4 ms·
Elusive ‘Einstein’ solves a longstanding math problem
- wslh 4y agohttps://archive.is/oLRS9 https://archive.is/oLRS9
- quickthrower2 4y agoI love that there is still room for a tinkerer to make discoveries like this.
- wellthatsfunny 4y ago"It's turtles, all the way down!"
- bertman 4y agoDiscussion from two weeks ago: https://news.ycombinator.com/item?id=35273707 https://news.ycombinator.com/item?id=35273707
- dang 4y agoThanks! Macroexpanded: Mathematicians discover shape that can tile a wall and never repeat - https://news.ycombinator.com/item?id=35273707 https://news.ycombinator.com/item?id=35273707 - March 2023 (156 comments)
- mg 4y agoImportant to note that the solution uses reflection. It uses the tile and a mirrored version of it. In my book, this means it is rather using two tiles than one. So the journey is not over. The question if a nonperiodic tiling of the plane is possible with one (non-mirrored) tile is still open.
- pyinstallwoes 4y agoChirality and handedness maybe?
- oceansea 4y ago[flagged]
- sacrosancty 4y ago[dead]
- SiempreViernes 4y agoSeems a bit stingy to say you don't get to flip the tiles over, though I guess it makes sense if you are talking tiles with only one "nice" side.
- brookst 4y agoAlso if you’re talking about the platonic ideal of a plane. There is no “flip” transformation purely within the 2D plane.
- zokier 4y agoWhat do you mean? Flipping is same as reflection, one of the rigid transformations > The rigid transformations include rotations, translations, reflections, or any sequence of these. [...] All rigid transformations are examples of affine transformations. https://en.wikipedia.org/wiki/Rigid_transformation https://en.wikipedia.org/wiki/Rigid_transformation
- brookst 4y agoI mean, cut out paper shapes and work on a table as a 2D plane. There is no way to get a reflected shape without lifting the shapes off the plane. And now I’m curious: what additional transformations could we do with 3D shapes if we could move them in a 4D space?
- scotty79 4y agoI think there's none. Flipping in 3d can reduce the shape to a line or a point or even make it disappear entirely from initial 2d space. Transformation in 4d would be capable of the same, but it could also remove the shape from 3d space that contains initial 2d space. But as you transform 2d shape it remains embedded in it's own local 2d space. So you are not really transforming the shape but it's associated 2d space. And since the only thing can do to 2d space to transform it into another 2d space without deforming it is translation, rotation and flipping going into higher dimensions doesn't give you anything extra.
- jeffparsons 4y ago> In my book, this means it is rather using two tiles than one. I expected to see more about this. Are a given 2D shape and its mirror image generally considered the same shape by... the people who study this stuff? That would surprise me. So much so that calling this an "aperiodic monotile" doesn't feel right.
- evanpw 4y agoYes, "same shape" in this context means "isometric". Rotations and reflections are considered differences in the way the shape is placed into the plane, not differences in the shape itself.
- jeffparsons 4y agoFascinating. Rotation feels like it should be "allowed" to me, because you can continuously rotate a 2D shape to any other orientation, in 2D, without it ever becoming not the same shape. Reflections feel like a totally different thing, because there's no continuous path to go from a shape to its reflection in 2 dimensions: you either have to have it instantaneously jump to its reflection, or introduce an extra imaginary "third dimension" for it to move through. I'm not arguing with the definitions, because that's pointless. I'm just trying to explain why I find it so surprising as a lay person that reflections would be admitted in this way.
- Jarmsy 4y agoThe more recent Quanta article covers it much better https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/ https://www.quantamagazine.org/hobbyist-finds-maths-elusive-...
- ourmandave 4y agoThe title suggests Will Hunting is a real person.
- itissid 4y agohttps://youtu.be/48sCx-wBs34 https://youtu.be/48sCx-wBs34 veritasium explanation of the tiling problem is excellent
- quesera 4y agoAnnoying that the headline writer went for the cheap "Einstein" as reference to clever human, ignoring that the guy (his name is Smith) found an "ein stein" (German: "one shape") that tiles aperiodically.
- codeflo 4y agoSorry to ruin your pun idea, but Stein doesn't mean geometric shape. The most common meaning is simply stone, but it can also mean board game piece. Words you might use for these shapes instead would be Form (meaning shape) or Kachel (meaning tile). Source: native speaker.
- quesera 4y agoSimplified for concise comment above, but translation taken from the article: > (The term “einstein” comes from the German “ein stein,” or “one stone” — more loosely, “one tile” or “one shape.”) I'm mostly reacting to the use of "Einstein" as a label, which I always dislike. The article seemed to hint at a reason for the pun/reference (Smith found a "one shape") but there no was justification for bringing Albert into the discussion. ... Edit: In retrospect, there's another way to read the article headline. If you ignore the capital E (or forgive it because of German Noun capitalization Rules), you might argue that the "Elusive 'einstein' (tile) answers/resolves a longstanding question in tile geometry". I think this would be too generous though. They did capitalize the E, and the verb makes more sense as a human action. Might be the result of multiple edits. NYT headline telephone game.
- codeflo 4y ago> (The term “einstein” comes from the German “ein stein,” or “one stone” — more loosely, “one tile” or “one shape.”) As I said, that "loose" meaning doesn't exist, but either way it would be "ein Stein". I guarantee that no one at the NYT bothered to ask a German speaker. However, I did check the German Wikipedia entry on tilings, which to my surprise does confirm that the word "einstein" is an established synonym for an aperiodic monotile. It also states that this usage isn't widespread in Germany itself. I presume that's because it doesn't actually make any sense in German.
- hexwab 4y agohttps://www.youtube.com/watch?v=FkZPMf73qYc https://www.youtube.com/watch?v=FkZPMf73qYc is a presentation by the authors. I haven't seen this linked widely.