[HN Gopher] Two algorithms for randomly generating aperiodic til...
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Two algorithms for randomly generating aperiodic tilings
Author : fanf2
Score : 137 points
Date : 2023-04-10 15:26 UTC (1 days ago)
(HTM) web link (www.chiark.greenend.org.uk)
(TXT) w3m dump (www.chiark.greenend.org.uk)
| mcphage wrote:
| There's another method, at least for the Penrose rhombus tilings:
| https://archive.bridgesmathart.org/2022/bridges2022-285.html...
|
| The author of this was one of the authors of the Hat tile paper.
| flangola7 wrote:
| I don't understand what's so magical about the hat. It's a non-
| square shape that repeats.
| brickers wrote:
| it's a non-square shape that does not and cannot repeat
| flangola7 wrote:
| How can you cover infinite surface without infinite number of
| objects?
| klyrs wrote:
| Technically the shape itself repeats like a mug, infinitely
| tiling the plane. However, that tiling is not overly
| repetitive -- if it's like the Penrose tiling, it can be
| self-similar in a handful of rotations about a single origin,
| but unlike a square tiling, does not admit infinitely many
| self-similarities.
| uoaei wrote:
| "Aperiodic" is the opposite of "periodic"
| pmarreck wrote:
| I think I am obsessed with aperiodic Penrose tilings. My 21 month
| old son's room has an area rug with such a tiling. I want to tile
| a bathroom in my house with them at some point. If you want to
| make my day, link a photo you have of one of these IRL.
|
| It is such a wicked combination of beauty and math, like
| fractals.
| madcaptenor wrote:
| Where can I get such a rug? I fixed up my home office a couple
| years ago and it's nice except I need a rug both to make it
| look nice and to absorb some sound.
| masfuerte wrote:
| Photos of a Penrose pavement here
| https://www.maths.ox.ac.uk/outreach/oxford-mathematics-alpha...
| and here https://hardscape.co.uk/inspire/case-studies/maths-
| institute...
| pmarreck wrote:
| https://media.giphy.com/media/14vFOciTnQjnl6/giphy.gif
| tiedieconderoga wrote:
| Depending on how much free time you have, you may or may not
| want to dive into the rabbithole of aperiodic tilings in
| medieval Islamic architecture.
|
| https://www.sciencenews.org/article/ancient-islamic-penrose-...
| cwmma wrote:
| A company tried to put it on toilet paper but Roger Penrose
| claimed ownership so they had to stop http://bit-
| player.org/2017/sir-roger-penroses-toilet-paper
| pmarreck wrote:
| Does he have ownership over ALL aperiodic tilings or just the
| ones he's come up with?
|
| Anyway, that's unfortunate. Can't he just get a cut or
| something?
|
| Also, this all really begs the question as to whether this is
| an "invention" or just "math". Imagine if Newton's heirs had
| to get a cut every time Newtonian physics was used (or
| calculus for that matter... splitting it with Leibniz's
| heirs)
| [deleted]
| woudsma wrote:
| Near my previous job there was a park with similar tiling [0].
| I'm not sure if the pattern qualifies as truly aperiodic(?) but
| it triggered my brain every day, I couldn't not look at it. I
| love these kinds of patterns, they always invoke my curiosity.
| Just like in music irregularity makes things interesting.
| Something for your brain to solve. You know there has to be logic
| behind a seemingly random pattern.
|
| [0]: https://www.dutchdesignawards.nl/gallery/funenpark/
| isaacg wrote:
| That pattern looks periodic, but it's super cool! I think the
| periodic unit is a flower shape consisting of 8 tiles: 2 in the
| middle making a hexagon, and then 6 more around the hexagon,
| like the petals of a flower.
| motohagiography wrote:
| Naively, and with general interest public forum curiousity, these
| tiling problems seem to be about iterating the proportions of the
| sides of the tiles to get symmetrical shapes, but given each tile
| is also necessarily a hamiltonian circuit between the
| angles/nodes of a shape and the tiles are aperiodic, the implied
| visual symmetry of the shapes doesn't seem meaningful.
|
| It seems like there would be infinite possible aperiodic tiles
| (with real valued side lengths), so long as the number of angles
| (or nodes/vertices) for a whole tile (like a triangle) has the
| same evenness or oddness as the number of vertices extending from
| the node as the number of sides of the shape it is a part of.
|
| So to completely tile a plane aperiodically, each node/angle of a
| triangle must have an odd number of "sides" from its adjacent
| tiles stemming from it to completely tile a plane, where each
| angle of a hexagon must be a node with an even number of sides
| connecting at its vertices. Once you are more than one "hop" away
| from another tile, you can have even or odd numbers of verticies.
|
| The perimeter of any plane with a complete aperiodic tiling must
| still be a hamiltonian path around its edge, therefore the graph
| of the verticies representing the angles the aperiodic tiles must
| also reduce to being made of other "shapes" with hamiltonian
| paths. It implies to me that for every tile that is odd-sided, it
| requires a complementary odd-sided shape somewhere in the tiling
| to form a hamilonian path of "hops."
|
| It's not a sufficient condition, but naively it looks like a
| necessary one. No math will get done here, but from a general
| interest reasoning perspective, I'd wonder if tiles and
| hamiltonian cycles are the same thing.
| fanf2 wrote:
| The hat is part of an infinite family of aperiodic tiles, as
| illustrated in the last image on this page:
| https://cs.uwaterloo.ca/~csk/hat/
|
| This adjustability was a surprise, we have not seen an
| aperiodic tiling like this before.
| eganist wrote:
| [flagged]
| lubesGordi wrote:
| [flagged]
| billfruit wrote:
| How exactly was the 'hat' pattern arrived at? Was it through
| experiment or through an analytic process.
| necubi wrote:
| There's a great story here:
| https://www.quantamagazine.org/hobbyist-finds-maths-
| elusive-....
|
| It was found by a hobbyist playing with PolyForm Puzzle Solver
| fanf2 wrote:
| This article has some of the story behind the paper
| https://www.theguardian.com/science/2023/apr/03/new-einstein...
|
| Short answer to your question: David Smith discovered the hat
| shape by experimentation.
| csense wrote:
| This website is a refreshing breath of fresh air. There's no
| "sign up with your email newsletter," it's a traditional webhost
| (not a Medium or a Substack), there's almost no styling / images
| / branding, it doesn't churn and stutter due to the behind-the-
| scenes gyrations of the latest monstrous multi-megabyte
| JavaScript framework with hundreds of dependencies.
|
| Simon Tatham (the author of this article) is the developer of
| PuTTY [1], an open source Windows native SSH client, and Simon
| Tatham's Portable Puzzle Collection [2], a bunch of simple games
| implemented in portable C (with OS-specific frontends, or you can
| play them in-browser via WebAssembly).
|
| The website has been updated in ways that improve it (this just-
| written article is a huge amount of informative content, and
| being able to play a puzzle in-browser via WASM is very welcome),
| but it never jumped on any of the 21st century design bandwagons
| that have taken over most of the WWW (many of which, I suspect,
| only exist to create jobs for web developers and branding
| consultants).
|
| [1] PuTTY https://www.chiark.greenend.org.uk/~sgtatham/putty/
|
| [2] https://www.chiark.greenend.org.uk/~sgtatham/puzzles/
| paulddraper wrote:
| Agreed.
|
| Also it could benefit from a modicum of CSS.
| wwarner wrote:
| I feel like any discussion of aperiodic tiling that doesn't
| mention de Bruijn is missing the mark. He showed that
| aperiodicity results from projecting wireframes in 4 or more
| dimensions onto a plane. The non-repetitive patterns appear for
| the same reason that irrational numbers appear in Euclidian
| geometry.
| fanf2 wrote:
| Isn't de Bruijn's method specific to Penrose tiles? The
| aperiodic monotile paper says in section 2 that it is an open
| question whether the cut-and-project method can construct hat
| tilings. So de Bruijn would have been no help to solve Simon
| Tatham's problem of how to generate hat tile puzzle grids. His
| two algorithms are the old one his Loopy puzzle uses for
| Penrose grids, and the new one it uses for hat grids.
|
| You can play Loopy on a hat tile grid here:
| https://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/loop...
| wwarner wrote:
| yes, agree, but you're proving my point because as you say
| the interesting thing about the hats is that they don't fit
| into the simple geometric explanation of other aperiodic
| tilings.
| powerset wrote:
| When I tried to give a quick explanation of what an aperiodic
| was to a friend recently, calling it a geometric version of
| irrational numbers was the easiest way to do it. Glad to hear
| my explanation had some substance to it, and wasn't just one of
| those "you can think of it like this, but that's not what it
| really is" explanations.
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(page generated 2023-04-11 23:00 UTC)