Talk:Pentagonal tiling

message: I'm curious what the criteria is for uniqueness? For instance, the blue (column 4, row 1), and pink (column 3, row 2) tilings are topologically identical, being distortions of the more symmetric Cairo pentagonal tiling. Tom Ruen (talk) 10:00, 17 November 2010 (UTC)
replies: text: Each one has a different set of conditions (in terms of certain sums of edge lengths and angles that have to equal each other) for the set of pentagons that can form that pattern, maybe? —David Eppstein (talk) 15:47, 17 November 2010 (UTC)
replies: text: These are not isohedral tilings (they are monohedral, not isohedral); only five of the types of pentagons admit isohedral tilings.  The figure of 14 applies to "types" of (convex) pentagons that tile; a "type" should indeed be understood to refer to the set of relations between edge lengths and angles that appear in the tiling (and it should be understood that you ignore types for which a proper subset of the relations is also sufficient to ensure the pentagon tiles).  If you want to count types of isohedral tilings by convex pentagons, Tilings and Patterns section 9.1 gives a notion of "polygonal isohedral type", of which there are 24 types using convex pentagons (all of which use pentagons belonging to one or more of the five types of convex pentagon admitting isohedral tilings).  Joseph Myers (talk) 22:02, 17 November 2010 (UTC)
thread_title:  Set of 14 isohedral pentagonal tilings 

message: Does the {5,∞} tiling actually exist, as suggested by the article?  This seems impossible as it would suggest each side must be infinitely long, but I'm not a topologist.  Am I correct in assuming this is in error?
If it does exist, can we get a disc model for it as we have for the {∞,4} tiling (as seen in Order-4 pentagonal tiling)? TricksterWolf (talk) 17:07, 1 January 2012 (UTC)
replies: text: Sure, it exists. Its faces are regular ideal pentagons, with all five vertices at infinity. It can be generated by starting with a single ideal pentagon and then repeatedly mirroring it across one of its edges. Unfortunately the Kaleidotile software that was used to generate these images only goes up to 8. —David Eppstein (talk) 18:42, 1 January 2012 (UTC)
replies: text:  I'm sure User:Tamfang can generate this tiling. He made a good sampling of uniform tilings with ∞ orders, but not (2 5 ∞) specifically. See his gallery at . This regular tiling :File:H2_tiling_25i-4.png, {5,∞} shows all the vertices at infinity! Tom Ruen (talk) 19:11, 1 January 2012 (UTC)
replies: text:  Wow, awesome.  :)  Thanks! TricksterWolf (talk) 21:16, 4 January 2012 (UTC)
thread_title:  {5,∞} tiling 

message: what are the tilings called where identical polygons (not necessarily regular) are used.  — Preceding unsigned comment added by 14.99.14.87 (talk) 10:14, 17 March 2014 (UTC)
replies: text: They are called monohedral. —David Eppstein (talk) 16:05, 17 March 2014 (UTC)
thread_title:  what are these type of tilings called 

message: Why is it unacceptable to include any use of the turn as a measurement of angle?  — Preceding unsigned comment added by 2001:48F8:7042:E46:6C2E:B9C0:D020:5835 (talk) 21:49, 2 August 2015 (UTC)
replies: text: Linking to the turn article, as a more precise term for describing a single complete revolution, as you did in this edit: not particularly problematic. Replacing standard angle measures (degrees or radians) by fractions of a turn, as promoted by the fringe tauist movement and as you did in this edit: not ok. We should not be in the business of promoting nonstandard measurement systems here. —David Eppstein (talk) 22:46, 2 August 2015 (UTC)
thread_title:  Angle measure

message: This article was nominated for enwiki's In the News front page section based on the discovery of the 15th convex tiling. The nomination included further sources that have not been added to this article. - Pointillist (talk) 22:55, 16 August 2015 (UTC)
thread_title:  Additional sources for 15th convex pentagon tiling 

message: Anyone else noticed that any pentagon that can be assembled in any other regular tilable pattern (triangle, quadrilateral, hexagon) qualifies ?  — Preceding unsigned comment added by Buz9er (talk • contribs) 01:07, 17 August 2015 (UTC)
replies: text: I just added a subsection on the pentagon's relation to the hexagon. - Catsquisher (talk) 22:24, 20 August 2015 (UTC)
thread_title:  Hexagonality 

message: — Preceding unsigned comment added by EdPeggJr (talk • contribs) 21:54, 3 August 2015 (UTC)
replies: text: I moved the above comment as it was added to the middle of an existing discussion with a new subject heading and without any appropriate indenting so may have caused confusion and is also very easy to miss  Nil Einne (talk) 19:16, 17 August 2015 (UTC)
thread_title:  Set of 15 isohedral pentagonal tilings 

message: I don't know why articles as in this would be featured in the Main Page's In the news section when most of the public does not pay attention to things such as "Mathematicians at the University of Washington Bothell discover a new type of pentagon tiling, the first new way to "tile the plane" with a single polygon since 1985." How many people will remember the fact that a group of people in a university discovered a 15th type of pentagonal tiling? No more than 1 million I think. If any reason for the featuring of the "Pentagonal tiling" article in there exists, please tell me. }I6ixce93IxI{ 22:46, 17 August 2015 (UTC)
replies: text: I certainly thought it was interesting and informative. Czolgolz (talk) 22:51, 17 August 2015 (UTC)
replies: text: Okay, thanks. =D This article is good. }I6ixce93IxI{ 22:53, 17 August 2015 (UTC)
replies: text: I think this is an important mathematical discovery for people worldwide too. Deryck C. 10:20, 18 August 2015 (UTC)
replies: text: I agree. :) }I6ixce93IxI{ 12:29, 18 August 2015 (UTC)
replies: text: In the future, if you want to contribute to decisions as to what articles appear on the main page In The News section, you can do so at WP:ITN/C.  Your input would be valued there!  --Jayron32 12:33, 18 August 2015 (UTC)
thread_title:  This article on in the news? 

message: The Guardian article on the newly discovered pattern in turn cites Ed Pegg, who drew the colourful image on this article. Is there some kind of circular reference going on? Can we find an additional source for the discovery of the latest tiling pattern? Deryck C. 10:21, 18 August 2015 (UTC)
replies: text: The Guardian article has the new 15th form with 3 colors (apparently for the 3-isohedral positions), while I agree the SVG with a summary of all 15 is a bit too small to clearly verify its correctness AND I'd prefer a version with isohedral colorings on all 15 forms. (As I suggested in previous topic above). Tom Ruen (talk) 10:43, 18 August 2015 (UTC)

text: This webpage has the 15 tilings, including k-isohedral colorings.  Tom Ruen (talk) 11:48, 18 August 2015 (UTC)

text: I replaced the monocolored example collection image with :File:15-monohedral pentagonal tiling types.png, collected from Jaap Scherphuis's Tiling Viewer applet, by permission. It's just a bitmap, but the Java source code is available, so I'll look into adding an SVG export option. Tom Ruen (talk) 15:10, 18 August 2015 (UTC)
replies: text: There's a very short preprint visible at https://twitter.com/gelada/status/626810879435083776 (dated July 29, from before the Guardian article and before Pegg's updated image). —David Eppstein (talk) 16:42, 18 August 2015 (UTC)
thread_title:  Circular reference? 

message: It would be nice if the Marjorie Rice article could be decorated by an image of the four (out of fifteen) pentagon tiles that she discovered. Since there's been quite a lot of activity here in reworking the pentagon tiling images, maybe someone here would be interested in putting together such an image? It might be as simple as picking out the correct four from :File:PentagonTilings.svg and arranging them into a square. —David Eppstein (talk) 21:59, 18 August 2015 (UTC)
replies: text: I can extract parts of that SVG, but is Schattschneider 1978 a reliable source for which of :File:PentagonTilings.svg should be attributed to Rice, or do we need diagrams that are closer to Schattschneider's figures 8 & 10? - Pointillist (talk) 00:09, 19 August 2015 (UTC)
replies: text: I think it's a good enough source. My understanding is that the ones we want are the bottom four tilings in the figure. —David Eppstein (talk) 00:31, 19 August 2015 (UTC)
replies: text:  I've done the bottom four in 2x2 and 4x1 layouts. Then I went back to Schattschneider and now I'm not sure those are the correct four. So I've done new versions. Can you pop over to :commons:File:PentagonTilings_Rice_2x2.svg to check whether the older or newer of the two versions in the right one? Same goes for :commons:File:PentagonTilings_Rice_4x1.svg. I'm no mathematician! Thanks - Pointillist (talk) 11:11, 19 August 2015 (UTC)

text: Visually comparing images does get tricky with geometric variations. I think k-isohedral colorings help distinguish them better. I expanded the history section into Pentagonal_tiling#Monohedral_convex_pentagonal_tilings with separate PNG example pictures of each of the 15 tilings (k-isohedral colorings) with single prototile diagrams, and diagrams. I'm glad if these images can be replaced by nicer SVG ones, but I'd prefer the k-isogonal colorings which help distinguish them, and made it easier to see the symmetry and fundamental lattices. Tom Ruen (talk) 12:03, 19 August 2015 (UTC)

text: I think the SVG images are the correct ones: Tom Ruen (talk) 12:14, 19 August 2015 (UTC)

text: A Java Applet here can draw them, with parametric sliders for variations.  I think one form for each is good, unless we moved them into a new article and added one section per tiling where multiple images for each could be shown? I'm looking into SVG export option to the applet source code. Tom Ruen (talk) 20:56, 19 August 2015 (UTC)
thread_title:  Rice tiling illustration 

message: For the new (15th) tiling, Vertex C in the image and text below should be 135 degrees instead of 105. If you look at the points in the tiling where the red, blue, and yellow pentagons meet, you'll see that 2C + E = 360. Since E is 90, C must be 135. This is also visually noticeable in the image (C and D are not the same angle). I would change this myself, but I'm not familiar with the editing protocols here. 2604:6000:1513:202C:4057:481E:72A:CE2C (talk) 22:47, 19 August 2015 (UTC)
replies: text: Good catch! I agree that looks correct, confirmed here . I had copied the numbers from a mistaken source. I made the correction to the image and text. Tom Ruen (talk) 23:12, 19 August 2015 (UTC)
thread_title:  Incorrect angle on 15th tiling 

message: With s = sqrt(3), use as a motif the pentagons defined by p15 / 8 and -p15 / 8. The offset vector between motifs is then {3 (1 + s)/2, 1}
  p15 ={{{5-5 s,1+s}, {5-5 s,5-3 s}, {1-s,-7+s}, {-1+s,-1-s}, {1-s,1+s}},
  {{-5-3 s,7-s}, {-7-s,1+s}, {5-5 s,5-3 s}, {5-5 s,1+s}, {1-5 s,5+s}},
  {{-1-3 s,11-s}, {5-5 s,13-3 s}, {1-s,1+s}, {5-5 s,1+s}, {1-5 s,5+s}},
  {{-1-3 s,3-s}, {1-5 s,-3+s}, {-11-s,1-3 s}, {-11-s,-3+s}, {-7-s,1+s}},
  {{-5-3 s,7-s}, {-11-s,9-3 s}, {-7-5 s,-3+s}, {-11-s,-3+s}, {-7-s,1+s}},
  {{-11-s,-3+s}, {-11-s,1-3 s}, {-7-5 s,-11+s}, {-5-7 s,-5-s}, {-7-5 s,-3+s}}} EdPeggJr (talk) 23:30, 19 August 2015 (UTC)
thread_title:  Optimal set of pentagons for the 15th tiling 

message: The latter is close to a regular pentagon (all interior angles obtuse or greater than 90°), while the tiled ones have acute angles, and another one looks like a right angle.
: The inconsistency is unintended, but not incorrect, with geometric variations on each type. I uploaded new images that are consistent. Tom Ruen (talk) 15:42, 20 August 2015 (UTC)
thread_title:  Illustrations don't match [2 pgg (22×) c=e B+D=180] 

message: Marjorie Rice presents, as a basis for her pattern "Roses", a beautiful tiling of a special case of pentagon #1 that is different from the one we depict here. Doris Schattschneider mentions that "a high school class in New South Wales, Australia, had made a project to discover equilateral pentagons that tessellate and had discovered many different types". That made me realize that we actually only list different pentagons here, not different tilings, as the article title suggests. Given that they even have different topologies (no small feat, seeing how many different tiling types can share the same topology: ), shouldn't we include those tilings here? — Sebastian 21:22, 20 August 2015 (UTC)
thread_title:  Roses and other distinct tilings from special cases 

message: Is it worth noting that some of the tiles (e.g. the bisected hexagon) can be arranged in ways that lack global translational symmetry.  For example, the bisected hexagon can tile the plane using the (Type I) hexagonal pattern, but one can also rotate individual pairs of pentagons (essentially rotating individual hexagons) in ways that allow one to break the global symmetry of the pattern.  It's not particularly interesting mathematically, but it might be worth noting global symmetry is not a required feature of a pentagonal tiling of the plane.  Right now all of the example images have large scale symmetry, which may mislead people to think this is absolutely required of any tiling, even though exceptions can be constructed.  Dragons flight (talk) 21:03, 22 August 2015 (UTC)
replies: text: I think that would be worth noting. In addition, I suggest adding examples of pentagonal tilings made up of more than one prototile.  — Preceding unsigned comment added by Catsquisher (talk • contribs) 21:16, 22 August 2015 (UTC)
thread_title:  Irregular tilings 

message: (see Illustrations don't match above)
Is it possible to make the large single cells below each pattern use the same shape (and orientation) as (one of) the cells in the pattern above? EG in "1", angles AED look similar but B&C are very different.
Also the length and angle text varies in size enormously and is often unreadable at the thumbnail size in the table. The angles for the cell in 6 are about right IMO. -- SGBailey (talk) 20:43, 23 August 2015 (UTC)
replies: text: Request noted. I agree the geometric variations should be consistent between the 3 images - full, single, and lattice. It was all done pretty quick. Ideally they can be made more carefully as SVG eventually also. Tom Ruen (talk) 22:45, 23 August 2015 (UTC)
thread_title:  Illustrations 

message: Would it be appropriate to make a Pentagonal Tiling Category with this article, the three Dual uniform tilings (one of which is a redirect to a section) and Marjorie Rice?Naraht (talk) 04:38, 24 August 2015 (UTC)
replies: text: Thank you for your suggestion. What use do you see in doing that? To me, the effort of maintaining these, and for our readers of finding these seems bigger than the benefits, given that :Category:Tessellation has no more than 32 entries, which still allows easy finding of those pages. — Sebastian 05:26, 24 August 2015 (UTC)
thread_title:  Pentagonal Tiling Category? 

message: It might be good to state how many degrees of freedom each tiling family has.  Ignoring orientation and scale, a general pentagon shape can be described by 6 parameters.   Type 15 has no degrees of freedom remaining, and so corresponds to only a single shape, but I think all of the rest have one or more degrees of freedom remaining (usually one or two), but it might be useful to say how many freedoms each family has.  In addition, it might be nice to list one or a few examples of side lengths and angles for each type.  Though the families are well characterized by the formulas stated it isn't entirely trivial to translate the formulas to specific examples, and that might be useful for people trying to explore this further.  Dragons flight (talk) 08:13, 26 August 2015 (UTC)
replies: text: I agree. The applet at http://www.jaapsch.net/tilings/ shows explicit degrees of freedom. (I had to download and run as a JAR since my browser complains about Java security) I hope we can expand new sections (or a related articles) with variations on each type. It does seem a bit open-ended how to best show multiple degrees of freedom, although one animation could exercise each degree independently or different ones in sequence. Tom Ruen (talk) 09:38, 26 August 2015 (UTC)

text: If 14 also has no degrees of freedom, then we should say what all of its angles are.  Is that actually implied by the formulas currently given in the text?  Dragons flight (talk) 17:02, 26 August 2015 (UTC)
replies: text: Confirmation on 14 having no degrees of Freedom http://www.scipress.org/journals/forma/pdf/2001/20010001.pdf Can't find a version of it with a doi. :( Also, from the text the size of C is cos–1((3(sqrt(57)) – 17)/16) *Not* a pretty number. Naraht (talk) 17:25, 26 August 2015 (UTC)

text: AH, I see this 2012 paper gives them rounded to 1/100ths:  p.94: 90°, 145.34°, 69.32°, 124.66°, 110.68°. Tom Ruen (talk) 17:32, 26 August 2015 (UTC)
replies: text: The two links above seem to be broken. Could someone clarify in the article which these degrees of freedom are for each of the types above? Which lengths? Which angles? Within which intervals? Thanks a lot! 130.238.112.129 (talk) 18:25, 24 July 2023 (UTC)
thread_title:  Family size and samples 

message: Interesting topic. It is not as obvious as it could be that the 15 illustrations are just representative examples of tiling patterns, and that edge lengths and angles can vary. This may be inferred from the formulas, but it I think it should be explained clearly in words too, in the introduction, for people who may not necessarily grasp what the formulas are saying. A related potential issue is the use of the word "type"  in "Fifteen types of convex pentagons are known to tile the plane monohedrally (i.e. with one type of tile)". It is not very clear what a "type" of tile is, or whether the word "type" is used with the same meaning in both places. A mention that mirror images are allowed would also be useful, as it is not necessarily obvious that an asymmetric tile is the "same" as its mirror image. 31.49.126.141 (talk) 11:47, 26 August 2015 (UTC)
replies: text: As discussed in the section above, 13 of the 15 are "types/examples" having degrees of freedom, the last two have no freedom and thus are different. I'm not quite sure how to phrase that.Naraht (talk) 17:43, 26 August 2015 (UTC)
thread_title:  Suggested clarifications 

message: The Type 5 examples include those where the Pentagons are *not* convex. IMO, we should not be including non-convex examples here.Naraht (talk) 16:04, 28 August 2015 (UTC)
replies: text: I understand. I'm in a place of indecision over that question. As well, degenerate cases also can be included or excluded: degeneracy either by edge lengths going to zero, or having two adjacent parallel edges, BOTH cases effectively changing pentagons into quads or triangles. But given well-defined parametric variations, there's reason at least to show them. The worst degeneracy cases have coinciding or intersecting nonadjacent edges, and being visually confusing and they are easier to reject. Tom Ruen (talk) 16:26, 28 August 2015 (UTC)
replies: text:  I can understand that and given that Type 1 has a very easy example where angle E is re-entrant. (A=60,B=90,C=90,D=60 and E=240), we certainly should mention it and some of them are really pretty looking. But the *entire* section is labelled Convex and the work on the 15 types has clearly been for Convex. Any idea if allowing for re-entrant angles gives additional tilings? (I'm sure that degenerate do, but I don't think it is that useful other than being shown as an edge case for a set of examples)Naraht (talk) 17:49, 28 August 2015 (UTC)

text:  If you're in a place of indecision let me vote to omit them.  I read the article and found the convex examples confusing, and detracting more than they were adding. -- I came here just to open the discussion and I'm glad there's already some thought to removing it.  -- please do so! 174.92.79.155 (talk) 16:50, 29 August 2015 (UTC)

text: I'm pretty sure that non-convex pentagons have more examples. In particular, consider any pentagon where two consecutive angles add to 2 and the two sides on either side of these two angles have equal lengths. Then such a pentagon can be combined with itself (by a rotation of ) to form a parallelogram, which can then be translated to tile the plane. But the condition on this pentagon that two angles add to 2 is not one that can be fulfilled by a convex pentagon, so this gives a different class of pentagon tilers than any of the convex ones. Even when the side lengths are not equal the  rotation gives a hexagon that meets the Conway criterion. (This example is simple enough that we can probably find a source mentioning it.) —David Eppstein (talk) 17:57, 28 August 2015 (UTC)
replies: text: See http://web.hku.hk/~amslee/mathbook/attachments/tile/ down near the bottom for further consideration of concave pentagon tiling with rotational, but no slide symmetry...Naraht (talk) 18:37, 28 August 2015 (UTC)
replies: text: I see expansion of our non-convex section on the Horizon. :)Naraht (talk) 18:43, 28 August 2015 (UTC)

text: I added my own "checkerboard" (Square tiling checkerboard as concave dodecagons.svg) on the right. We can call those dodecagon faces, but there's no limit to such expansions. Tom Ruen (talk) 20:03, 28 August 2015 (UTC)

text: The Java application has parameters for all the types, and I've not yet attempted a summary graphic of variations for types 1, 2, and avoided the cases on my first pass for types 3,4. On a quick look, types 3,10,11,12,14,15 are only convex, and 1,2,4,5,6,7,8,9,13 can be concave. Tom Ruen (talk) 19:12, 28 August 2015 (UTC)

text: Interesting. These are different from the ones I've mentioned: the left and right ones have degree three at some or all of the concave angles, and the middle one, while closely resembling "Pentagonal checkerboard II", is different as the two equal sides are non-adjacent in PC2 and adjacent in the one you link. It would be good if we could find sources for more of these; I think there's room in the article for some of them but I don't think we should add them with only the applet as source. —David Eppstein (talk) 03:04, 5 September 2015 (UTC)
thread_title:  Type 5 examples -Convexity 

message: ====From the 15====
From the examples given in the article, both 5 & 6 have non-convex versions, type 1 definitely has one and type 14 and 15 don't (since they are fully constrained). Which of the others can the degrees of Freedom give a non-convex one?Naraht (talk) 18:49, 28 August 2015 (UTC)
replies: text: The Java application has parameters for all the types, and I've not yet attempted a summary graphic of variations for types 1, 2, and avoided the cases on my first pass for types 3,4. On a quick look, types 3,10,11,12,14,15 are only convex, and 1,2,4,5,6,7,8,9,13 can be concave. Tom Ruen (talk) 19:12, 28 August 2015 (UTC)

text: Interesting. These are different from the ones I've mentioned: the left and right ones have degree three at some or all of the concave angles, and the middle one, while closely resembling "Pentagonal checkerboard II", is different as the two equal sides are non-adjacent in PC2 and adjacent in the one you link. It would be good if we could find sources for more of these; I think there's room in the article for some of them but I don't think we should add them with only the applet as source. —David Eppstein (talk) 03:04, 5 September 2015 (UTC)
thread_title: Expansion

message: From the examples given in the article, both 5 & 6 have non-convex versions, type 1 definitely has one and type 14 and 15 don't (since they are fully constrained). Which of the others can the degrees of Freedom give a non-convex one?Naraht (talk) 18:49, 28 August 2015 (UTC)
replies: text: The Java application has parameters for all the types, and I've not yet attempted a summary graphic of variations for types 1, 2, and avoided the cases on my first pass for types 3,4. On a quick look, types 3,10,11,12,14,15 are only convex, and 1,2,4,5,6,7,8,9,13 can be concave. Tom Ruen (talk) 19:12, 28 August 2015 (UTC)
thread_title: From the 15

message: In addition we've got David's split parallellogram and the one in the image where those two sides that aren't part of the 2 angles. We've got (from the HK link) those that form Polygons and expansions of Polygons (with rotational symmetry), we've also got those tilings derived from Reptiles.Naraht (talk) 18:49, 28 August 2015 (UTC)
replies: text: Interesting. These are different from the ones I've mentioned: the left and right ones have degree three at some or all of the concave angles, and the middle one, while closely resembling "Pentagonal checkerboard II", is different as the two equal sides are non-adjacent in PC2 and adjacent in the one you link. It would be good if we could find sources for more of these; I think there's room in the article for some of them but I don't think we should add them with only the applet as source. —David Eppstein (talk) 03:04, 5 September 2015 (UTC)
thread_title: Other than the 15

message: These remarks relate to the section 'Monohedral convex pentagonal tilings'.
The definition of transitivity classes does not permit glide reflections to be ignored, but coloring the tiles may change the transitivity classes as a color-preserving symmetry must map each tile onto one of the same color.
The colored tilings should be described as such or, preferably, removed. (These are not the tilings where color is used only to distinguish transitivity classes.) Their inclusion is a space-consuming distraction superfluous to an account of monochromatic tilings. Also, it may be considered to be unsourced original research. Kinewma (talk) 00:37, 19 September 2015 (UTC)
replies: text: I'm happy if you can provide a better description. I agree including two distinct (isohedral) colorings (with and without glides) takes more space right now. An alternative presentation might only include the larger number of colors, but state that glide reflections map chiral pairs in different shades of the same color. What's uploaded now is close to that and colors could be tweaked for clarity. Tom Ruen (talk) 00:55, 19 September 2015 (UTC)
thread_title:  Glide reflections and colored tilings 

message: In the Talk section of the German article came a "Thank you" for the extensive artistic graphic work to which I served myself (edit history respected). Herewith I pass this "Thank you" on to the real artists.
Side question: Besides the (probably older then Reinhardt (?)) tiling in the streets of Cairo: Do yo know of any "natural appearances" of those pentagons? Crystals, quasicrystals, proteins, other mineralogical / chemical / biological findings? ! Bikkit ! (talk) 08:10, 19 September 2015 (UTC)
thread_title:  Forwarded "Thank you" 

message: Many of the so-called-examples given for the different types are not even pentagons, let alone convex pentagons.  For instance, only 2 of the 8 illustrations for type 5 are true examples of the type.  This page is getting pretty complicated, so eliminating the extraneous examples might help prevent confusion.  --Lasunncty (talk) 08:05, 6 October 2015 (UTC)
replies: text: I note that the image in question was one of 's. He contributes many good images to the project, and has a lot of expertise on tilings and polyhedra to contribute, but he also tends to stuff the articles he works on full of huge, sloppily-sourced or unsourced image galleries of dubious relevance to the article and dubious mathematical rigor, far out of proportion to the actual text of the articles. This article, more than many of them, is one where a lot of images can be of great benefit in understanding the subject, which is why I've been staying quiet about this, but it seems to have gone the same way. —David Eppstein (talk) 03:58, 7 October 2015 (UTC)

text: A new editor added a few nice smooth and slow animations that should help show geometric variations (with nonconvex prototiles as well as colinear edge cases). OTOH, the variations of Types 1 and 2 degrees of freedom variations are vast and I've not been brave enough to try anything there. Tom Ruen (talk) 21:38, 7 October 2015 (UTC)

text: I decided to put the new animations by User:Another_Matt into the summary tables and remove my static examples. Tom Ruen (talk) 05:48, 11 October 2015 (UTC)
replies: text: Thanks for working on this.  The animations really make a difference.  --Lasunncty (talk) 08:02, 11 October 2015 (UTC)
thread_title:  "examples" 

message: These comments relate to the section "Monohedral convex pentagonal tilings".
I feel that the relationship between pentagon type and tiling topology is not always explained as clearly as it could be. For example, in the description of the later types, there seems to be almost an implicit assumption of one-to-one correspondence, although this was shown not to be true for some earlier cases. It would be nice if the article could explicitly state how many tiling topologies exist (or are known to exist) for each pentagon type (and vice versa if appropriate), or to state that this is unknown if that is the case. 
It is not clear to me why an extremely broadly defined "type 1" pentagon (and to a lesser extent "type 2") is then split into multiple more tightly-defined cases with different tiling topologies. Why aren't these separate types in their own right? It seems arbitrary.
A reference is made to a "standard tiling exhibited by all members", but it is not clear in the multi-topology cases which topology is "standard", or who defines what is "standard".
I cannot make any sense of the statement "B. Grünbaum and G. C. Shephard have shown that there are exactly twenty-four distinct 'types' of isohedral tilings of the plane by pentagons. All use Reinhardt's tiles." What is a "type" of tiling here? What about the isohedral examples demonstrated that do not use Reinhardt's tiles? Are they not "types" of tiling? 86.152.163.198 (talk) 01:10, 9 October 2015 (UTC)
replies: text: I largely agree. It is confusing to connect completely different tiling topologies with the same pentagonal type, and no topology can be considered standard, except perhaps arbitarily what forms have been previously presented. My only goal was to show some example tiling variations, within different symmetries, without considering them complete or representative for the full set of possibilities. Tom Ruen (talk) 10:29, 10 October 2015 (UTC)

text: All isohedral (=tile-transitive) tilings use tile types 1-5, usually with additional conditions necessary for the tiling. The statement that B. Grünbaum and G. C. Shephard 'have shown that there are exactly twenty-four distinct "types" of tile-transitive tilings by pentagons according to their classification scheme' is on page 33 of D. Schattschneider's 'Tiling the Plane with Congruent Pentagons'. There's a link to the article at http://www.maa.org/programs/maa-awards/writing-awards/tiling-the-plane-with-congruent-pentagons (JSTOR access not needed). Kinewma (talk) 07:31, 26 October 2015 (UTC)
replies: text:  Thanks for the link. I hadn't been able see this important article before. Tom Ruen (talk) 11:44, 26 October 2015 (UTC)

text:   The last page contains an example tiling with 6-fold symmetry. I colored it. I also see they put dots in the middle of mirror image tiles. I imagine it's nontrivial to prove such a tiling can continue forever. Tom Ruen (talk) 01:55, 28 October 2015 (UTC)
thread_title:  A few comments 

message: Hi Tom, apropos of the recent activity on pentagonal tiling, I thought you might be interested in the tiling at :File:Exotic pentagonal tiling.png, if you haven't seen the pattern before. I don't particularly propose adding it to the article ... I just uploaded it for interest. Another Matt (talk) 01:51, 23 October 2015 (UTC)
replies: text: Are there any sources for aperiodic pentagonal tilings? Tom Ruen (talk) 09:08, 24 October 2015 (UTC)

text: I outlined a "base" edge in color, and then colored seemingly 22 radial subtilings. Tom Ruen (talk) 09:45, 24 October 2015 (UTC)
thread_title:  Aperiodic pentagonal tiling

message: Below is the pentagon and its tiling. It seems different compared to Type2 [with regards to which two edges are equal], and from Type4 [where two angles need to be 90 degrees, which results in edge to edge tilings]. Thoughts?
:
 
Satyrchary (talk) 01:43, 20 July 2016 (UTC)
thread_title:  Is this a new tile type and tiling? 

message: I suppose the answer will be "because the source says so", but why does Type 1 have two conditions listed in the first table? B+C=180 and A+D+E=360 are the same condition. kennethaw88 • talk 22:36, 10 May 2017 (UTC)
replies: text: Yes, its redundant but harmless. Not everyone instantly thinks about the sum of all the angles of a pentagon. Tom Ruen (talk) 22:44, 10 May 2017 (UTC)
thread_title:  Type 1 

message: By Michaël Rao, dated May 1, 2017. Here:  at 
Alas, I had put a lot of effort into a very similar approach, but have let the project languish. Beaten to the punch!
This evidently closes out the question of which convex polygons tile the plane, first posed a century ago. Of course the result has yet to be independently verified. Peer review will be somewhat complicated. Not only must the math and logic be checked, but so must the 5,000-line search program. Bobhearn (talk) 15:55, 22 May 2017 (UTC)
replies: text: This new paper lists 24 types - I see, types 16-24 are subsets of the first 15 or degenerate. It's surprising there are no news articles about it yet, since the pentagonal problem was in the news when type 15 was discovered. Tom Ruen (talk) 18:51, 22 May 2017 (UTC)
replies: text:  Well, it's not published yet. Also not as exciting, even though really, it's a more significant result IMO. Bobhearn (talk) 23:26, 22 May 2017 (UTC)
thread_title:  Claimed exhaustive proof that there are only 15 types of monohedral convex tiling pentagons 

message: Since there is a special section Pentagonal tiling#Nonperiodic monohedral pentagons tilings shouldn't one oppose to them the foregoing 15 types as «Periodic monohedral pentagonal tilings» ? (Because the former «tile the plane» as the latter do!) --Nomen4Omen (talk) 10:31, 24 August 2017 (UTC)
thread_title:  '''Periodic''' monohedral pentagonal tiling 

message: Above, there is some discussion about chiral pairs. Obviously, the type-1 tile allows both, tilings with chiral pairs (e.g. Lattice_p5-type1_cm.png) and tilings without (e.g. Lattice_p5-type1.png).
For tile type-2 only primitive units with chiral pairing are shown (Lattice_p5-type2.png and Lattice_p5-type2b.png). For tiles type-4 and type-5 only primitive units without chiral pairing are shown (Lattice_p5-type4.png and Lattice_p5-type5.png). Do these types of tiles force that there is not a tiling without chiral pairing (type-2) or not a tiling with chiral pairing (type-4 and type-5) ? --Nomen4Omen (talk) 16:04, 25 August 2017 (UTC)
thread_title:  Which types allow tilings with chiral pairs, which don't ? 

message: As I understand Proof by intimidation then proofs by intimidation should be avoided. Unfortunately, in Pentagonal tiling#Stein (1985) Type 14 one can read "Other relations can easily be deduced." about type 14. This may be easy for some people, but it is not for me (and probably for several other readers). It would be really great if any of you experts could help and explain or add a link to an explanation. Thanks a lot for that! 130.238.112.129 (talk) 18:12, 24 July 2023 (UTC)
thread_title:  easy or not