Talk:Pentagonal tiling

Set of 14 isohedral pentagonal tilings

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)

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)

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)

{5,∞} tiling

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)

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)

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 [1]. 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)

Wow, awesome.  :) Thanks! TricksterWolf (talk) 21:16, 4 January 2012 (UTC)

what are these type of tilings called

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)

They are called monohedral. —David Eppstein (talk) 16:05, 17 March 2014 (UTC)

Angle measure

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)

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)

Additional sources for 15th convex pentagon tiling

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)

Hexagonality

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)

I just added a subsection on the pentagon's relation to the hexagon. - Catsquisher (talk) 22:24, 20 August 2015 (UTC)

Edge-to-edge tilings?

I looked at the 15 tilings listed and recolored as if non-edge-to-edge tilings represented multiple edges, and find 6 of 15 are pure edge-to-edge, while others are mixed. Is this useful to distinguish? I did it quick with MSPaint, with some filling unclear on tiles along boundaries. Tom Ruen (talk) 02:14, 17 August 2015 (UTC)

I like the idea, and it makes for some beautiful coloring. — Sebastian 16:01, 20 August 2015 (UTC)

Set of 15 isohedral pentagonal tilings

— Preceding unsigned comment added by EdPeggJr (talk • contribs) 21:54, 3 August 2015 (UTC)

Reddit discussion - 15th Pentagon has a write-up by Casey Mann and links to the necessary data. The motif of the tiling uses 12 pentagons. — Preceding unsigned comment added by EdPeggJr (talk • contribs) 22:03, 3 August 2015 (UTC)

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 [2] Nil Einne (talk) 19:16, 17 August 2015 (UTC)

This article on in the news?

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)

I certainly thought it was interesting and informative. Czolgolz (talk) 22:51, 17 August 2015 (UTC)

Okay, thanks. =D This article is good. }I6ixce93IxI{ 22:53, 17 August 2015 (UTC)

I think this is an important mathematical discovery for people worldwide too. Deryck C. 10:20, 18 August 2015 (UTC)

I agree. :) }I6ixce93IxI{ 12:29, 18 August 2015 (UTC)

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)

Showing the families

I think this page would be considerably more useful if we could copy the equations and the examples from http://www.mathpuzzle.com/tilepent.html . How close can we get without a problem? The equations can't be trademarked, right?

Depending on the source there are different constraint sets (equation sets) mapped to different type ids. For instance, the types 2 and 3 described within source Reinhardt, 1918 do not correspond to the (brief) constraints stated for the same ids used by wolfram and mathpuzzle.com. However, it seems as if Wikipedia, Wolfram and Mathpuzzle agree on a specific mapping between a constraint set to a specific type id (in the range 1-15). Also, some types are a subset of others, i.e. having more or more specific constraints added compared to another. Some specific instances of a pentagon fit more than one type, or the other way around: formulas of different types may produce the same pentagon shape, that is. As for the mathpuzzle.com examples one needs to work with the sources to find out whether the brief equation sets really resemble (all of) the class the example picture stands for. "Have fun" and please sign your comments next time. --Cmuelle8 (talk) 07:37, 18 August 2015 (UTC)

That's a good article. I was also interested in isohedral colorings, and used images from it to draw some, so 5 seem to be 1-isohedral, 7 are 2-isohedral, and 3 are 3-isohedral. I moved #12 to the top because it has a topological hexagonal tiling like the first three. Tom Ruen (talk) 09:13, 18 August 2015 (UTC)

For type #1 - if D==E then it's topology is both, prismatic and hex, isn't it? --Cmuelle8 (talk) 20:20, 19 August 2015 (UTC)

Circular reference?

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)

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)

This webpage has the 15 tilings, including k-isohedral colorings. [3] Tom Ruen (talk) 11:48, 18 August 2015 (UTC)

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)

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)

Rice tiling illustration

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)

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)

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)

@David Eppstein, 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)

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)

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

Rice pentagonal tilings

Index	9	11	12	13
PNG
SVG

It might be helpful to use code for these pentagons. http://community.wolfram.com/groups/-/m/t/550169 has exact values for the motif and offset for the fifteenth pentagon, and http://demonstrations.wolfram.com/PentagonTilings/ has code for all of the pentagons. All of the Marjorie Rice tilings can be modified, so it would be better to show a range or an animation for each tiling. EdPeggJr (talk) 20:33, 19 August 2015 (UTC)

A Java Applet here can draw them, with parametric sliders for variations. [4] 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)

Incorrect angle on 15th tiling

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)Anonymous

Good catch! I agree that looks correct, confirmed here [5]. 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)

Optimal set of pentagons for the 15th tiling

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)

Meaning of bottom part of each box?

What do the entries like pgg (22×) mean? If they aren't explained, they should be deleted.Naraht (talk) 02:21, 20 August 2015 (UTC)

I added a description above the first table: Tom Ruen (talk) 10:53, 20 August 2015 (UTC)

The wallpaper group symmetry of these tilings are rotational p2 (2222), p3 (333), p4 (442), p6 (632), and pgg (22×), the last one which includes glide reflections. <Tom Ruen>

If the colored tilings are not removed immediately (as they make the article harder to follow and are unsourced and original research), at least their alleged symmetry groups should be corrected to the color types. A color type identifies all the symmetries of a plane pattern. Color types are not (yet) described in Wikipedia but in the book 'The Symmetries of Things' (but its list of 3-fold color types misses 22*//o ). Kinewma (talk) 20:33, 28 September 2015 (UTC)

The representation of color-symmetries in 'The Symmetries of Things' is slightly different, more towards repeated elements in complex patterns than solid colored domains, but could apply here in the case of coloring with chiral pairs distinct. So that notation shows A // B, where A is the uncolored symmetry, and B is the (usually lower) colored symmetry. ... No what Conway is doing is more complicated than that. A / B for 2-color symmetry anyway.

Anyway, so could keep the higher-colored images, and show a combined symmetry notation.

So we could reduce this duplication: p1 (°) p2 (2222) cm (*×) pmg (22*) pgg (22×)

Into: *× / ° 22* / 2222 22× / 2222 cm / p1 pmg / p2 pgg / p2

If we did this, I agree it would be good to add color-symmetry notation somwhere on wikipedia, possibly at Orbifold notation since that's what Conway uses. Tom Ruen (talk) 03:36, 29 September 2015 (UTC)

I reduced them to single images. I put a second row for the symmetry in cases where there are chiral pairs and they are considered distinct. I don't see Conway's slash notation helpful for now. Tom Ruen (talk) 06:50, 29 September 2015 (UTC)

Illustrations don't match [2 pgg (22×) c=e B+D=180]

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)

Roses and other distinct tilings from special cases

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[6] 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: Hexagonal tiling#Topologically equivalent tilings), shouldn't we include those tilings here? — Sebastian 21:22, 20 August 2015 (UTC)

Irregular tilings

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)

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)

Here's some pentagonal examples with more than 1 type of pentagon, made from k-uniform tiling duals. They should be colored by pentagon type or k-isohedral positions. Tom Ruen (talk) 14:54, 23 August 2015 (UTC)

2-uniform duals		3-uniform duals			4-uniform duals

Wrong isohedral colorings in some types

I think some of the isohedral colorings are wrong, Observe that whenever a symmetry group contains glide reflections (that is, types 2, 7, 8, 9, 11, 12, 13 and 15, whose group of symmetry is pgg) there are chiral pairs of tiles and the number of orbits "ignoring glide reflections" MUST be twice the number with glide reflections considered (that is, the number of p2-orbits is twice the number of pgg-orbits). In the text it was said several times that a 2-isohedral tiling became 3-isohedral when glide reflections are neglected, which is wrong. In all these cases the 3 must be a 4. After I edited to correct this Tom Ruen put back one of the 3's. I think he is mislead by the pictures, which sometimes do not split colors of a pgg orbit into two colors for the p2 orbits.

For example, in type 7 (the one that Tom corrected back) the "chains" of yellow pentagons in the following picture should alternate colors; each connected chain of yellow pentagons stays of one color, but colors alternate from one chain to the next.

I have corrected (back) the text, but someone please correct the pictures... — Preceding unsigned comment added by Pacosantosleal (talk • contribs) 17:19, 23 August 2015 (UTC)

Let me be more precise on what is needed.

1) The following pictures (types 7, 9, 11 and 13, "chiral coloring") should be changed to use four colors. In each of them, one of the classes of the 2-isohedral coloring has not been split into its two chiral classes.

2) A chiral version of the following pictures (which do not take chirality into account despite the file names) should be made and incorporated into the tables (types 8 and 12):

Pacosantosleal (talk) 17:35, 23 August 2015 (UTC)

Got it. Thanks! I expanded the 4- Rice tilings for 4-colors after looking more carefully at the fundamental domains. Tom Ruen (talk) 18:11, 23 August 2015 (UTC)

Okay, I think I've got them all. Tom Ruen (talk) 18:19, 23 August 2015 (UTC)

Everything looks fine now. Thank you! Pacosantosleal (talk) 19:24, 23 August 2015 (UTC)

Illustrations

(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)

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)

Pentagonal Tiling Category?

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)

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)

Family size and samples

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)

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)

p.s. According to the Java applet at least, this is the number of degrees of freedom for each type. (Ignoring rigid scale and orientation) That actually doesn't look to bad after 1 and 2. Tom Ruen (talk) 16:41, 26 August 2015 (UTC)

Type	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Degrees	5	4	1	2	2	1	1	1	1	1	1	1	1	0	0

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)

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)

AH, I see this 2012 paper gives them rounded to 1/100ths: [7] p.94: 90°, 145.34°, 69.32°, 124.66°, 110.68°. Tom Ruen (talk) 17:32, 26 August 2015 (UTC)

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)

Suggested clarifications

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)

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)

By the way, the colourings are not explained as clearly as they could be either. "the tiles are also colored by their k-isohedral positions within the symmetry" is hard to understand for lay people, and seems crying out for an ordinary-language "in other words , ..." explanation. Phrasings like "contains glide reflections which if ignored, contain chiral pairs of tiles, colored here as yellow and green" I can't understand at all. I cannot fathom why some patterns have multiple colourings and some only one, and whether the initial explanation of what the colouring means applies to all colourings or only some. 31.49.126.141 (talk) 20:12, 26 August 2015 (UTC)

Type 5 examples -Convexity

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)

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)

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)

Pentagonal checkerboard.svg: Non-convex pentagonal tiling with two consecutive angles adding to 2π

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)

I'm pretty sure that non-convex pentagons have more examples. In particular, consider any pentagon where two consecutive angles add to 2π 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 π) to form a parallelogram, which can then be translated to tile the plane. But the condition on this pentagon that two angles add to 2π 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 π 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)

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)

I see expansion of our non-convex section on the Horizon. :)Naraht (talk) 18:43, 28 August 2015 (UTC)

Square tiling checkerboard as concave dodecagons.svg: Square tiling example as nonconvex dodecagons

p.s. David's Pentagonal checkerboard.svg example offers another "degeneracy" that can exist on concave cases, having two prototiles in contact on more than 1 edge (i.e. having valence-2 vertices). Such cases can't happen on parametric variations of convex ones. Tom Ruen (talk) 19:25, 28 August 2015 (UTC)

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)

Expansion

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)

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)

Other than the 15

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 2π 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)

Pentagonal checkerboard II.svg: another non-convex pentagonal tiling

Pentagonal checkerboard II.svg is another one. It works for non-convex pentagons in which two non-adjacent angles add to 2π and two edges adjacent to these angles have equal lengths (with the equal edges either both clockwise from the equal angles or both counterclockwise from them). I've drawn it in this example with two parallel sides but that's not a requirement for this pattern; it also works when the sides are not parallel. I don't know of a source for this (if I did I would add it to the article). —David Eppstein (talk) 21:49, 3 September 2015 (UTC)

There's a scarily huge number of nonconvex pentagonal tilings and variations. Here are three interesting cases, with p1, pg, pgg symmetry, 1 degree of freedom, known as NC5-73a and NC5-112b, NC5-71c in Jaap's applet. Tom Ruen (talk) 22:51, 3 September 2015 (UTC)

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)

Mystery convex monohedral type

I saw this link added and removed [8] and looked at the generalized form, and I can't see it matching any of the 15 types. It has p2 symmetry and 2-isohedral, so not types 1-5, and its length and angle constraints don't clearly match any of the others. Tom Ruen (talk) 18:38, 3 September 2015 (UTC)

The tiles are type 1. There are many examples of tiles that can be arranged in additional ways beyond the main tiling of its type, but that does not in itself constitute a new type of tile. Dragons flight (talk) 18:44, 3 September 2015 (UTC)

I'm not at all sure of that. All examples for type-1 are 1-isohedral. If type-1 allows variations that are 2-isohedral, that contradicts my intuition of what is possible. This tiling DEMANDS one angle be 90°, and can't exist otherwise. Tom Ruen (talk) 18:55, 3 September 2015 (UTC)

Jaap Scherphuis, author of the applet used in most of the images here said it is type-1, but he classified under a name N5-1u, one of 81 parametric cases he lists as type 1, 2-isohedral. So anyway, it shows the wikipedia article has to be expanded on the definitions of types 1 (and 2) apparently. Tom Ruen (talk) 19:25, 3 September 2015 (UTC)

Jaap also replied: Tom Ruen (talk) 19:41, 3 September 2015 (UTC)

You need to differentiate between a tile and a tiling. The tile used in this tiling is a type 1 tile - it has two parallel sides. In order for it to be able to create this particular tiling, this tile is more restricted than the normal type 1 tile, in that on top of the two parallel sides (i.e. C+D=180), it also has one angle equal to 90 and another equation on the angles (2B+C=360). There are many beautiful tilings in my applet that use a type 1 tile with extra restrictions, but each of those tiles is also able to form the standard "unrestricted" general type 1 tiling, so are not considered a new pentagon type.

Lattice or translation cluster?

What's a 2-tile lattice? Would 2-tile translation cluster be a better description? Kinewma (talk) 04:24, 5 September 2015 (UTC)

It might help to link Lattice (group). This page from WolframAlpha calls the set a primitive unit [9] There's also Primitive cell. Tom Ruen (talk) 04:55, 5 September 2015 (UTC)

Jaap defines a primitive unit or unit parallelogram as 'A section of the tiling (usually a parallelogram or a set of neighbouring tiles) that generates the whole tiling using only translations, and is as small as possible.' This is exactly what we want. A lattice is a set of points, not a set of neighbouring tiles. I'll change lattice to primitive unit. Kinewma (talk) 07:43, 5 September 2015 (UTC)

I changed to primitive cell since it is named that way in Wikipedia. Tom Ruen (talk) 08:12, 5 September 2015 (UTC)

Primitive cell is not the term we want. Wikipedia says that a 2-dimensional primitive cell is a parallelogram, but the section of the tiling we want to describe is not. D. Schattschneider says that a lattice unit is a parallelogram, and that primitive cell is the term crystallographers use for a lattice unit. She does not use the term primitive unit but does say "A smallest region of the plane having the property that the set of its images under this translation group covers the plane is called a unit of the pattern." See Schattschneider, Doris (1978), "The Plane Symmetry Groups: Their Recognition and Notation" (PDF), American Mathematical Monthly, 75 (6): 439–450. Kinewma (talk) 04:29, 22 September 2015 (UTC)

We can look further for usages for a best term, but the set of tiles is directly related to the fundamental parallelogram, as this shows for type 15 tiling. every area outside the parallelogram could be cut and pasted inside the parallelogram on the opposite side. Tom Ruen (talk) 04:17, 22 September 2015 (UTC)

The parallelogram is a primitive cell which, being in a pgg tiling, is four times the area of a fundamental domain. A fundamental parallelogram is something else, defined on the complex plane. Kinewma (talk) 05:33, 22 September 2015 (UTC)

I'm not sure what we're arguing but the term fundamental parallelogram is also used as a translational domain, even if the wlink talks explicitly about the complex plane. the number of fundamental domains within the translational domains of a specific wallpaper group is show in diagrams, like here Wallpaper_group#Group_p1_.28o.29, or for 22x File:Wallpaper group diagram pgg.svg, the yellow fundamental domain is 1/4 of the full rectangle as you say. Tom Ruen (talk) 07:00, 22 September 2015 (UTC)

Illustrate more tilings by subsets of pentagons?

Subsets of type 1 and type 2 convex pentagons give the following tilings not yet illustrated:

• Type 1 cm (IH22) a=e, b=d, B+C=180°, 2-tile translation cluster;
• Type 1 pmg (IH24) b=d, B+C=180°, 4-tile translation cluster;
• Type 1 pgg (IH25) a=e, b=d, B+C=180°, 4-tile translation cluster;
• Type 2 pgg (IH27) b=d, c=e, B+D=180°, 4-tile translation cluster.

For details, see the IHxx pages at http://freespace.virgin.net/tom.mclean/html/pentagon_templates.html Kinewma (talk) 04:44, 5 September 2015 (UTC)

I see IH1-IH93 are 93 edge-to-edge isohedral tilings, from "Tilings and Patterns, 8.2. The classification of isohedral tilings", pp.282-295, expressing curved-edge tiles. I also see that website uses translation cluster, but its not clear anyone else uses that term. Tom Ruen (talk) 05:53, 5 September 2015 (UTC)

Here's IH21-29 (pentagonal isoheedral tilings), scanned and colorized by translational positions for reference: Tom Ruen (talk) 07:59, 5 September 2015 (UTC)

I added examples for type 1, IH22,24,25. Tom Ruen (talk) 09:33, 6 September 2015 (UTC)

Mathematical Tourist

There once was a lady called Rice
whose pentagons tiled really nice.
The result of her hobby
you can find in that lobby.
Consider it, once, maybe twice... ! Bikkit ! (talk) 08:02, 14 September 2015 (UTC)

Nice Limerick, but I'm confused on something. The tiling in the lobby clearly has places where six of the Pentagons come together, which means an angle of 60 degrees (and this would be a fixed angle for that particular tile (some angles aren't fixed but that one would be). But none of Rice's tiles have a fixed 60 degree angle. I understand there might have been some changes adopting it to a floor tile, but until we can tie to a specific one of the tiles (9,11,12, or 13), I'm not sure it belongs on this page. (The wikipedia article on Rice, sure.)Naraht (talk) 13:29, 14 September 2015 (UTC)

Rice's tiling is apparently a special version of type 5. Jaap's applet has it as a 3-isohedral variation, labeled N5-1r5, with no degrees of freedom (a=b=c, d=e, A=60, B=120, C=90, D=120, E=150). Tom Ruen (talk) 22:20, 14 September 2015 (UTC)

Which doesn't show up anywhere in our "degrees of freedom" examples. In our Type 5 examples, all of the 60 degree angles for the pentagons are in the 6 way meeting points with the other 6 degree angles (so *all* of the Pentagons are in the "flowers", in the floor tile, they aren't. (However it does build 18 pentagon "flowers") Naraht (talk) 18:22, 15 September 2015 (UTC)

Indeed, another example of a type 5 we could document. (I also discovered it fits within a regular triangular tiling, with each irregular pentagon combining 2 full triangles, and 1/3 of a triangle divided in the center, shown here with overlay [10].) Tom Ruen (talk) 19:13, 15 September 2015 (UTC)

Two points. This article could easily be broken into Pentagonal tiles and Pentagonal tiling. And documenting all of the possible Pentagonal Tilings (especially with types 1-5) would both be massive *and* could *easily* stray into WP:OR.Naraht (talk) 19:55, 15 September 2015 (UTC)

This tiling does not use a Type 5 tile, but a Type 1 tile (two parallel sides, i.e. two adjacent corners add to 180). - Jaap. 83.84.24.226 (talk) 21:20, 15 September 2015 (UTC)

Okay, I see. It is consistent with both Type 1 and 5. (a=b=c, d=e, A=60°, B=120°, C=90°, D=120°, E=150°)

Type 1: (B+C=180°, A+D+E=360°) - consistent, with a bit of label renaming.

Type 5: (a=b, d=e, A=60°, D=120°) - consistent. Which one takes priority? I'd judge as the type with the constraints as closest. Tom Ruen (talk) 22:42, 15 September 2015 (UTC)

I don't think there is such a things as priority, The Dual Uniform tilings include *both* that they fulfill. My question is whether it is possible to have one that falls into more than two types.Naraht (talk) 00:36, 16 September 2015 (UTC)

It may be OR, but seems useful and direct to draw a hierarchical graph of the 15 types from most general to most constrained types. Tom Ruen (talk) 06:02, 16 September 2015 (UTC)

I imagine more of a Venn diagram. There are no types that completely overlap, but many types (other than 14/15) permit overlaps with other types. Dragons flight (talk) 06:24, 16 September 2015 (UTC)

BY DEFINITION there are no types that perfectly overlap. There are tilings that have types that are strict subsets of the types of other tilings, but they are omitted from the list of 15, because that list is about the types of tilers, not about all the possible tilings, so it doesn't include redundant types. As for making a Venn diagram or any other kind of inclusion diagram: without sourcing that would be original research and we shouldn't do that here. —David Eppstein (talk) 07:25, 16 September 2015 (UTC)

Agreed no perfect subset overlap, but some special cases can overlap, while others can never overlap. Don't worry, if I attempted something, and it made some sense, I'd add it to the talk page here. ;) Tom Ruen (talk) 07:37, 16 September 2015 (UTC)

p.s. I mostly gave up, but see any "intersection" of 2 or more pentagon types leads to a new "type" that is more constrained than any of the original types, but its worse than that. It's possible two types can combine in different "orientations" leading to different combined constraints, even if most orientations won't be compatible at all, they still must be checked. So a more meticulous person than me might be able to sort through all that. Tom Ruen (talk) 05:32, 17 September 2015 (UTC)

Glide reflections and colored tilings

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)

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)

Forwarded "Thank you"

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)

"examples"

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)

For example, 4 of the block of 8 so-called type 4 examples are not tilings by pentagons. Kinewma (talk) 03:46, 7 October 2015 (UTC)

I note that the image in question was one of Tomruen'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)

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)

Type 4	Type 5	Type 6	Type 7	Type 8
Type 9	Type 10	Type 11	Type 12	Type 13

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)

Thanks for working on this. The animations really make a difference. --Lasunncty (talk) 08:02, 11 October 2015 (UTC)

A few comments

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)

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)

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)

Thanks for the link. I hadn't been able see this important article before. Tom Ruen (talk) 11:44, 26 October 2015 (UTC)

Hirschhorn 6-fold rotational symmetry monohedral pentagonal tiling

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)

Aperiodic pentagonal tiling

Aperiodic pentagonal tiling

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)

Are there any sources for aperiodic pentagonal tilings? Tom Ruen (talk) 09:08, 24 October 2015 (UTC)

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

Is this a new tile type and tiling?

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)

Edited 02:49, 20 July 2016 (UTC)

Oops - I just realized this is a Type1 pentagon (E+A+B=360 degrees), so it is NOT a new type.

Type 1

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)

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)

Claimed exhaustive proof that there are only 15 types of monohedral convex tiling pentagons

By Michaël Rao, dated May 1, 2017. Here: [11] at [12]

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)

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)

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)

Periodic monohedral pentagonal tiling

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)

Sorry, got it. The nonperiodic tilings are only special tilings of an already described type of tile (with periodic tilings), namely type-1. --Nomen4Omen (talk) 09:45, 25 August 2017 (UTC)

Which types allow tilings with chiral pairs, which don't ?

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)
Another observation regarding chiral pairing: type-15's primitive unit Lattice_p5-type15.png has 4 clockwise and 4 counter-clockwise tiles. Moreover, its mirror image occurs in the (only) tiling.
Not so with type-14's primitive unit Lattice_p5-type14.png: There, the 2 red and 2 yellow tiles have the same orientation and the 2 blue tiles are reflections of both. This means that the mirror image of this primitive unit cannot occur in the tiling of type-14. (So there are kind of 1½ tilings of type-14.)
A 3rd observation regarding chirality: The primitive unit of type-10 looks very similar to type-14 with the 2 red tiles being reflections of the 2 blue and the 2 yellow ones. But here, the equations do not change when exchanging B<->E, C<->D, b<->a, c<->e. So the degree 1 of freedom can be used to find the reflected primitive unit in the type-10 tiling.
Are there a technical terms for these behaviors? --Nomen4Omen (talk) 17:50, 25 August 2017 (UTC)

easy or not

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)