[HN Gopher] Infinite tiling pattern could end a 60-year mathemat...
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Infinite tiling pattern could end a 60-year mathematical quest
Author : pseudolus
Score : 50 points
Date : 2023-06-02 12:32 UTC (1 days ago)
(HTM) web link (www.nature.com)
(TXT) w3m dump (www.nature.com)
| headalgorithm wrote:
| Some recent and related discussions:
|
| https://news.ycombinator.com/item?id=36119920
|
| https://news.ycombinator.com/item?id=36056488
|
| https://news.ycombinator.com/item?id=35273707
|
| https://news.ycombinator.com/item?id=35264965
|
| https://news.ycombinator.com/item?id=35242458
| Solvency wrote:
| For real, generally curious why this little tile keeps being
| posted here.
| throw_pm23 wrote:
| A rare case of cutting edge math research, but understandable
| to the general public, at least in its results.
| [deleted]
| gliptic wrote:
| Only the first link refers to this, the others are the
| previous "hat" tile.
| foobarbecue wrote:
| So the polygon hat from a few weeks ago only worked as an
| aperiodic tile if you were allowed to tile it with a flipped
| version? I wasn't aware of that caveat at the time. I guess that
| sort of detail doesn't make it into the popular press.
| ColinWright wrote:
| The cool thing is that the ratio of flipped to non-flipped is
| p^4, where p is the golden ratio: (1+sqrt(5))/2
| IanCal wrote:
| Is it that the shape _can_ be tiled so it doesn 't repeat a
| pattern or can you only tile it like that?
| enriquto wrote:
| It must be the latter. Many trivial shapes admit non-periodic
| tilings. For example a 2x1 rectangle: you tile the plane with
| squares, and divide each square in two in a non-periodic way.
| mabbo wrote:
| > and divide each square in two in a non-periodic way.
|
| That doesn't actually sound simple. Can you provide an
| example of how you would do that?
| enriquto wrote:
| You traverse the squares in a spiral pattern starting from
| a central one. Then each square is divided vertically or
| horizontally according to the binary digits of pi (or
| whatever).
| zuminator wrote:
| It doesn't even need to be (pseudo-) random. You can
| trace a spiral pattern out from the origin like so: 1
| vertical up, turn right, 1 horizontal right, turn down, 2
| vertical down, turn left, 2 horizontal left, turn up, 3
| vertical up, turn right, 3 horizontal right, turn down,
| etc. The resulting spiral pattern will never repeat.
| BurningFrog wrote:
| For each 2x2 block, decide randomly/arbitrarily to split it
| horizontally or vertically.
| cvoss wrote:
| This solution won't be accepted without a lot more work.
| Flipping a coin is not guaranteed to produce an aperiodic
| tiling, since, in the trivial case where the coin comes
| up heads indefinitely, you'll be producing a periodic
| tiling. (You will argue it's not likely to do this, and
| I'd agree. In fact, I'd conjecture you'll only produce a
| periodic tiling with measure zero probability.) But
| what's required is to actually show an aperiodic tiling.
| If you have a proof that coin flipping will produce such
| a tiling with non-vanishing probability, then,
| presumably, embedded in your proof, you will likely find
| a decision procedure for producing an aperiodic tiling
| deterministically.
| IanCal wrote:
| That makes sense but now I'm more confused about Penrose
| tilings. My understanding was that there were rules about
| placement to avoid periodic tilings.
|
| Thinking out loud, is this down to local Vs global rules?
| Otherwise I'm a bit confused as to why Penrose tiles were a
| big deal.
|
| To be clear, I'm fully aware the confusion is because I'm
| missing something.
| firstlink wrote:
| An aperiodic tile (a) admits a tiling (b) does not admit a
| periodic tiling.
| cypherpunks01 wrote:
| Does an aperiodic tiling mean that if you take the entire
| infinite tiled plane, and while sliding around a copy of the
| plane, there are zero other places you can slide it to that
| matches the original?
|
| I'm just a bit confused by the opt-repeated claim that the tiling
| "never repeats itself" and I'm not sure if I'm understanding the
| brief translational symmetry explanation correctly.
| iamgopal wrote:
| My question is, It repeats aperiodically and predictably, like
| a Fibonacci ? Or aperiodically and unpredictably ? Like a prime
| numbers ?
| ndsipa_pomu wrote:
| It'll be predictable. Other aperiodic tilings (e.g. Penrose
| tiles) look to have regular patterns from a quick glance, but
| those patterns won't have translational symmetry despite
| looking vaguely symmetrical.
| fanf2 wrote:
| The proof that the tiling is aperiodic comes with a recursive
| algorithm for generating tilings.
| IIAOPSW wrote:
| Yeah. Its like an irrational number but in 2d.
| mrfox321 wrote:
| Yes, that's the definition of aperiodic tiling.
|
| There is no translational symmetry.
| jameshart wrote:
| no _sliding_ symmetry - you can combine translation and
| rotation, and you won 't find a match.
| amluto wrote:
| There could be a (finite) rotational symmetry. The Penrose
| tilings, for example, have a five-fold rotational symmetry.
| gilleain wrote:
| Yes, exactly.
|
| Also this video:
|
| https://m.youtube.com/watch?v=IfVwelta1fE
|
| Helped me understand an example of a monotile that _can_ tile
| aperiodically but also has a periodic tiling.
| Glyptodon wrote:
| If that's the case, does it have practical near repeats as
| the difference in rotation between an selected origin tile
| and another tile somewhere across infinity becomes bound to
| approach zero? I'd have guessed that with only 360 degrees of
| rotation at very large numbers you'd be bound to get a tile
| that has nearly the same rotation if not the same rotation as
| another tile. I'd have guessed that for a single tile
| aperiodic tiling there'd be a 2nd non
| rotational/translational requirement: that the if rotation is
| the same, the adjoining tiles of every case of the same
| rotation would not be the same, but that seems like it'd
| require infinite positions for two adjacent tiles to
| interlock, which also seems like at some point would mean two
| non identical positions are functionally identical for
| practical purposes unless the shape allows for infinite
| permutations of adjacency to a single tile that aren't
| effectively just the same at limit. Very curious about how
| I'm understanding or not understanding this . Do you get
| large subsheets as it extends to infinity that differ
| distinctly from other subsheets by single tiles, with some
| kind of corellary that subsheets can repeat but there is no
| subsheet that con cover the plane in totality besides the
| whole itself?
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