beginning of that section, since all of these statements apply only in that case. This result can be found in my Ph.D. thesis, but was known long before then. I suspect the first person to state this was Kepler, 1619, in "Harmonice Mundi". The statement you began with, about three polygons meeting at a vertex and one has an odd number of sides, is a very limited argument. As you say, it only eliminates certain uniform tilings, and doesn't apply to the general situation. It's a statement often used for elementary tiling sources that are limiting themselves to the consideration of uniform tilings, but it adds nothing here. It should probably be deleted. Chaveyd (talk) 13:49, 27 April 2014 (UTC)

What does this mean?

"3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4"

What is claimed to have these two different types of vertices? If it's the tiling, there isn't only one possible tiling with 3.4.3.12; in fact there's a 2-uniform tiling with 3.4.3.12 and 12.12.3, not 3.3.4.3.4 at all. Similarly with much of the rest of this list.... I would edit it, but I'm not 100% sure I'm right in my guess about what the list is trying to say. 91.105.25.26 01:53, 20 August 2007 (UTC)

Yes there are problems with the list as its far from complete. One way it could be considered is as a list of vertex configs which yield a uniform tiling, and explination as to why the other configs do not yield a unifiorm one. It could do with an expansion listing the other tilings which can contain a particular type of vertex. --Salix alba (talk) 08:16, 20 August 2007 (UTC)

I believe I've cleaned up this rather confusing wording. And, along the way, corrected two errors. When the original editor wrote phrases like " 3².4.12 - not uniform, has two different types of vertices 3².4.12 and 3² ", he presumably meant that trying to build a uniform tiling with the first vertex type forced you to include the second vertex type. Of course once you discard the constraint of "uniform", very little is forced, and there are infinitely many tilings that contain the first vertex. It appears, though, that the original author (Salix alba) meant to include the "simplest" tilings containing that vertex, i.e. to list the 2-uniform tilings that can be constructed using that particular vertex. He admits (above) that his list is incomplete, and I have corrected that. The two corrections are: (1) He lists two 2-uniform tilings using the vertex 3².6²; there are three; (2) He listed 3.4.3.12 as requiring the vertex 3².4.3.4, but this actually generates a 3-uniform tiling that also includes the vertex type 3⁶. There are, however, three different 2-uniform tilings that contain that vertex type, and I have listed them all.

It's interesting to note that of the four vertex types which can occur in edge-to-edge tilings by regular polygons, but cannot occur in uniform tilings, the next section of this article shows examples of three of them, but does not have the (unique) example of a 2-uniform tiling with vertices of type 3.4.3.12 (an example is shown here). We should probably include a picture of that tiling as well, but it would be best if it were in a format comparable with the others shown there. I'll ask the creator of those tiling drawings if he would be willing to make one more. Chaveyd (talk) 15:34, 27 April 2014 (UTC)

The semiregular tiling 3^2.4.3.4 is said to have symmetry "p4", but it must be "p4g", according to the information on the specific page for that tiling[1] — Preceding unsigned comment added by 193.147.19.10 (talk) 17:46, 11 April 2016 (UTC)

How many??

The following proves there are infinitely many tilings made out of regular polygons:

• A hexagon is always interchangeable with a flat pyramid (6 triangles meeting at a vertex)
• A dodecagon is always interchangeable with a flat cupola (a hexagon surrounded by squares on each edge and triangles at each vertex)

But, is the following question known:

How many tilings of regular polygons are there that don't contain in them any flat pyramid or flat cupola?? Georgia guy (talk) 22:55, 10 October 2010 (UTC)

Still infinitely many (even uncountably many), even if you require the polygons to meet edge-to-edge, because you can tile the plane by any sequence of square strips and equilateral triangle strips. —David Eppstein (talk) 22:59, 10 October 2010 (UTC)

Added Reference link

The references for this page have long included a link to a 1989 paper of mine. Since that source is difficult to access, I have added a direct link from that reference to my web page from which one can download a copy of the paper. I hope doing so this is not a violation of Wikipedia rules. I was not the person who originally added this link (which I'm sure would be a violation), I only added the link from that reference to my page. Chaveyd (talk) 05:53, 26 June 2013 (UTC)

It seems unproblematic to me but I fixed the formatting. —David Eppstein (talk) 06:09, 26 June 2013 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Euclidean tilings by convex regular polygons/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

For areas that should be covered but aren't yet, see my previous comments at Talk:Tiling by regular polygons#Cleanup and topics not covered. Joseph Myers 02:14, 20 May 2007 (UTC)

Substituted at 18:16, 17 July 2016 (UTC)

Some of your 2005 suggestions for things to be covered have been, but most have not. It seems to me that you are correct that we should include sections on:

Equitransitive tilings; and

Equitransitive unilateral tilings by squares;

Tilings by star polygons and by hollow self-intersecting polygons don't really belong here, because this article restricts itself to convex regular polygons. That might be a better addition to the article on Star polygons.

Similarly, the dual tilings (Laves tilings) probably don't belong here, but we should certainly link to that article from this one. I'm not sure I'll have the time in the near future to add those sections, but is anyone else willing to do that? Darrah (talk) 04:19, 19 July 2016 (UTC)

I guess I agree with Darrah. Incidentally, I don't have any general sources for the dual k-uniform tilings, but I have generated images, collected here on a userpage: User:Tomruen/uniform_tilings. Tom Ruen (talk) 06:13, 19 July 2016 (UTC)

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Sub page for "k-uniform tilings" too?

The section about "Archimedean, uniform or semiregular tilings" have a sub page where it lists those examples, but for the "k-uniform tilings" section all examples are listed here, making it a huge article. Surely this should be improved by a sub-page too? Rixn99 (talk) 08:50, 16 December 2018 (UTC)

Dual uniform tilings

Tomruen: credit for my initiative is very kindly appreciated. Also please kindly change your .svg's so they reflect the uniformity. Some of your tilings, for example, are only red and no other color. Thanks for your consideration!

P.S. Please allot a consistent, markedly different, color for every planigon/semiplanigon. When the uniformity is higher than the number of planigons (vertex types), please use darker shades to denote secondary polygons.

Edit: sorry ClueBot NG, but I did not know this was vandalism. Tomruen overwrote my images first. User:Harry Princeton

I don't think duals belong in this article. It is already pretty huge. There are already images online here User:Tomruen/uniform_tilings if we have sources to defend them in an article possibly elsewhere. I agree having different colors for each shape is nice, while my SVG are colored by the number of sides alone. Tom Ruen (talk) 22:13, 25 June 2019 (UTC)

Thanks so much for your kind and quick reply, Tom Ruen. Yes, your page is a very nice reference for duals for most uniform tilings :).

While publishing articles about dual uniform tilings is not necessary, it is suggested that you make User:Tomruen/uniform_tilings more accessible to others. Once upon a time people will be curious about the duals of the uniform tilings. It would benefit Wikipedia articles to either include all dual uniform tilings or none (e.g. Laves tilings). By the way, I have realized the 4 semiplanigons independent of you (and also coined the term). Harry Princeton (talk) 23:04, 25 June 2019 (UTC)

P.S. I will only do the duals of the Krotenheerdt tilings in Euclidean tilings by convex regular polygons.

P.P.S I have added a section of Krotenheerdt uniform 6 and 7 tilings in your page, if that is helpful :) User:Tomruen/uniform_tilings. Harry Princeton (talk) 04:31, 26 June 2019 (UTC)

21 possible vertex polygons labeled by their face configurations: 3 regular ones in yellow, 8 semiregular in magenta, 4 demiregular in brown, and 6 unuseable ones in orange.

I continue to think this article is already much too big to include the duals, but here's an adjusted graphic from the scattered unlabeled one. I grouped rows by sides, and put the unusable orange triangles ones on the bottom. Tom Ruen (talk) 19:15, 13 July 2019 (UTC)

Thanks for your kindly update. I have since revised the image above. Tomruen: 'There are already images online here User:Tomruen/uniform_tilings if we have sources to defend them in an article possibly elsewhere.' It has already been proven for uniform tilings (or the proof remained to be demonstrated); the Conway operator of dual involves connecting the centroids of regular polygons to form new faces around each original vertex (interchange faces and vertices). Such connecting edges are perpendicular bisectors among the shared edges since centroids lie on edge perpendicular bisectors (see the dihedral group), and these shared edges form faces which are planigons/semiplanigons around each shared vertex. This perpendicularity (or orthogonality) perhaps begat 'ortho' in Conway's mind when he invented Conway polyhedron notation (no verifiable source, but the operation's name and behavior is unambiguous), or the superimposition of the dual tiling on the original tiling. In fact, you had used them for Planigon without collaborating with me. — Preceding unsigned comment added by Harry Princeton (talk • contribs) 20:58, 15 July 2019 (UTC)

Verifiability

Tomruen, I am not sure if http://probabilitysports.com/tilings.html is a verifiable source for enumerating Euclidean uniform tilings (particularly, 4-uniform and 5-uniform). Indeed, it is a strong source, but I want to be careful, since it's the only one and it is not explicitly published. — Preceding unsigned comment added by Harry Princeton (talk • contribs) 00:11, 21 July 2019 (UTC)