A 13-sided shape called 'the hat' forms a pattern that never repeats:
A 13-sided shape known as "the hat" has mathematicians tipping their caps.
It's the first true example of an "einstein," a single shape that forms a special tiling of a plane: Like bathroom floor tile, it can cover an entire surface with no gaps or overlaps but only with a pattern that never repeats.
"Everybody is astonished and is delighted, both," says mathematician Marjorie Senechal of Smith College in Northampton, Mass., who was not involved with the discovery. Mathematicians had been searching for such a shape for half a century. "It wasn't even clear that such a thing could exist," Senechal says.
Although the name "einstein" conjures up the iconic physicist, it comes from the German ein Stein, meaning "one stone," referring to the single tile. The einstein sits in a weird purgatory between order and disorder. Though the tiles fit neatly together and can cover an infinite plane, they are aperiodic, meaning they can't form a pattern that repeats.
With a periodic pattern, it's possible to shift the tiles over and have them match up perfectly with their previous arrangement. An infinite checkerboard, for example, looks just the same if you slide the rows over by two. While it's possible to arrange other single tiles in patterns that are not periodic, the hat is special because there's no way it can create a periodic pattern.
Identified by David Smith, a nonprofessional mathematician who describes himself as an "imaginative tinkerer of shapes," and reported in a paper posted online March 20 at arXiv.org, the hat is a polykite — a bunch of smaller kite shapes stuck together. That's a type of shape that hadn't been studied closely in the search for einsteins, says Chaim Goodman-Strauss of the National Museum of Mathematics in New York City, one of a group of trained mathematicians and computer scientists Smith teamed up with to study the hat.
It's a surprisingly simple polygon. Before this work, if you'd asked what an einstein would look like, Goodman-Strauss says, "I would've drawn some crazy, squiggly, nasty thing."
[...] And the hat isn't the end. Researchers should continue the hunt for additional einsteins, says computer scientist Craig Kaplan of the University of Waterloo in Canada, a coauthor of the study. "Now that we've unlocked the door, hopefully other new shapes will come along."
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https://phys.org/news/2023-06-aperiodic-tile-hat-true-chiral.html
Mathematicians from Yorkshire University, the University of Cambridge, the University of Waterloo and the University of Arkansas have one-upped themselves by finding a close relative of "the hat," a unique geometric shape that does not repeat itself when tiled, that is a true chiral aperiodic monotile
[...] Just three months ago, the same four mathematicians announced what has come to be known in the field as the "einstein" shape—a single shape that could be used for aperiodic tiling all by itself. They called it "the hat."
[...] But others in the field pointed out that the shape described by the team was not, technically, a single aperiodic tile—it and its mirror image are two unique tiles and both are needed to create the shape described by the team. Apparently agreeing with the assessment of their colleagues, the four mathematicians took another look at their shape and found that by slightly modifying it, the need for its mirror would no longer exist and it indeed represented the true einstein shape
More information:David Smith et al, A chiral aperiodic monotile, arXiv (2023). DOI: 10.48550/arxiv.2305.17743
Journal information: arXiv
Recently: Mathematicians Have Finally Discovered an Elusive 'Einstein' Tile
One wonders if they have developed special search tools to look for these shapes, or if they just sit around over beers sketching on napkins?
(Score: 0) by Anonymous Coward on Sunday March 26 2023, @02:42PM (8 children)
I'm thinking about coding something for this. Maybe someone should manufacture the tiles.
(Score: 2) by RamiK on Sunday March 26 2023, @04:01PM (3 children)
Code yourself a pair of scissors and a printer.
compiling...
(Score: 0) by Anonymous Coward on Sunday March 26 2023, @06:22PM
Print the scissors.
(Score: 0) by Anonymous Coward on Monday March 27 2023, @03:20AM (1 child)
I want to make an infinite tiling screensaver or something with this shape.
(Score: 2) by RamiK on Monday March 27 2023, @11:09AM
The pattern isn't periodic so it's literally the worst option for live wallpapers or parallax... I mean, it's not to say you can't optimize the shader down to minimal overhead with a big texture and only computing the visible sections... Well, have at it I guess: https://f-droid.org/en/packages/de.markusfisch.android.shadereditor/ [f-droid.org]
compiling...
(Score: 4, Insightful) by JoeMerchant on Sunday March 26 2023, @04:15PM (3 children)
Someone definitely should be manufacturing tiles as we speak.
However, I have a problem believing that a single shape can completely tile an infinite surface with no repetition. The number between repetitions may be quite large, but the first tile has 13 sides, the next tile also has 13 sides, there are only 13 rotations it can be placed in (26 if flipping is allowed) on each available face, so 169 (338) possible orientations between the first two, and subsequent tiles are much more constrained in their placements.
This may be a shape that requires a maximal number of orientations before repetition, but not infinite. There are no irrational numbers at play in the angles, it will repeat eventually.
🌻🌻 [google.com]
(Score: 5, Insightful) by JoeMerchant on Sunday March 26 2023, @04:38PM
O.K. mathematics is all about the definitions, and the patterns do repeat, but they don't completely repeat... The "tiles form metatiles" and these (repeating) tiles form up into larger metatiles etc. and that progression is non-repeating on the largest scale due to the increasing complexity of the larger and larger meta-tiles.
So, you start with a 13 sides tile (itself a composite of 8 identical "kite shapes" which are subdivisions of a hexagon) and when you tile a surface with this 13 sides shape (which only has six "valid" orientations, 12 if you need to flip it) you eventually create a larger "metatile" which can then be used to tile the surface. These metatiles repeat until they create a larger (more complex) metatile which again may be used to tile the surface, the larger meta-tiles also repeat in different orientations.
I would be curious if, digging deeper, the non-repeating behavior only holds true in x-y pattern searching and consideration of 120 degree rotations does show repetition.
🌻🌻 [google.com]
(Score: 2) by krishnoid on Sunday March 26 2023, @07:50PM (1 child)
Unless the angles are irrational :-P Wait, would that make a difference?
(Score: 1, Interesting) by Anonymous Coward on Sunday March 26 2023, @08:02PM
You do the math. https://www.youtube.com/watch?v=sj8Sg8qnjOg [youtube.com]
(Score: 1) by khallow on Sunday March 26 2023, @02:57PM (7 children)
(Score: 3, Informative) by JoeMerchant on Sunday March 26 2023, @04:47PM (6 children)
Looking at this image: https://www.sciencenews.org/wp-content/uploads/2023/03/032323_ec_einstein-tiles_feat-680x383.jpg [sciencenews.org]
I don't see any mirrored shapes.
🌻🌻 [google.com]
(Score: 3, Informative) by khallow on Sunday March 26 2023, @07:04PM (5 children)
(Score: 2) by JoeMerchant on Sunday March 26 2023, @07:40PM (2 children)
Agree, the hat is a reflection of the shirt, but I doubt that reflections are required to achieve infinite tiling.
🌻🌻 [google.com]
(Score: 3, Informative) by Anonymous Coward on Sunday March 26 2023, @08:26PM (1 child)
Unfortunately, this isn't the ultimate solution because it turns out that you do need to reflect the tile for it to work, so although it is "ein Stein," the fact that you occasionally need to reflect the tile doesn't make it the "holy grail" solution they're looking for, so it is still unknown whether a single shape can tile the infinite plane in a non-repeating manner. So if you want to make some custom tiles for your bathroom, you're going to need to make two tiles, the "normal" and the reflected.
(Score: 2) by JoeMerchant on Tuesday March 28 2023, @09:55PM
I don't think the reflection is necessary for it to work:
https://gizmodo.com/new-13-sided-shape-the-hat-tiles-aperiodic-monotile-1850268575 [gizmodo.com]
🌻🌻 [google.com]
(Score: 2) by JoeMerchant on Tuesday March 28 2023, @10:03PM
This article has a larger example:
https://gizmodo.com/new-13-sided-shape-the-hat-tiles-aperiodic-monotile-1850268575 [gizmodo.com]
🌻🌻 [google.com]
(Score: 3, Informative) by JoeMerchant on Tuesday March 28 2023, @10:08PM
Ok even the blue and whites the dark blue is a mirror of the light blue and most other shapes.
🌻🌻 [google.com]