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The "Einstein Tile" A geometric shape that does not repeat itself when tiled

FirePro9FirePro9 Posts: 455
edited March 2023 in The Commons

Apparently the elusive "Einstein Tile" has been discovered and it is a 13-sided polykite shape that when tiled never repeats itself. This would seem to be good news for us, eventually.

https://phys.org/news/2023-03-geometric-tiled.html

https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile

 

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Post edited by FirePro9 on
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Comments

  • algovincianalgovincian Posts: 2,576

    That's really cool! Thanks for posting.

    - Greg

  • HylasHylas Posts: 4,772

    huh cool

  • LeanaLeana Posts: 11,028

    Interesting smiley

    I wonder how the guy came up with that. It's actually quite a simple shape, which is the most surprising.

  • WendyLuvsCatzWendyLuvsCatz Posts: 37,816

    designed by obviously someone who has never had to clean the mouldy grout on the bathroom floor

  • ColinFrenchColinFrench Posts: 635

    Thanks for the links FirePro9, it's a fascinating discovery!

  • memcneil70memcneil70 Posts: 3,745

    That would make a great quilt!

    Thanks

  • LeatherGryphonLeatherGryphon Posts: 11,173
    edited March 2023

    Being a Roger Penrose fanboi, I knew about Penrose Tilescool, but this is even simpler.yes  I wonder if having a prime number of sides(13) has any bearing on the mathematics of why it works.indecision

    Post edited by LeatherGryphon on
  • FishtalesFishtales Posts: 6,043

    Is it just me or can nobody else see the pairs and triples in those images?

     

  • WendyLuvsCatzWendyLuvsCatz Posts: 37,816
    edited March 2023

    Fishtales said:

    Is it just me or can nobody else see the pairs and triples in those images?

     

    not adjacent 

    they shaded ones oriented the same way to illustrate that I think 

    it may repeat a couple randomly but it doesn't continue in a pattern 

    Post edited by WendyLuvsCatz on
  • TotteTotte Posts: 13,510

    Fishtales said:

    Is it just me or can nobody else see the pairs and triples in those images?

     

    Those are quite "repetitive" and they all look like t-shirts hanging to dry.....

    tiles.jpg
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  • LeatherGryphonLeatherGryphon Posts: 11,173
    edited March 2023

    Repetition may seem to appear locally, but it's an illusion caused by our eye/brain's innate (but flawed) raison d'etre, to try to find patterns.surprise   I'm sure that close examination of all the other tiles around/inside a "repeating" pattern would find differences.  If they've proved mathematically that it doesn't repeat, then it doesn't repeat.cheeky  Of course that also depends upon one's defintion of repetition.frown

    I think they're stylized teapots.indecision

    Post edited by LeatherGryphon on
  • FeralFeyFeralFey Posts: 3,897
    edited March 2023

    Very cool! Thanks for sharing. I love a bit of arty science. Now to scurry off and see how we can use this in 3D...

     

    edited to add: I realize that the einstein tile is a 2D construct, and therefore it's application in 3D would be limited to texture tiling - pretty much how it would work in the design industry with fabrics, flooring, wallpapers. But I think it's still a fascinating concept and I'm looking forward to see where this may lead.

     

    Post edited by FeralFey on
  • LeatherGryphonLeatherGryphon Posts: 11,173

    WendyLuvsCatz said:

    designed by obviously someone who has never had to clean the mouldy grout on the bathroom floor

    yes 

  • richardandtracyrichardandtracy Posts: 5,062
    WendyLuvsCatz said:

    designed by obviously someone who has never had to clean the mouldy grout on the bathroom floor

    Use black grout. Then no-one, not even the householder, will see the mould. ;)

  • TaozTaoz Posts: 9,731

    LeatherGryphon said:

    Being a Roger Penrose fanboi, I knew about Penrose Tilescool, but this is even simpler.yes  I wonder if having a prime number of sides(13) has any bearing on the mathematics of why it works.indecision

    Interesting question.  But the big question is I imagine, how to make the textures seamless. 

  • AlmightyQUESTAlmightyQUEST Posts: 1,961

    Totte said:

    Fishtales said:

    Is it just me or can nobody else see the pairs and triples in those images?

     

    Those are quite "repetitive" and they all look like t-shirts hanging to dry.....

    The headlines on the articles are confusing if not misleading. It's not the shape that doesn't repeat, the point is that it is a single shape, it's that the pattern of how the shapes interlock doesn't repeat. This doesn't mean the same configuration exist either, it means that it won't occur at regular intervals as other patterns would. For example, you wouldn't be able to predict the same configurations repeating every X number of horizontal iterations of a pattern.

    Cool mathematic stuff being hidden behind poorly worded headlines.

  • LeatherGryphonLeatherGryphon Posts: 11,173
    edited March 2023

    Taoz said:

    LeatherGryphon said:

    Being a Roger Penrose fanboi, I knew about Penrose Tilescool, but this is even simpler.yes  I wonder if having a prime number of sides(13) has any bearing on the mathematics of why it works.indecision

    Interesting question.  But the big question is I imagine, how to make the textures seamless. 

    Um..., I think in this case, seamless is hopeless.devil 

    Post edited by LeatherGryphon on
  • TaozTaoz Posts: 9,731

    LeatherGryphon said:

    Taoz said:

    LeatherGryphon said:

    Being a Roger Penrose fanboi, I knew about Penrose Tilescool, but this is even simpler.yes  I wonder if having a prime number of sides(13) has any bearing on the mathematics of why it works.indecision

    Interesting question.  But the big question is I imagine, how to make the textures seamless. 

    Um..., I think in this case, seamless is hopeless.devil 

    Probably.  I'd rather think that fibonacci and other "natural" algorithms is the way to go - to imitate nature's seemingly chaotic randomness in a way that takes as little resources as possible.  Procedural shaders also work quite well in many cases, like these:

    https://www.daz3d.com/oso-shader-pack-1-for-iray

  • FirePro9FirePro9 Posts: 455

    For fun I modeled 3 objs of the Einstein Tile "Hat".  Per Wikipedia ( https://en.wikipedia.org/wiki/Einstein_problem ) the Hat is formed from 8 copies of a 60-90-120-90 kite, glued edge-to-edge and uses a mirror image as well (see attached image). 

     

    Einstein Tiles - showing mirrored tiles.png
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    Einstein tile - Hat OBJs.zip
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  • LinwellyLinwelly Posts: 5,795

    FirePro9 said:

    Apparently the elusive "Einstein Tile" has been discovered and it is a 13-sided polykite shape that when tiled never repeats itself. This would seem to be good news for us, eventually.

    https://phys.org/news/2023-03-geometric-tiled.html

    https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile

     

    I would need a larger example to see if it's repetitive or not, for now I can see some repeating I marked one block of repe4at in green and the other in red, would need a larger one to see how the rest sorts in there but I have the impression it does sort

    Einstein tiles.png
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  • Richard HaseltineRichard Haseltine Posts: 96,850
    edited March 2023

    Linwelly said:

    FirePro9 said:

    Apparently the elusive "Einstein Tile" has been discovered and it is a 13-sided polykite shape that when tiled never repeats itself. This would seem to be good news for us, eventually.

    https://phys.org/news/2023-03-geometric-tiled.html

    https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile

     

    I would need a larger example to see if it's repetitive or not, for now I can see some repeating I marked one block of repe4at in green and the other in red, would need a larger one to see how the rest sorts in there but I have the impression it does sort

    There wil be local repeats, but if you draw a bugger box then eventurally you will reach a point at which the pattern does not repeat. Just as an irrational number may recur for a bit but will then stop recurring.

    Post edited by Richard Haseltine on
  • LinwellyLinwelly Posts: 5,795

    Richard Haseltine said:

    Linwelly said:

    FirePro9 said:

    Apparently the elusive "Einstein Tile" has been discovered and it is a 13-sided polykite shape that when tiled never repeats itself. This would seem to be good news for us, eventually.

    https://phys.org/news/2023-03-geometric-tiled.html

    https://www.sciencenews.org/article/mathematicians-discovered-einstein-tile

     

    I would need a larger example to see if it's repetitive or not, for now I can see some repeating I marked one block of repe4at in green and the other in red, would need a larger one to see how the rest sorts in there but I have the impression it does sort

    There wil be local repeats, but if you draw a bugger box then eventurally you will reach a point at which the pattern does not repeat. Just as an irrational number may recur for a bit but will then stop recurring.

    I don't have the time or the engergy to follow up on the math of the article, that's why I thought it was a bit sad that there wasn't a larger sample which would visualise the thing. I can imagine it being like irrational numbers

  • AlmightyQUESTAlmightyQUEST Posts: 1,961
    If you are interested, the first article links to the journal at the ends of the article which appears to be freely available at the moment, with more examples and explanations.
  • LinwellyLinwelly Posts: 5,795

    AlmightyQUEST said:

    If you are interested, the first article links to the journal at the ends of the article which appears to be freely available at the moment, with more examples and explanations.

    I might take a plunge there, this weekend

  • LinwellyLinwelly Posts: 5,795
    edited April 2023

    ok I found this article in German:

    https://www.spektrum.de/news/hobby-mathematiker-findet-lang-ersehnte-einstein-kachel/2124963

    which contains this graphic which adresses the repetitive aspect I noticed and shows why it's not repetitive in the end

    https://static.spektrum.de/fm/912/f2000/output2.png

     

    Also now I want a Penrose tiling for my DAZ rooms ... pretty please

    Post edited by Linwelly on
  • FirePro9FirePro9 Posts: 455

    Linwelly, that is an interesting article because it says that the lengths of the sides of the "Hat" can be varied, producing an infinite number of Einstein tiles.  They include an animated GIF of the Hat shape morphing between a 7-sided comma shape and a 6-sided chevron shape. The chevron shape seems so much simpler, not sure why they are fixated on the Hat shape.

     

    einstein 13 sided hat.jpg
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    Einstein Tile - 7 sided comma.jpg
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    Einstein Tile - 6 sided chevron.jpg
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  • LinwellyLinwelly Posts: 5,795

    as I understood that gif both ends of the scale where the shape looses sides it as well looses it's function as einstein tile since they both clearly look repetitive

  • FirePro9FirePro9 Posts: 455

    Linwelly said:

    as I understood that gif both ends of the scale where the shape looses sides it as well looses it's function as einstein tile since they both clearly look repetitive

     Oh, that makes sense, thanks for the clarification. 

  • ChumlyChumly Posts: 793

    The cool thing, for our purposes though, is that if each of those tiles was a Patch of Grass, or a section of cobblestone, it would look a lot better and natural than the normal tiling Daz uses now and which the eye can find the pattern pretty quickly.

  • ArtiniArtini Posts: 8,836
    edited April 2023

    How do we generate such tiles for Daz Studio, then?

    I also see how the pattern repeats, but maybe it is just related to the examples provided.

    In Unity there are some methods to avoid repetitive texturing on the terrains or the other objects

    and they work pretty well.

     

    Post edited by Artini on
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