More tab-in-slot Penrose tiles

[pcallahan]

I think I have found about the right size for cutting Penrose tiles out of ~0.25mm thick cardstock. The long edges of the kites and darts are 25mm or a little under an inch. Smaller than that, they're hard to fit together. I can make them bigger, and they get a little easier to work with. I just prefer them to be smaller for making more elaborate patterns.

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Here are a lot of them assembled into a symmetric patch:

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It's actually a little more interesting viewed from below.

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It takes some care to keep the ordering of the tabs consistent and some of these are probably not. This is supposed to be hidden.

[Entity Valkyrie 2]

Gorgeous! The design resembles the flag of Greenland

[breaker's glider gun]

I think I have found about the right size for cutting Penrose tiles out of ~0.25mm thick cardstock. The long edges of the kites and darts are 25mm or a little under an inch. Smaller than that, they're hard to fit together. I can make them bigger, and they get a little easier to work with. I just prefer them to be smaller for making more elaborate patterns.

what did you use to make them? I'm curious what made them so regular.

[pcallahan]

what did you use to make them? I'm curious what made them so regular.

I cut them out on a Cricut. It's a die cutter that can be used for cutting many materials as long as they're thin enough.

I didn't take a picture of these tiles on the mat, but here's an example of printer paper with top and bottom (flipped) of aperiodic monotiles that I posted recently.

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The tab-in-slot tiles were much simpler to make because I only needed to cut one layer.

[pcallahan]

Here's a more elaborate tab-in-slot design, but it requires no gluing, only folding, as shown below. When making a tiling, long tabs go under a fold while the short ones (Bezier bumps) fit back into the slots that the long tabs protrude from. The long side of each kite or dart is 2cm, so the tiles are small but still usable. Unlike the single layer design, the tilings are easily disassembled. They still hold well when putting them together but can be a little fiddly and requires some practice. If I can do it, just about anyone can. They pass a drop test as long as you don't throw it hard intentionally and even then it will mostly hold.
Design:

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Top:

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Bottom:

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[pcallahan]

I imagine this has been observed before, but I haven't seen it, and I don't see any items for sale based on it, e.g. on Etsy, where one can find many versions of Penrose tiles as well as the newer aperiodic monotile.

I was thinking about the deflation rule for kites and darts and how much simpler it is to express in terms of half-kites and half-darts, which are themselves Robinson triangles (though the Robinson deflation rule looks a little different). Aside: once I started on that, I found that I could build larger tilings in Inkscape without having to write a Python script.

You could make a set of kites and darts by splitting them into triangles, obviously. However, you would need a way to specify right and left triangles to insure aperiodicity. So this results in 4 kinds of pieces, which is a bit inelegant.

However, if you build 3-dimensional pieces, you can get back to having just two. Are there any other writeups of this? I don't have a 3D printer or an Etsy store, but this is interesting enough to produce physically.

To get the 2 3D kite and dart halves, all you do is take the 2D tiles for kites and dart halves, pair them up and glue them back to back. You need to do it consistently so they follow placement rules. However, there are just 2 triangular shapes, and to form kites and darts, you pair the tile with itself in flipped position. This picture shows the layers diagrammatically, the resulting kite and dart, and the half kite, half dart deflation that can now be done directly with these tiles. If you had enough of them, you could build a tiling bottom-up.

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I cut those tiles out of 1mm matboard on a Cricut die cutter, which is a slow process requiring 8 passes with the knife blade and also requires the size to be larger than I like. Then I glued them back to back as shown. (Time to get that 3D printer. Any recommendations? Creality Ender looks affordable.)

Here's half of generation one of "sun." I miscalculated and forgot I needed 4, not 2 triangles to make a single dart or kite. But it's also a nice proof of concept since you can make half layouts.

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Finally, if you like Penrose rhombs better than kites and darts (I don't, to be honest), you can also make those out of the same triangles. The same tiling can be decomposed into either kites and darts or fat and skinny rhombs.

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Note: One thing I would do differently next time is make the connections more visually distinct. They consist of circular arcs of radius 10mm, 12.5mm, 15mm, and 17.5mm respectively, and will not fit if placed incorrectly, but it is too easy to try. A year spent handling Swiss coinage should have warned me this was a bad idea.

Finally, I have worked through this by example, not proof. Please let me know if there is any reason this shouldn't work or may permit a periodic tiling. I believe it works.

[pcallahan]

I wanted to verify that these were consistent with Robinson substitution rules, which split rhombs into triangles. It seemed clear that they should be, but I found the picture confusing, since I'm accustomed to thinking in terms of kites and darts, and the sizes of the two component triangles looked backwards. This image shows how all three tilings can be done with the 3D triangle pieces. Recall that while there appear to be four distinct pieces in this picture, there are actually only two and they can be flipped to reveal different faces. In the context of this diagram, the tiles must not be flipped but only rotated, since only one side is exposed and the other is glued to the opposite face.

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[pcallahan]

Mostly just to keep my notes in a place I won't lose them. We can use just one size of circular arc, since its position off-center and the length of the edge creates 4 unambiguous connections. These tiles can't be flipped. They are intended to be glued back-to-back, so only one side can ever be facing up. The other is interior. There is no longer the potential for confusion when the half-kite composite piece is upside down, because one side has "tabs" on both long sides and the other has "blanks" on both. The latter makes the physical piece more fragile. This might be an issue while cutting, but not after they're glued. I'll have to cut these on mat board to verify that they they are as usable as the last set. The smaller arcs may cause them to fall apart easily.

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Added: This idea was staring right at me.

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If I cut the tiles as shown out of card stock (not mat board) and fold on the dotted line I will get the same tiles, admittedly very thin. But if I glue layers above and below so the tabs are hidden, it might be usable. It's hard to tell without trying. Note: the mat board appears to be foldable after all, though scoring would help.

Added: Hiding the tabs with thin card stock poses a problem, because it might be possible to flip the tiles and force the wrong edges unless the layer is glued securely enough to block the tab from being inserted between. This problem could be reduced by moving the tabs so they are not mirror images, though I think it will not be as aesthetically pleasing.

Added: Folding isn't a win since it tends to affect the alignment unless the paper is too thin to be useful. I do have some other approaches I'm looking into.

[pcallahan]

After rereading this old post of mine on CGOL still life tiles viewtopic.php?f=7&t=2188&p=158213#p158213 I briefly wondered if I was missing something and if there could be 2D tiles for kite half and dart half respectively that enforced boundary conditions. That post shows how to construct an edge that is both distinct and flippable, though it requires a disconnected tile.

After some thought, I see that solves a different problem: flipping around a line perpendicular to the edge. When you are flipping around the the edge itself, you require that points at the same distance and displacement along the edge have the same absent/present value, so there is no way for it to fit against itself flipped unless the edge is a straight line (assume the edges themselves are closed and allowed to overlap). 3D tiles solve this by surfacing the opposite contour when rotated in 3D space and are not subject to the constraints of really flipping a 2D shape.

[pcallahan]

I got a little fancier with the connectors. Here are the kite and dart halves in tinkercad. https://www.tinkercad.com/things/b8DADd ... e-and-dart The reason for the explicit layering is that I am prototyping this with glued card stock. Those weren't too hard to make once the layers are cut, just time consuming. Here are some screenshots if you aren't able to connect to tinkercad for some reason.

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I may redo the whole thing purely in tinkercad with smooth connectors. These are based on flat tiles imported from Inkscape that I send to the Cricut.

[pcallahan]

Again, mainly for personal notes, and a trivial observation. I think it's useful to build up the kite deflation step from the dart deflation step That is: kite_i + dart_i to dart_{i+1} and then kite_i + dart_{i+1} to kite_{i+1}. This may make the connection to Fibonacci numbers a little more apparent since it composes two successive generations. I think it makes coding more modular as well, though I haven't tried. Here's what it looks like (made in Inkscape).

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The red arrow emphasizes that the component is placed without scaling. Each arrow could also be filled in with a linear transformation, I think setting the origin at the lower left point of each triangle would keep it simple (I have Python code to do this but it is based on the deflations from the Wikipedia page).

Another useful thing from a programming perspective is that you get a binary expansion tree, and it is probably easier to work with without special cases. What I would like to do, which I have not so far, is to maintain the adjacencies as I go, and not have to rely on geometry for it.

Finally, every time I reread the Wikipedia page I chew over this statement:

The half kite and half dart deflation are useful only in the context of deflating a larger pattern as shown in the sun and star deflations. They give incorrect results if applied to single kites and darts.

Can anyone explain what this means? Indeed, you will never get a symmetric sun or star this way, but I don't see how it is "incorrect". You will still create a larger and large usable patch of a Penrose tiling. Am I missing something?

Added: Adjacency should be definable in terms of lowest common ancestor, but I have to admit I am having trouble formalizing it.Two tiles adjacent on a cut will have a lowest common ancestor that produced this cut. But merely checking for lowest common ancestor is not enough because many pairs of tiles that are not adjacent will also have that LCA.

OK, one way is just to keep track of boundary tiles as three ordered lists and then stitch them together for each deflation. That clearly works, but I wonder if there is a discrete coordinate system that would make adjacency more apparent. If you maintain real coordinates of tile centers, you can look at Euclidean nearest neighbors, but that seems inelegant to me.

[pcallahan]

I finally got a 3D printer, and amazingly, it isn't all that finicky or hard to use. It's a Flashforge Adventurer 3 Pro 2, chosen for economy and "beginner" recommendations. I have seen some negative reviews but it works so far. I did have to recalibrate it because it was squishing my tiles by 0.3mm in height.

The first thing I realized was that my cardstock designs all have horizontal overhang and can't be printed. So I worked on a connector that has a sloped tab. It's over 50° from vertical and 45 is the recommended safe limit. However, it still prints on both sides. This shows the design and some actual photos.

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...which is all to the good, but then I suddenly realized I don't need any overhang. Robinson triangle tilings are two colorable, and as a result you can make tiles where one side is all "tabs" and the other is all "blanks". There is still the problem of fitting all those blanks in place, but the close placement isn't as big a problem printing a top layer as it is cutting material with a Cricut. Here's a picture of the triangles from Tinkercad. The tab shapes are intentionally asymmetric to prevent illegal placement even with forcing.

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It took a little trial and error. In fact, the first version could be forced in a way to form an illegal parallelogram with two kite halves that would tile periodically. Note that you need to leave a little space for tabs to fit. I added 0.5mm, which seems to work well and does not wiggle. I did the geometry in Inkscape and then imported the four layers into Tinkercad.

Finally, here's generation 1 of "sun" composed of 30 of these tiles: 20 kite halves and 10 dart halves.

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I have removed two pieces to show them separately, but they all fit well and the assembly is liftable. Each layer is 1.5mm thick, for 3mm total thickness, and they handle well. They are 40mm on the long side. The whole thing weighs 32g, so each piece averages a little over a gram. I haven't tried to weigh them separately. The weight, like area, should differ by the golden ratio. Each takes about 5 minutes to print, making Cricut rather competitive on speed. However, I don't need to do any additional assembly, and they are easier to remove.

Next step is to get another color of PLA. I have several other tiling projects include CGOL still life tiles, that would benefit from 3D printing.

[pcallahan]

Another note to myself.

File under: why did I ever find Penrose tiles confusing? If you start with Robinson triangles, aka kite and dart halves, all you have to do is build them in succession. You begin with a dart half of unit area (as defined for this purpose), and a kite of area (sqrt(5)+1)/2, or about 1.618, i.e. the golden ratio phi. The kite half is also known as the Golden triangle. Now to make the next one in the sequence, just attach the previous two together along matching sides, rotating and flipping as needed.

Oops, that's ambiguous... but not for long. By the time you have applied the rule twice, there is no longer any question of which sides match. So after that, you can just keep copy-pasting, rotating and flipping as needed, and attaching sides. This can be done easily in a vector editor such as Inkscape. You just have to rotate by 36° at a time. Here are the first few steps.

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A couple of things become apparent in this process. The area increases by a factor of phi at each step and the total number of triangles follows the Fibonacci sequence. The maximum side length increases by a factor of phi on the kite half to dart half transition.

This is classical Euclidean geometry with some combinatorics, and it kind of makes me wonder how nobody discovered Penrose tiles a long time ago. I know that Kepler had some ideas, and Girih tiles use deflation. But this is simple enough that I believe that the above picture and its implications would have been immediately comprehensible to anyone who had explored Golden triangles.

Granted, it may not have been obvious that an aperiodic tiling could be forced with boundary rules or even that this was an interesting question. But kite and dart tilings still might have been discovered using this approach.

[pcallahan]

Hidden tab version of flippable triangles. These actually print with a flatter overhang than usually recommended, with a slope of 2/3 or about 56.3° from vertical. This makes it possible to go out 3mm in a 2mm layer. I used a total thickness of 4.5mm but made the tiles smaller to use less material.

I think the jigsaw style pieces handle better, but these are better if you want a layout without contours. I have another idea of how to make kites and darts instead of Robinson triangles. These won't have to be flippable. However, I really like the triangles because deflation is so easy to explain.

Design of pieces from Tinkercad.

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Closeup of pieces.

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Sun expanded two steps (30 pieces).

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I am waiting to get another color of filament and then I'll make more tiles where kites and darts are different colors.

[pcallahan]

I think I like the jigsaw triangles better. They are less finicky for compensating for thickness, and hold together a lot better. I have another idea for non-flippable kites and darts, but I still have to work on it. Here are 60 triangles, with equal numbers of white and blue. It does not decompose into kites and darts, but it does decompose into rhombs.

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[pcallahan]

Here are conventional kite and dart tiles with jigsaw connections. The only novel part is that they are hidden on one side. It makes it a little tricky to assemble, but you just have to rotate the tile to connect from above and below simultaneously. Once it's together, it holds very well, since an interior tile cannot be removed without removing its neighbors first. Getting the spacing took some trial and error and I'm not quite done. If they hold too tightly, the whole thing buckles up. I finally came up with an approach similar to spacers on floor tiles. I remove about 0.1mm around each kite (and eventually darts the next time I print more), or alternatively you can think of it as extending the tab slightly. When exactly placed, the tiles have that gap around them. Since 3D printing isn't quite that precise, it just leaves some wiggle room. Here's the design as a screenshot from Tinkercad

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Here's what the tiling looks like underneath, assembled with jigsaw connections.

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And here's what it looks like from above, showing only the kite and dart shapes.

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These tiles are pretty compact, with a long side of 20mm. They each take a little over 3 minutes to print. I just weighed the tiling and it is 34g of PLA plastic, for 20 darts and 50 kites.

[pcallahan]

More pieces for a bigger layout, top and bottom. I may stop here unless I find a friend to give these to. They seem to work OK. There's also an opportunity to put a texture on top, such as the braid I have shown in another thread.

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These work well enough, though it would be different experience entirely if they were injection molded with the precision of Legos.

This picture shows how to connect a dart to a pair of kites to form an ace. Similar approaches can be used to add tiles in other places. The whole thing is liftable, at least in the interior, so it can be done without disturbing the placed tiles if you're a little bit careful.

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