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Single aperiodic tiling
Posted: March 23rd, 2023, 6:06 pm
by pcallahan
Did anyone else catch this news?
https://www.newscientist.com/article/23 ... er-repeat/ "Mathematicians discover shape that can tile a wall and never repeat: Aperiodic tiling, in which shapes can fit together to create infinite patterns that never repeat, has fascinated mathematicians for decades, but until now no one knew if it could be done with just one shape"
Unfortunately, NewScientist is paywalled. There are some other links here:
https://cs.uwaterloo.ca/~csk/hat/
I haven't read the proof.
Re: Single aperiodic tiling
Posted: March 23rd, 2023, 6:19 pm
by muzik
The Wikipedia page has already been updated to include it:
https://en.wikipedia.org/wiki/Einstein_problem
Re: Single aperiodic tiling
Posted: March 23rd, 2023, 6:39 pm
by confocaloid
The Aperiodical:
https://aperiodical.com/2023/03/an-aper ... le-exists/
Now the question, are there any gliders on this tiling?
Re: Single aperiodic tiling
Posted: March 24th, 2023, 2:31 pm
by pcallahan
I don't have a 3D printer, but I see people are printing these already. I can make some out of layered cardstock on a Cricut. Maybe I ought to join the fun and do that this weekend. They look easy to cut and might not need any complicated jigsaw connections to hold together.
Re: Single aperiodic tiling
Posted: March 24th, 2023, 5:12 pm
by pcallahan
For anyone who hasn't already made this themselves, here's an SVG of the base tile. I drew it in Inkscape using 10mm as the length of the long side of a kite. I used snap-to (midpoint, object center, cusp node) on a regular hexagon to split out a kite shape, copied and assembled those with snapping, and then traced the outline. Some of the numbers in the path can be simplified to integers.
Code: Select all
<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<svg
width="420mm"
height="297mm"
viewBox="0 0 420 297"
version="1.1"
id="svg8"
inkscape:version="1.0.1 (c497b03c, 2020-09-10)"
inkscape:export-filename="tile.svg"
inkscape:export-xdpi="96"
inkscape:export-ydpi="96"
xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
xmlns="http://www.w3.org/2000/svg"
xmlns:svg="http://www.w3.org/2000/svg"
xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
xmlns:cc="http://creativecommons.org/ns#"
xmlns:dc="http://purl.org/dc/elements/1.1/">
<defs
id="defs2" />
<sodipodi:namedview
id="base"
pagecolor="#ffffff"
bordercolor="#666666"
borderopacity="1.0"
inkscape:pageopacity="0.0"
inkscape:pageshadow="2"
inkscape:zoom="0.35"
inkscape:cx="400"
inkscape:cy="560"
inkscape:document-units="mm"
inkscape:current-layer="layer1"
inkscape:document-rotation="0"
showgrid="false"
inkscape:showpageshadow="2"
inkscape:pagecheckerboard="0"
inkscape:deskcolor="#d1d1d1" />
<metadata
id="metadata5">
<rdf:RDF>
<cc:Work
rdf:about="">
<dc:format>image/svg+xml</dc:format>
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rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
<dc:title />
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<g
inkscape:label="Layer 1"
inkscape:groupmode="layer"
id="layer1">
<path
style="fill:#9fcbe8;fill-opacity:1;stroke:#000000;stroke-width:0.2;stroke-dasharray:none;stroke-opacity:1"
d="m 7.8573872,29.907665 11.5470048,4e-6 2.886749,-5.000002 8.660258,4.999999 8.660249,-4.999997 -2.886749,-4.999997 -5.773505,-3e-6 2e-6,-9.9999982 -8.660256,-5 -2.886746,5 -5.773505,-4e-6 1e-6,9.9999992 -8.6602528,5 2.88675,4.999999"
id="path1150" />
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</svg>
I built a patch from the NS article picture to make sure it works. One thing I did not realize before doing that is that the dark blue tile has to be flipped.
- path1150-81-5.png (48.8 KiB) Viewed 6097 times
Re: Single aperiodic tiling
Posted: March 25th, 2023, 12:31 am
by hkoenig
On a Mac, I was able to easily import your SVG file into Sketch, and play around with it. Thanks.
I would like a set similar to the acrylic Penrose Kites & Darts for sale at
the pentagon universe. I really like the feel of these. If someone does a Kickstarter project or something similiar, I would interested in knowing about it.
Re: Single aperiodic tiling
Posted: April 6th, 2023, 10:57 pm
by PHPBB12345
Can be update to
HandWiki?
Re: Single aperiodic tiling
Posted: April 10th, 2023, 1:17 pm
by dvgrn
hkoenig wrote: March 25th, 2023, 12:31 am
On a Mac, I was able to easily import your SVG file into Sketch, and play around with it. Thanks.
I would like a set similar to the acrylic Penrose Kites & Darts for sale at
the pentagon universe. I really like the feel of these. If someone does a Kickstarter project or something similiar, I would interested in knowing about it.
This was mentioned on Discord but not on the forums, I think: Rob Fathauer is selling manufactured sets of einstein tiles with the shirt graphic, shipping in about a week. I
pre-ordered a set; they're supposed to start shipping around the end of this week.
Re: Single aperiodic tiling
Posted: April 13th, 2023, 11:33 am
by hkoenig
A number of people with 3-D printers and laser cutters and such have been selling tile sets made from various materials. I bought a 3-D printed plastic set from a seller on Etsy.com that are about 2.5cm across and a couple of millimeters thick, with a price per tile of about US$0.40. I find it a lot easier to work with these than trying to manipulate images in a drawing program.
What would be nice would be a set with different colored sides, which would help emphasize the location of the reflected tiles. Or pre-grouped metatiles to go with the individual ones.
Re: Single aperiodic tiling
Posted: April 13th, 2023, 12:14 pm
by pcallahan
hkoenig wrote: April 13th, 2023, 11:33 am
What would be nice would be a set with different colored sides, which would help emphasize the location of the reflected tiles. Or pre-grouped metatiles to go with the individual ones.
I have enough ways to make my own that I won't be ordering any from a 3rd party. See
viewtopic.php?f=12&t=5980 for a description of two of these.
Actually, I find it easier to work with them virtually in Inkscape, doing precise rotations and placement. The big problem is just that they are vaguely symmetric-looking, making it very confusing to keep track of which side is which.
My latest thought is to add hex and triangular grid lines to them like this: If I get ambitious, I may make more with layered cardstock and glue printed cardstock on top and bottom (Cricut has a "print and cut" feature.) Considering the amount of time I would spend manipulating them, a few minutes with glue is a small amount of added effort.
- g1695.png (178.81 KiB) Viewed 5778 times
Update Maybe I'll print and cut top and bottom layers for the 14 layered tiles I made already. That's not too much work and they should look pretty good with the hole covered up. The biggest problem I have is keeping them from sliding around. The foam-on-rubber-cement tiles are easier for that.
Re: Single aperiodic tiling
Posted: April 14th, 2023, 12:12 am
by pcallahan
The top and bottom labels came out well with print and cut on ordinary printer paper.
- 20230413_204709.jpg (88.05 KiB) Viewed 5743 times
I've already peeled them off the mat. Before I glue them to the tiles, I should probably fill in the +-shaped holes, maybe with Elmer's glue and corn starch.
Update I started working on these. I don't expect to finish tonight, but here are two tiles with top and bottom layers and twelve more with holes filled and drying.
- 20230413_215043.jpg (376 KiB) Viewed 5736 times
I should coat them to make them last, maybe with Mod Podge.
I don't think I'll ever be crazy enough to make more tiles this way. One practical approach would be to print them on magnetic sheets, which worked very well for Moore-neighborhood tiles, the difference being that I am no longer limited to cutting with scissors. They would not be flippable, so you're stuck with two physical tiles. I have not tried cutting magnetic sheet with a Cricut. I believe it's possible, but I'm not sure it's recommended.
Update 2 Here are 5 arranged in a patch.
- 20230413_221622.jpg (718.76 KiB) Viewed 5733 times
I'll finish the other 9 later. It'll probably have to wait until Sunday at the earliest.
Re: Single aperiodic tiling
Posted: April 17th, 2023, 12:31 am
by pcallahan
My 14 hat tiles, labeled and coated in mod podge.
- 20230416_211907.jpg (186.24 KiB) Viewed 5670 times
The mod podge is still a little tacky, which is an advantage for getting the tiles to stay put on the table top. I'm not sure how long that will last. I believe it eventually cures.
I will not make any more tiles using this technique unless I can find someone willing to pay about $15 per tiny tile.
Re: Single aperiodic tiling
Posted: June 30th, 2024, 2:51 pm
by pcallahan
I finally got back to this with the goal of adding hidden tabs and slots to hold the tiles together. The trick was where to put the tabs and slots. It turns out that you only have to put them on the long kite edges, alternating tab and slot. However, to make the tile flippable, you need to reverse the tab and slot edges between top and bottom. When long edges are joined, there are interlocking tabs and slots on each side.
Printables link:
https://www.printables.com/model/928547 ... -hidden-ta
The tiles hold together well with this method, but they are not easy to 3D print because of overhang. I handled this by printing each tile in two parts held together with a dovetailed peg, as follows.
- 20240630_094820.jpg (1.5 MiB) Viewed 5030 times
They are printed with grooves facing down and assemble like this.
- Screenshot 2024-06-30 at 11.49.04 AM.png (1.54 MiB) Viewed 5030 times
The result tiling looks like this, and is liftable.
- 20240630_095442.jpg (1.34 MiB) Viewed 5030 times
Re: Single aperiodic tiling
Posted: July 19th, 2024, 9:56 am
by pcallahan
Here's a new
printable tile, this time done as a single piece. There are circles on the underside to indicate flipped tiles. They hold together with interlocking tabs and holes. The holes are hidden except for small slits on the faces of the tiles. The tabs stick out of the concave “L” boundaries of each tile, since these must always match a convex boundary on an adjacent one. Holes are placed in convex boundaries and fit tabs if present but allow these boundaries to face each other.
- Screenshot 2024-07-19 at 6.48.55 AM.png (775.17 KiB) Viewed 4718 times
These are patches of the tiling face up and face down.
- 20240718_085021.jpg (1.97 MiB) Viewed 4718 times
- 20240718_085036.jpg (1.99 MiB) Viewed 4718 times
Re: Single aperiodic tiling
Posted: July 30th, 2025, 10:05 am
by pcallahan
Update I don't even like these tiles very much compared to Penrose tiles but I've now done four 3D print variations.
These are the smallest printable monotiles I have made so far,.
https://www.printables.com/model/136836 ... ein-requir
- 20250729_003713.jpg (1.56 MiB) Viewed 450 times
They are around the size of a US dime and hold together well with interlocking tabs. Unlike my other projects, they require superglue for assembly. A form is included for aligning them during gluing.
- 20250727_075806 (1).jpg (347.08 KiB) Viewed 450 times
In March, I also improved the one piece solution, but it has to be larger so the snap tabs work. To print in two colors, you have to switch filaments during printing.
https://www.printables.com/model/122521 ... les-with-h
- 20250310_183241.jpg (316.83 KiB) Viewed 450 times
I could try making spectre tiles next, the ones that don't require flipping. Purely in terms of aesthetics I find them even less interesting though.