5 ms·

What can you do with an Einstein (aperiodic monotile)?

43 points by robinhouston 3y ago

Terr_ 3y ago
I don't know WTF the New York Times is thinking, making their paywall cover everything. No first paragraph, not a subtitle, even the the title itself is truncated. How is anyone supposed to get interested-enough in their content to register?
- filoleg 3y ago
  Just checked, you are totally right, I straight up cannot see anything aside from the title background picture they have for the article. Ironically, reader mode on iOS safari lets me see a good chunk of the article including the title.
- mmh0000 3y ago
  Whaaa... You don't find the only uncovered text "What Can You Do" enthralling? I want to know what I can do! //On a side note: https://archive.is/fp84g https://archive.is/fp84g
  - Terr_ 3y ago
    "He says... 'What would a computer do with a lifetime supply of chocolate?' ... I am now telling the computer EXACTLY what he can do with a lifetime supply of chocolate!!" [0] [0] https://www.youtube.com/watch?v=gP-eR8a3vIU https://www.youtube.com/watch?v=gP-eR8a3vIU
neonate 3y ago
http://web.archive.org/web/20231211110223/https://www.nytimes.com/2023/12/10/science/mathematics-tiling-einstein.html http://web.archive.org/web/20231211110223/https://www.nytime...
quickthrower2 3y ago
https://archive.is/fp84g https://archive.is/fp84g <- images load on this one
quickthrower2 3y ago
You can SVG it! <polygon points="89.3,146.9 8.5,194 35.3,240.5 143.3,240.5 170.2,193.6 250.8,240.7 331.7,193.9 305,146.9 251,146.9 251.7,53.4 170.1,6.5 143.3,53.4 89.3,53.4 "/>
- l0b0 3y ago
  OK, follow-up puzzle: Can you specify the exact einstein polygon with only integer coordinates?
  - javajosh 3y ago
    trivial. you can multiply by any n and then set the viewport attribute to compensate.
    - l0b0 3y ago
      No, I mean mathematically. I had a skim through the original paper[1], but unfortunately it's not easy to figure out what the coordinates are. If any of them are related as a factor of an irrational number, I don't think it's possible. [1] https://arxiv.org/abs/2303.10798 https://arxiv.org/abs/2303.10798
      - jameshart 3y ago
        The points of the Einstein tile all fall on a hex grid - see the illustration here: https://gametek.substack.com/p/the-einstein-tile https://gametek.substack.com/p/the-einstein-tile So in cartesian coordinates, there are √3 components that will always crop up. But there is always an integer number of √3's in each x and y coordinate, if that helps - every vertex can be written as (a + b√3, c + d√3) for integers a, b, c and d. Oh, and that goes for every vertex of every tile in the infinite tiling, not just for a single tile.
        basil-rash 3y ago
        > Oh, and that goes for every vertex of every tile in the infinite tiling, not just for a single tile. Could it possibly not? i.e. is there any way to tile n-gons in the (A + B*sqrt(3))[] kernel that has vertices not in that kernel? What comes to mind is the counterexample of plain unit squares tiled with strips at irrational offsets from one another. But what about for n != 4? Does every tiling have some vertex-vertex co-location?
jaymzcampbell 3y ago
...Play Tetris, of course! He named his monotile variant of the game Montris. (Another entrant had a similar idea, called Hatris.) There's no follow up comment but I'm wondering if they know such a game existed, by Alexey no less, of that exact name, which involves stacking... hats. [1] The British in me still really loves the pun in Einstein - "ein Stein" meaning "one stone", nothing to do with he-of-relativity fame. [2] [1] https://en.wikipedia.org/wiki/Hatris https://en.wikipedia.org/wiki/Hatris [2] https://www.theguardian.com/science/2023/apr/03/new-einstein-shape-aperiodic-monotile https://www.theguardian.com/science/2023/apr/03/new-einstein...
heinrich5991 3y ago
https://archive.is/fp84g https://archive.is/fp84g
- oniony 3y ago
  Already posted
unholiness 3y ago
I used a variation of the Einstein to make a D&D puzzle. Players had to use a limited number of tiles to make a sufficiently-difficult hedge maze. https://www.threads.net/@nonarbitrarity/post/CuYKHNSMEhw https://www.threads.net/@nonarbitrarity/post/CuYKHNSMEhw Playing with Einsteins, they really feel like a leap forward in puzzle technology. The aperiodicity means, no matter how many you put together, you've never quite figured out how you'll get the next bit to work without gaps. Endlessly entertaining.

43 points by robinhouston 3y ago

I don't know WTF the New York Times is thinking, making their paywall cover everything. No first paragraph, not a subtitle, even the the title itself is truncated. How is anyone supposed to get interested-enough in their content to register?

filoleg 3y ago
Just checked, you are totally right, I straight up cannot see anything aside from the title background picture they have for the article. Ironically, reader mode on iOS safari lets me see a good chunk of the article including the title.
mmh0000 3y ago
Whaaa... You don't find the only uncovered text "What Can You Do" enthralling? I want to know what I can do! //On a side note: https://archive.is/fp84g https://archive.is/fp84g
- Terr_ 3y ago
  "He says... 'What would a computer do with a lifetime supply of chocolate?' ... I am now telling the computer EXACTLY what he can do with a lifetime supply of chocolate!!" [0] [0] https://www.youtube.com/watch?v=gP-eR8a3vIU https://www.youtube.com/watch?v=gP-eR8a3vIU

neonate 3y ago

http://web.archive.org/web/20231211110223/https://www.nytimes.com/2023/12/10/science/mathematics-tiling-einstein.html http://web.archive.org/web/20231211110223/https://www.nytime...

quickthrower2 3y ago

https://archive.is/fp84g https://archive.is/fp84g <- images load on this one

quickthrower2 3y ago

You can SVG it! <polygon points="89.3,146.9 8.5,194 35.3,240.5 143.3,240.5 170.2,193.6 250.8,240.7 331.7,193.9 305,146.9 251,146.9 251.7,53.4 170.1,6.5 143.3,53.4 89.3,53.4 "/>

l0b0 3y ago
OK, follow-up puzzle: Can you specify the exact einstein polygon with only integer coordinates?
- javajosh 3y ago
  trivial. you can multiply by any n and then set the viewport attribute to compensate.
  - l0b0 3y ago
    No, I mean mathematically. I had a skim through the original paper[1], but unfortunately it's not easy to figure out what the coordinates are. If any of them are related as a factor of an irrational number, I don't think it's possible. [1] https://arxiv.org/abs/2303.10798 https://arxiv.org/abs/2303.10798
    - jameshart 3y ago
      The points of the Einstein tile all fall on a hex grid - see the illustration here: https://gametek.substack.com/p/the-einstein-tile https://gametek.substack.com/p/the-einstein-tile So in cartesian coordinates, there are √3 components that will always crop up. But there is always an integer number of √3's in each x and y coordinate, if that helps - every vertex can be written as (a + b√3, c + d√3) for integers a, b, c and d. Oh, and that goes for every vertex of every tile in the infinite tiling, not just for a single tile.
      - basil-rash 3y ago
        > Oh, and that goes for every vertex of every tile in the infinite tiling, not just for a single tile. Could it possibly not? i.e. is there any way to tile n-gons in the (A + B*sqrt(3))[] kernel that has vertices not in that kernel? What comes to mind is the counterexample of plain unit squares tiled with strips at irrational offsets from one another. But what about for n != 4? Does every tiling have some vertex-vertex co-location?

jaymzcampbell 3y ago

...Play Tetris, of course! He named his monotile variant of the game Montris. (Another entrant had a similar idea, called Hatris.) There's no follow up comment but I'm wondering if they know such a game existed, by Alexey no less, of that exact name, which involves stacking... hats. [1] The British in me still really loves the pun in Einstein - "ein Stein" meaning "one stone", nothing to do with he-of-relativity fame. [2] [1] https://en.wikipedia.org/wiki/Hatris https://en.wikipedia.org/wiki/Hatris [2] https://www.theguardian.com/science/2023/apr/03/new-einstein-shape-aperiodic-monotile https://www.theguardian.com/science/2023/apr/03/new-einstein...

heinrich5991 3y ago

https://archive.is/fp84g https://archive.is/fp84g

oniony 3y ago
Already posted

unholiness 3y ago

I used a variation of the Einstein to make a D&D puzzle. Players had to use a limited number of tiles to make a sufficiently-difficult hedge maze. https://www.threads.net/@nonarbitrarity/post/CuYKHNSMEhw https://www.threads.net/@nonarbitrarity/post/CuYKHNSMEhw Playing with Einsteins, they really feel like a leap forward in puzzle technology. The aperiodicity means, no matter how many you put together, you've never quite figured out how you'll get the next bit to work without gaps. Endlessly entertaining.