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Mathematicians Discovered a New 13-Sided Aperiodic Monotile

19 points by galaxyLogic 4y ago

acadapter 4y ago
Would this shape be covered by copyright? Or is it an uncopyrightable mathematical idea? For example, would it be possible for various floor tile suppliers to cut their material in this shape?
- SideQuark 4y ago
  Penrose Tiles were patented and used successfully to stop a manufacturer from using them. So this is patentable (and likely already filed I'd guess). https://morris.cymru/en/2009/01/penroses-patent-and-the-battle-of-the-tissue-tiles-contains-mathematics/ https://morris.cymru/en/2009/01/penroses-patent-and-the-batt...
- sacrosancty 4y ago
  [dead]
- kadoban 4y ago
  I don't think a copyright would make sense, in general you can't copyright just a shape. It has to be part of something with creative input. You could trademark it as part of a logo, but that does not seem to apply here (there's no business trying to use it as their mark). Maybe a design patent, or normal patent? (I'm not a lawyer) Another complication if they tried to get others not to use it: I believe they found a whole family of shapes, not just one shape. Even if they can keep one, seems like it'd be harder to keep them all to themselves. (not sure they've shown any desire to prevent others from using them either, this is just hypothetical).
  - DougBTX 4y ago
    > It has to be part of something with creative input. I honestly don’t know how copyright rules should or do apply here, but gosh, there must have been some creative input to produce this tile over the decades it took to make!
    - kadoban 4y ago
      Yeah that's a good point that I'm not really able to resolve (my knowledge of IP law tends to end not much past what applies to software). I _think_ it wouldn't really count if it's not visible in the output though, the shape itself.
  - acadapter 4y ago
    Perhaps you're right - Penrose tiles were protected by patent and trademark. Wikipedia treats them as copyright-free.
  - galaxyLogic 4y ago
    A basic rule is that you cannot patent a mathematical formula. But what you can patent is a novel way of "doing" anything. So if this is a novel way of covering your floor then my guess you could at least try to patent it.
waynecochran 4y ago
I haven't read the paper -- was 13 sides proved to be minimal? Or is the next quest to find a tile with fewer sides that tile the plane aperiodically?
- fsckboy 4y ago
  the point is that this is the first (so, only known) tile shape that is also aperiodic. Previously aperiodic tiling schemes (like Penrose) required several tiles of different shapes. any additional single tile shape that was aperiodic would be interesting regardless of how many sides it has. I supposed if there turned out to be many of them there would be a grouping based on how many sides.
  - fsckboy 4y ago
    I need to correct myself! What has been discovered is a family of shapes that are all aperiodic in the same pattern. https://cs.uwaterloo.ca/~csk/hat/examples/animation.gif https://cs.uwaterloo.ca/~csk/hat/examples/animation.gif https://cs.uwaterloo.ca/~csk/hat/animation.mp4 https://cs.uwaterloo.ca/~csk/hat/animation.mp4
- galaxyLogic 4y ago
  What would be interesting is a 3D mono-tile to cover 3D space. A SUPER LEGO. Assuming such a block also had like 13 or so sides I think you could keep such "bricks" together without glue and build houses and anything. Then disassemble if needed and build a different house.
beefman 4y ago
Dupe: https://news.ycombinator.com/item?id=35273707 https://news.ycombinator.com/item?id=35273707 https://news.ycombinator.com/item?id=35264965 https://news.ycombinator.com/item?id=35264965 https://news.ycombinator.com/item?id=35242458 https://news.ycombinator.com/item?id=35242458
fnordpiglet 4y ago
The article states this has practical applications but I didn’t notice any mentioned. I can imagine there’s theoretical and aesthetic applications, but what utility does it serve?
- galaxyLogic 4y ago
  One special feature of such tiles is that when you put them together they can't "slide" past another. Rectangular tiles can. So a floor covering made of these titles is inherently more stable than rectangular tiles I would guess. Or let's say you make a roof-covering with these tiles. Once the tiles are together I believe you can pull the whole covering by grabbing it from one side and it will all follow, even if the titles are not glued together. That would not be the case with rectangular titles.
  - fnordpiglet 4y ago
    But wouldn’t a simple interlock (like a puzzle piece) achieve the same result? I wonder if it is possible to use this to make self similar non repeating patterns (like say wood grains) without requiring more than N patterns, where N is some sufficiently high number and each tile pattern is designed to continue any side. I also wonder if instead of slippage it’s more like distribution of pressure or some other force related factor not concentrating in any area. I really wish they had explained the applications better it’s an interesting result.
    - galaxyLogic 4y ago
      > But wouldn’t a simple interlock (like a puzzle piece) achieve the same result? Yes but puzzle pieces are of different shapes whereas now we have a "monotile". It is very simple to produce all the monotiles you will ever need because they are all of the same shape. Consider the economy of LEGOs if a single LEGO-shape was enough for everything. That would make LEGO pieces cheaper. Puzzle-pieces would also take considerable effort and expense to put together. There is a reason why puzzles are called, puzzles. With monotiles putting them together is trivial. Now I have no idea what practical applications actually are planned for monotiles in practice. But an invention is still an invention not because it does something that has never been done before, but because it can or may do it more economically, and can be a better solution for some specific situations.

19 points by galaxyLogic 4y ago

Would this shape be covered by copyright? Or is it an uncopyrightable mathematical idea? For example, would it be possible for various floor tile suppliers to cut their material in this shape?

SideQuark 4y ago
Penrose Tiles were patented and used successfully to stop a manufacturer from using them. So this is patentable (and likely already filed I'd guess). https://morris.cymru/en/2009/01/penroses-patent-and-the-battle-of-the-tissue-tiles-contains-mathematics/ https://morris.cymru/en/2009/01/penroses-patent-and-the-batt...
sacrosancty 4y ago
[dead]
kadoban 4y ago
I don't think a copyright would make sense, in general you can't copyright just a shape. It has to be part of something with creative input. You could trademark it as part of a logo, but that does not seem to apply here (there's no business trying to use it as their mark). Maybe a design patent, or normal patent? (I'm not a lawyer) Another complication if they tried to get others not to use it: I believe they found a whole family of shapes, not just one shape. Even if they can keep one, seems like it'd be harder to keep them all to themselves. (not sure they've shown any desire to prevent others from using them either, this is just hypothetical).
- DougBTX 4y ago
  > It has to be part of something with creative input. I honestly don’t know how copyright rules should or do apply here, but gosh, there must have been some creative input to produce this tile over the decades it took to make!
  - kadoban 4y ago
    Yeah that's a good point that I'm not really able to resolve (my knowledge of IP law tends to end not much past what applies to software). I _think_ it wouldn't really count if it's not visible in the output though, the shape itself.
- acadapter 4y ago
  Perhaps you're right - Penrose tiles were protected by patent and trademark. Wikipedia treats them as copyright-free.
- galaxyLogic 4y ago
  A basic rule is that you cannot patent a mathematical formula. But what you can patent is a novel way of "doing" anything. So if this is a novel way of covering your floor then my guess you could at least try to patent it.

waynecochran 4y ago

I haven't read the paper -- was 13 sides proved to be minimal? Or is the next quest to find a tile with fewer sides that tile the plane aperiodically?

fsckboy 4y ago
the point is that this is the first (so, only known) tile shape that is also aperiodic. Previously aperiodic tiling schemes (like Penrose) required several tiles of different shapes. any additional single tile shape that was aperiodic would be interesting regardless of how many sides it has. I supposed if there turned out to be many of them there would be a grouping based on how many sides.
- fsckboy 4y ago
  I need to correct myself! What has been discovered is a family of shapes that are all aperiodic in the same pattern. https://cs.uwaterloo.ca/~csk/hat/examples/animation.gif https://cs.uwaterloo.ca/~csk/hat/examples/animation.gif https://cs.uwaterloo.ca/~csk/hat/animation.mp4 https://cs.uwaterloo.ca/~csk/hat/animation.mp4
galaxyLogic 4y ago
What would be interesting is a 3D mono-tile to cover 3D space. A SUPER LEGO. Assuming such a block also had like 13 or so sides I think you could keep such "bricks" together without glue and build houses and anything. Then disassemble if needed and build a different house.

beefman 4y ago

Dupe: https://news.ycombinator.com/item?id=35273707 https://news.ycombinator.com/item?id=35273707 https://news.ycombinator.com/item?id=35264965 https://news.ycombinator.com/item?id=35264965 https://news.ycombinator.com/item?id=35242458 https://news.ycombinator.com/item?id=35242458

fnordpiglet 4y ago

The article states this has practical applications but I didn’t notice any mentioned. I can imagine there’s theoretical and aesthetic applications, but what utility does it serve?

galaxyLogic 4y ago
One special feature of such tiles is that when you put them together they can't "slide" past another. Rectangular tiles can. So a floor covering made of these titles is inherently more stable than rectangular tiles I would guess. Or let's say you make a roof-covering with these tiles. Once the tiles are together I believe you can pull the whole covering by grabbing it from one side and it will all follow, even if the titles are not glued together. That would not be the case with rectangular titles.
- fnordpiglet 4y ago
  But wouldn’t a simple interlock (like a puzzle piece) achieve the same result? I wonder if it is possible to use this to make self similar non repeating patterns (like say wood grains) without requiring more than N patterns, where N is some sufficiently high number and each tile pattern is designed to continue any side. I also wonder if instead of slippage it’s more like distribution of pressure or some other force related factor not concentrating in any area. I really wish they had explained the applications better it’s an interesting result.
  - galaxyLogic 4y ago
    > But wouldn’t a simple interlock (like a puzzle piece) achieve the same result? Yes but puzzle pieces are of different shapes whereas now we have a "monotile". It is very simple to produce all the monotiles you will ever need because they are all of the same shape. Consider the economy of LEGOs if a single LEGO-shape was enough for everything. That would make LEGO pieces cheaper. Puzzle-pieces would also take considerable effort and expense to put together. There is a reason why puzzles are called, puzzles. With monotiles putting them together is trivial. Now I have no idea what practical applications actually are planned for monotiles in practice. But an invention is still an invention not because it does something that has never been done before, but because it can or may do it more economically, and can be a better solution for some specific situations.