Mathematicians Invent New 'Einstein' Shape (theguardian.com) 50
One of mathematics' most intriguing visual mysteries has finally been solved -- thanks to a hobbyist in England. From a report: The conundrum: is there a shape that can be arranged in a tile formation, interlocking with itself ad infinitum, without the resulting pattern repeating over and over again? In nature and on our bathroom walls, we typically see tile patterns that repeat in "a very predictable, regular way," says Dr Craig Kaplan, an associate professor of computer science at the University of Waterloo in Ontario. What mathematicians were interested in were shapes that "guaranteed non-periodicity" -- in other words, there was no way to tile them so that the overall pattern created a repeating grid. Such a shape would be known as an aperiodic monotile, or "einstein" shape, meaning, in roughly translated German, "one shape" (and conveniently echoing the name of a certain theoretical physicist).
"There's been a thread of beautiful mathematics over the last 60 years or so searching for ever smaller sets of shapes that do this," Kaplan says. "The first example of an aperiodic set of shapes had over 20,000 shapes in it. And of course, mathematicians worked to get that number down over time. And the furthest we got was in the 1970s," when the Nobel-prize winning physicist Roger Penrose found pairs of shapes that fit the bill. Now, mathematicians appear to have found what they were looking for: a 13-sided shape they call "the hat." The discovery was largely the work of David Smith of the East Riding of Yorkshire, who had a longstanding interest in the question and investigated the problem using an online geometry platform. Once he'd found an intriguing shape, he told the New York Times, he would cut it out of cardstock and see how he could fit the first 32 pieces together. "I am quite persistent but I suppose I did have a bit of luck," Smith told the Guardian in an email.
"There's been a thread of beautiful mathematics over the last 60 years or so searching for ever smaller sets of shapes that do this," Kaplan says. "The first example of an aperiodic set of shapes had over 20,000 shapes in it. And of course, mathematicians worked to get that number down over time. And the furthest we got was in the 1970s," when the Nobel-prize winning physicist Roger Penrose found pairs of shapes that fit the bill. Now, mathematicians appear to have found what they were looking for: a 13-sided shape they call "the hat." The discovery was largely the work of David Smith of the East Riding of Yorkshire, who had a longstanding interest in the question and investigated the problem using an online geometry platform. Once he'd found an intriguing shape, he told the New York Times, he would cut it out of cardstock and see how he could fit the first 32 pieces together. "I am quite persistent but I suppose I did have a bit of luck," Smith told the Guardian in an email.
How about non-repeating articles? (Score:5, Funny)
https://science.slashdot.org/s... [slashdot.org]
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But that's "last month", so it isn't a repeat. At least, not in the normal /. world...
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Slasheimers
Re:How about non-repeating articles? (Score:4, Funny)
https://science.slashdot.org/s... [slashdot.org]
Oh no! The infinitely reporting tile is creating infinitely repeating articles!
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https://science.slashdot.org/s... [slashdot.org]
Oh no! The infinitely reporting tile is creating infinitely repeating articles!
Without a repeating pattern in the articles, mind you.
Re:How about non-repeating articles? (Score:4, Insightful)
Some problems are unsolvable.
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(I shouldn't need to explain Fortran terminology here, should I?)
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Won't or Can't? (Score:1)
Could you create a repeating pattern with this shape if you wanted to?
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Not without creating gaps between shapes.
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We're developers, we're used to forcing things to (mostly) work when the deadline approaches.
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Except, in this case, if there are gaps, you aren't actually "tiling" at that point.
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They are Air Tiles, my new invention.
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> Could you create a repeating pattern with this shape if you wanted to?
I cant imagine that would be useful, because a basic proportional rectangle already has that property.
You can make an infinitely random rectangular pattern, just like shuffling a pile of dominoes. but you can also make it regular too, but choice.
For this shape to be mathematically interesting, it would have to be impossible to intentionally tile it regularly.
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The question was meaning, does the shape *force* a non-periodic tiling, or does it require the the right choices at the right time to make it non-periodic?
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i would welcome a more concise definition of "repeating pattern", because right in the demo picture given in the guardian i can clearly see a repeated pattern involving a dozen pieces that have exactly the same layout (the cluster with 3 clear figures in the middle of 1st quadrant, repeated halfway between quadrants 2 and 4) . maybe this repetition isn't "regular" if you make the series bigger? but it is a repetition, right?
ofc i assume this has been proven mathematically (probably with math way beyond my c
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The correct statement is that there's no way of tiling an infinite plane which has translational symmetry.
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Could you create a repeating pattern with this shape if you wanted to?
That I'm not sure about. But what I am sure about is that you would need special software to lay it, because it not only has the ability to have no repeating section, it also has the ability to make it so you can't add onto it any more. You could easily be laying tiles and have them fit nicely, only to find that you've created a design that doesn't actually work. Imagine getting 2/3rds of a room done, and discovering because of the way you permuted it ten tiles in you can't go any further. A shape like thi
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you would need special software to lay it, because it not only has the ability to have no repeating section, it also has the ability to make it so you can't add onto it any more. You could easily be laying tiles and have them fit nicely, only to find that you've created a design that doesn't actually work. Imagine getting 2/3rds of a room done, and discovering because of the way you permuted it ten tiles in you can't go any further. A shape like this that admits infinite many tilings that work, also admits infinite that don't actually mesh.
Has it been confirmed that it is possible to create an array of tiles such that it can no longer be extended? That is, if I start in the center of a wall, I might hit a layout such that I cannot reach any of the edges of the finite space?.
If so, would be interesting to determine the smallest "terminal" set of these tiles, the minimum arrangement beyond which no further tiles can be fit into place anywhere on the perimeter.
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The sticking point for me (not necessarily for my correspondent) was that this novel "einstein" was composed of multiple quadrilaterals with angles 60,90,120 and 90 so they have an underlying trigonal not-quite-symmetry, while Penrose's "kite and dart" and "2 rhombii" tilings exhibit an underlying p
Not sure why... (Score:2)
since I have a full head of hair all my immediate people do as well but the first thing I thought of was natural looking hair implantation. The next was how do these new einstein shapes work as a polygon in the 3D graphics world for more natural looking pieces.
In TFA they contrast with the patterns seen in nature but in reality the patterns always work as ways to model nature but fail when looking at any actual instance of nature... are there einstein tiles which reconcile the two? That opens too many possi
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> since I have a full head of hair
Lucky bastard!
> [could be used for] natural looking hair implantation.
Something that repeats on a larger scale, say every 13 tiles, would probably still work in practice, as the hair won't grow out fully symmetric or even anyhow. And roughly 20% of the hair follicles typically don't survive transplant. Repeating at a larger scale thus wouldn't show.
(Some ads are misleading because they count "follicular units", which are units of grouped strands, and if any one strand
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"Something that repeats on a larger scale, say every 13 tiles, would probably still work in practice, as the hair won't grow out fully symmetric or even anyhow. And roughly 20% of the hair follicles typically don't survive transplant. Repeating at a larger scale thus wouldn't show."
Maybe but presumably they already do things like this and the results are questionable... but then I guess if it is working well for some we wouldn't really know. lol
That might be "good enough" but why bother if you can just tile
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> the results are questionable
What, you've been sniffing people's hair?
Welcome to Slashdot, Mr. Biden! We don't like sniffers, but we dislike grabbers even more.
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Plus the original one was sort of hat shaped. What gets better than that?
Is this patented? (Score:1)
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That was always a *stupid* patent. Maths should never be patentable, its inherent to the universe. I'm not sure what Roger Penrose was thinking with that one, or for that matter the Patent office for granting it.
However, that particular patent was for a two shape tiling.
Whats interesting with this one is its a one shape tiling.
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I'm not sure when - or indeed, if - he actually took out a patent, or if he used some other aspect of IP law (IANAL, and I don't care enough to research it), but he either got a licensing deal on the specific design (design .NE. patent), or forced the arse-wipe company to change their de
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Perfect for public transport textiles (Score:1)
"Shape vs. Stone" (Score:1)
> "einstein" shape, meaning, in roughly translated German, "one shape"
"Stein" is English "stone" according to Google Translate.
DANGER! (Score:2)
Keep this away from politicians, they'll somehow use it to make gerrymandering even worse!
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But it's non-repeatable... so it'll only work for the first state to try it.
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which means they'll find infinite ways to screw it up.
The "hat" is actually 2 shapes, not 1 (Score:3)
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This is an imporant distinction. It leaves the harder problem still unsolved.
Its einstein not Einstein (Score:2)
There is no capitalization in "einstein shape" as it is an adjective.
Does this shape have spiky hair? (Score:2)
I mean, if an "einstein shape" is true to the original, it would have to, right?
Online geometry platform (Score:2)
https://www.geogebra.org/geome... [geogebra.org]
Pretty cool. Wish I was creative enough to actually need it to prove something geometrically.
But I have no idea if that was the tool they are actually talking about.
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Somehow missed that they linked it in the article.
So what's the algorithm for placing them? (Score:2)
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The original paper is available as a preprint on arxiv [arxiv.org] (PDF). It demonstrates how to generate patches by subdivision.
Penrose tiling (Score:2)
Can anybody explain to me how this is different from Penrose Tiling?
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There are various tiles which generate the Penrose tiling or something equivalent to it, but all of them use at least two distinct shapes of tile. This uses one tile (allowing for flipping the tile as not making a distinct one).
Einstein shape? (Score:1)
I was expecting something more like this. [flaticon.com]