[HN Gopher] Mathematicians discover shape that can tile a wall a...
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Mathematicians discover shape that can tile a wall and never repeat
Author : iamben
Score : 423 points
Date : 2023-03-23 12:33 UTC (10 hours ago)
(HTM) web link (www.newscientist.com)
(TXT) w3m dump (www.newscientist.com)
| sys42590 wrote:
| So it would be possible to make a new tiling for Tatham's Loopy
| puzzle [0] that would surely look nice.
|
| [0]:
| https://www.chiark.greenend.org.uk/~sgtatham/puzzles/js/loop...
| 6nf wrote:
| Flipping the shape is cheating imo. Might as well use Penrose for
| any actual tile work.
| bassrattle wrote:
| I'd really like to see this applied to 3D world modeling. If a
| landscape were tiled with a textured material in this way,
| perhaps it would look more natural.
| andrethegiant wrote:
| Oh shit new shape just dropped
| brobdingnagians wrote:
| Opportunity for a startup to start selling these, however niche
| that might be...
| hgsgm wrote:
| Look for it at Cherry Arbor Design
| https://cherryarbordesign.com/collections/all
| xhkkffbf wrote:
| A wall? How about a floor? I wanted to cover my kitchen in
| Penrose tiles but they didn't seem to be available. Anyone know
| where to get some?
| cwmoore wrote:
| Are there numeric indexes for locations in an aperiodic monotile
| covering a plane such as there are for the xyz in a tiled web
| map?
| 1MachineElf wrote:
| I wonder what applications there are for this in video games.
| Many games that attempt to show the outside world often suffer
| from repeating patterns in things like terrain, which would never
| happen in real life. Everything from 2D isometric games like
| Command & Conquer to 3D open worlds like Skyrim have this
| problem. Could that problem be solved by tile shapes that never
| produce repeating patterns?
| cadooo wrote:
| My immediate thought was how soon do I see this in a board
| game. Hexs are often used to create the game board. This could
| add a lot of variability to board setup.
| robinsonb5 wrote:
| Similarly, I found myself wondering about applications in
| halftone patterns for printing.
| TinkersW wrote:
| There are already techniques for removing the repeating
| textures(sometimes this is called texture bombing). Maybe this
| can be used to improve them, but only if it can be cheaply
| calculated on the fly.
| wetmore wrote:
| Preventing texture repetition is definitely one area you'll see
| techniques like this, e.g. via Wang tilings. Here is another
| example: https://iquilezles.org/articles/texturerepetition/
| 1MachineElf wrote:
| Fascinating article and code examples. Thanks for the link.
| patrickwalton wrote:
| I would love this, but also it would be way more expensive to
| tile, because you can't manufacture a consistent sheet of tiles.
| No two sheets would be alike!
| mnw21cam wrote:
| Can someone ELI5 why this is different from Penrose tiling?
| timmg wrote:
| As I understand it: they found a _single_ shape. Where existing
| Penrose tilings were composed of two (or more) shapes.
| phkahler wrote:
| But in one sense this is actually 2 shapes that are mirror
| images. It's still really cool, but I don't think it is
| ultimately what we've been looking for. As proof that it's
| not, we all know that a paper presenting one that doesn't
| need its mirror image to tile the plane aperiodically would
| still be a big dea.
| pohl wrote:
| _...they found a single shape_
|
| Kind of, though, right? One could also look at it as they've
| found two shapes that happen to be reflections of each other.
| Jarmsy wrote:
| Rigid transformations of a single shape
| https://en.wikipedia.org/wiki/Rigid_transformation
| dekhn wrote:
| It's not really a single shape since the tiling contains the
| shape's reflection which is normally considered a different
| shape since its handedness changes.
| sojuz151 wrote:
| It is just a single tile type, not two
| phkahler wrote:
| The two Penrose tiles are also an affine transformation of
| each other.
| OscarCunningham wrote:
| I don't think they can be if you want to enforce the
| matching rules using the tile's shape.
| hgsgm wrote:
| > matching rules using the tile's shape.
|
| That excludes affine transformation.
| Jarmsy wrote:
| Whereas in this new tiling there's a single shape and its
| _rigid_ transformations.
| lsaferite wrote:
| That's covered in the article.
| [deleted]
| ggrelet wrote:
| It has a paywall.
| schiffern wrote:
| https://archive.is/RW7Wy
| jlund-molfese wrote:
| But almost every HN post to pay-walled content includes an
| archive.is link to bypass the paywall.
| lsaferite wrote:
| They didn't indicate they couldn't read the article due to
| a paywall. They just asked for an ELI5 on something that's
| covered in the 4th paragraph of the article. My reaction
| would be different if you said you couldn't read the
| article and asked the same question.
| nashashmi wrote:
| somewhere in this arrangement of tiles is a picture of the world.
|
| --veritasium
| hgsgm wrote:
| Is the proven? Aperiodic is not the same as "full measure".
|
| 101001000100001... is aperiodic but doesn't contain every
| finite string.
|
| But you could say that becaus it contains an infinite set of
| distinct finite substrings, it can be put in bijection with any
| countable set of objects. That's not a "picture" in common
| parlance, via any sort of structured encoding, it's just an
| index.
| rascul wrote:
| Might be interesting to consider if/when such tiles are available
| for purchase.
| bitterlesson wrote:
| If you have access to a laser cutter, you can make them out of
| wood or acrylic. You may find a laser cutter at your local
| library or maker space. I'd be happy to make tiles for you at
| the cost of materials and shipping.
| bookofjoe wrote:
| https://archive.ph/RW7Wy
| EGreg wrote:
| So the Wang hypothesis was disproven, then?
| snowram wrote:
| Sounds like the start of BLIT by David Langford.
| tagami wrote:
| On an infinite plane, math is telling us that there is no
| pattern. Is this correct?
| DFHippie wrote:
| Somebody needs to manufacture a cookie cutter in this shape.
| shireboy wrote:
| I'm confused - I see multiple repeating patterns. three light
| blue hats in triangle around dark blue. grey boomerang pattern.
| two white tiles with same rotation and layout. I'm sure I
| misunderstand what is meant by "pattern that never repeats", but
| please dumb this down for me.
| bqmjjx0kac wrote:
| The claim is that this tesselation is aperiodic, meaning that
| the entire pattern does not have translational symmetry.
| pfortuny wrote:
| Whatever the size of a pattern you find, you cannot tile the
| plane with _translations_ (just translations) using that
| pattern: you need to rotate it. The definition is just that.
|
| So, there are "patterns" (as a matter of fact, the elementary
| tile is repeated infinitely on the tiling) but you cannot fill
| the plane with mere translations of one.
|
| See:
| https://personal.math.ubc.ca/~cass/courses/m308-02b/projects...
| taneq wrote:
| It only repeats for a little bit.
| wccrawford wrote:
| It looks to me like it repeats in 1 direction pretty quickly,
| but in other directions doesn't repeat at all... At least, as
| far as I can tell from the samples given.
| jsd1982 wrote:
| I was about to ask the same question but it appears the repeats
| are not exactly identical at the edges.
|
| Still, can one prove this is aperiodic from geometry alone? It
| seems rather difficult to actually prove that fact. Feels
| intuitive that there must be a period somewhere, however large
| it may be, on the infinite 2D plane.
| HelloNurse wrote:
| For most sets of shapes a periodic tiling is possible, but by
| no means guaranteed.
|
| For example, consider rectangles with sides of 1 and 3 units:
| they can cover the plane periodically (e.g. in a simple
| rectangular grid), but also aperiodically, because you can
| form a square grid of square 3 by 3 units "metatiles", each
| encoding one bit of information in the vertical or horizontal
| orientation of the narrow rectangles; then it's easy to break
| symmetry by orienting metatiles so that for all integers m
| and n some metatile differs from the metatile m rows and n
| columns away, so the period cannot be m rows and n columns.
| didericis wrote:
| I'm assuming the infinite sequence of segments along a straight
| vertical line at each "x-coordinate" (not sure how to say this,
| but you can see vertical lines made of blocks, mean those) were
| proved to be unique and non repeating
| mbi wrote:
| "An aperiodic tiling is a non-periodic tiling with the
| additional property that it does not contain arbitrarily large
| periodic regions or patches" [1]
|
| So, it has "islands" of repenting combinations of tiles, but
| these islands do not repeat / translate in a regular way.
|
| 1: https://www.wikiwand.com/en/Aperiodic_tiling
| msm_ wrote:
| For example, you can't replace these tiles with just
| rectangles. For regular shapes you can, for example if you tile
| your bathroom with squares there will be a repeating 2x2 square
| pattern too. Similarly, hexagons and triangles can be replaced
| with rectangular tiles (where every tile is the same). You
| can't do this for this pattern.
| sophacles wrote:
| Those configurations appear often, but they don't appear with a
| period. Look at a chessboard.. i can make it out of individual
| black and white tiles, or out of pre-assembled units of 1 white
| and one black tile (or a strip of length 4 or squares of size
| 4, etc). I can make a chess board by grouping the base units
| into a pattern, then only using that pattern. I can put that
| pattern on a wheel and roll it along a surface forever to get
| an infinite chessboard pattern.
|
| There's not a way to do that sort of grouping for these tiles.
|
| Compare pi. I can find the numbers representing my name in
| ascii an infinite number of times in the digits of pi, but if i
| find it once, there's no information in where to find it again,
| i can't just move forward n digits to find it, then another n
| digits to find it again, and ao on.
| paulrpotts wrote:
| Thank you for explaining "aperiodic" in a way that makes
| sense to non-geometers!
| OscarCunningham wrote:
| If you were to make two copies of the pattern and superimpose
| them on each other, they would never match perfectly no matter
| how you rotated them and shifted them around.
|
| However, they might match in small patches, just never across
| the entire infinite plane.
| fsckboy wrote:
| wait, you must mean if you created a copy on top of another
| copy, they would match like that, but there's no combination
| of shifting, rotation, mirroring, etc that would also match?
| (unless it you shifted it back to the starting orientation)
|
| cuz it would be mind blowing if you made a copy and it wasn't
| a copy... pauli whackamole exclusion tiling
| uptownfunk wrote:
| https://archive.is/RW7Wy
| swayvil wrote:
| It's based on a sorta chunkified kisrhombille tiling (which is
| pretty sane). Which is based on 1-2-sqrt3 triangles. (Which are
| deeply humdrum).
|
| So we have a serious "infinite chaos out of plain order"
| situation here. Which I call impressive.
|
| We have like 10 different chaoses, depending on how you do your
| first tile. What would a superposition look like?
|
| And it's pretty easy to organize, given that it's based on the
| chunkykisrhombille.
|
| Hmmm. What powers would it give us, bigstructurewise?
| aliljet wrote:
| What a fantastically enjoyable read. I really want to understand
| how this question arises and what's actually being tested and
| invented in coming to the solution to this. And, side note, this
| is absolutely going to be the strategy I use to paint one of my
| office walls.
| bitsinthesky wrote:
| Now, how many colors would you need so that no two adjacent
| regions have the same color?
| thefringthing wrote:
| All four, since it's fairly easy to find an odd wheel in the
| tiling.
| pmayrgundter wrote:
| That suggests that the layout at some radius R in the distance is
| unpredictable without "running" it, in the Wolfram computational-
| irreducible sense.
|
| They say they have "a new kind of geometric incommensurability
| argument", and there are many statements about the related
| undecidability of related tiling classes.. but not really
| grokking this.
|
| Anyone know?
| hooverd wrote:
| Floor installers HATE this one simple shape!
| OJFord wrote:
| Just the one? I have no idea how this is described
| mathematically, but just looking at the image in the article, the
| shape spans three hexagons, comprising 2/6 sectors of two of them
| and 4/6 of the third. I have no idea what I'm talking about, but
| it seems like it 'ought' to scale to larger (or at least some
| larger) polygons, or number of them spanned, even excluding
| trivial multiples (or 6/6 covered ones inserted in the middle)
| that are effectively the same shape.
| Someone wrote:
| There probably are countless others, but this is the first that
| we know of.
|
| And I wouldn't know whether trivial multiples that still tile
| the plane non-periodically exist. Once you pick a multiple,
| even the claim that any of these basic structures in the plane
| is part of the multiple you picked doesn't seem to have an
| obvious, trivial (1) proof to me, let alone the additional
| requirement that you can find non-overlapping ones.
|
| (1) I'm trying, likely unsuccessfully, to dodge the problem of
| triviality in mathematics
| (https://en.wikipedia.org/wiki/Triviality_(mathematics)) here
| odgaus wrote:
| On their project page [1] they even mention that there is a
| (infinite) family of shapes
|
| >> The hat is one member of a continuous family of shapes
| that are all aperiodic, and that all tile the plane in the
| same way.
|
| [1] https://cs.uwaterloo.ca/~csk/hat/
| dclowd9901 wrote:
| From a maths standpoint, I'm curious what the analogy to
| numbers is. Would this tile be like an irrational number? A
| prime number? But in 2d space?
| hgsgm wrote:
| In 1D you can't have a single shape tile aperiodically, since
| there is no room for variation.
|
| You could make a 2D diagram of the integers with they prime
| factorizations, which is aperiodic, but nearly periodic, but
| requires an infinite set of different "tiles".
|
| Perhaps you could take an irrational or transcendebtal
| number, and take its multiples or powers mod 1, to get an
| aperiodic nearly periodic sequence.
| penteract wrote:
| In the paper(linked in other comments), they say it's part of a
| family of such tiles which can be generated by changing some of
| the edge lengths.
| justinator wrote:
| Something tells me if one was to look closely you'd find this
| tile in Clark Richert's work, probably while he was living in an
| artist commune in South Colorado in the 60's (like they did with
| the Penrose Tile, which he got sued for - and won, since he
| showed prior art).
|
| https://www.google.com/search?q=%22Clark+Richert%22+Art&tbm=...
| swayvil wrote:
| That is a seriously chewed cookie.
| ubj wrote:
| > Until now, it wasn't even clear whether such a single shape,
| known as an einstein (from the German "ein stein" or "one
| stone"), could even exist.
|
| The actual topic of the article was impressive, but this little
| fact about the meaning of "ein stein" was pretty interesting as
| well. TIL.
| wongarsu wrote:
| Then you might also like Spielberg being German for "play
| mountain" or "game mountain", Adelson being German for "son of
| nobility" (though the son ending is more common in Nordic
| countries, the meaning is likely the same), Zuckerberg being
| German for "sugar mountain", Rosenberg being German for "rose
| mountain" or Friedman being old German for "peaceful man" or
| "protecting man".
| xdennis wrote:
| Also, it's pronounced neither "steen" nor "stayn", but "shtayn"
| or /StaIn/ in IPA.
|
| (I was once hearing someone talk about privacy and how people
| like Tsucabuc are destroying it. I never heard about him but
| apparently he is one of the owners of a large social media
| company. Then he mentioned Facebook and I realized he was
| pronouncing Zuckerberg in German.)
| chaxor wrote:
| Useful to remember when someone says you're 'dumb as a rock'
| wussboy wrote:
| I suppose you'd need to respond with, "Which rock?
| Specifically."
| kzrdude wrote:
| Penrose tilings have 5- or 10-fold symmetry right, what does this
| have? Maybe triangular symmetry? In their coloring I see lots of
| three-studded shuriken-like shapes.
| OscarCunningham wrote:
| This colouring makes the 3-fold symmetry more visible:
| https://mathstodon.xyz/@Danpiker/110062396666001681
| hgsgm wrote:
| It's obviously hexagonal (which includes triangular) just by
| looking at the tiles and the angles of the edges.
| satvikpendem wrote:
| There's a great Veritasium video about aperiodic tiling:
| https://www.youtube.com/watch?v=48sCx-wBs34
| LorenDB wrote:
| In the second image, the tiles look like West Virginia.
| zokier wrote:
| I notice that each tile has 5 or 6 neighbors. This reminds me of
| "football" tiling[1] so I wonder how would this hat tiling look
| on non-euclidean geometry, e.g. on a spehere
|
| [1]
| https://commons.m.wikimedia.org/wiki/File:Comparison_of_trun...
| beeforpork wrote:
| Definitely nice for bathroom tiles! Staring at the wall while
| doing your business and failing to find a repeating pattern --
| wonderful!
| scythe wrote:
| The unique feature of the new tile isn't that it will tile
| aperiodically. It's that it will _only_ tile aperiodically. If
| you just want an aperiodic tiling, you can achieve it with 2x1
| rectangles. There 's a big list of such patterns here:
|
| https://tilings.math.uni-bielefeld.de/
|
| examples:
|
| https://tilings.math.uni-bielefeld.de/substitution/domino-va...
|
| https://tilings.math.uni-bielefeld.de/substitution/tetris/
| remram wrote:
| Thanks for posting this. It should be higher.
|
| I can't believe NewScientist left that out of their headline.
| Asooka wrote:
| >Bielefeld university
|
| I'm not clicking that, I don't trust I'll be able to come
| back.
| teach wrote:
| For the downvoters, I'm pretty sure this is supposed to be
| a veiled reference to the
| https://en.wikipedia.org/wiki/Bielefeld_conspiracy
| kzrdude wrote:
| I've heard of this injoke a lot, but it doesn't seem to
| have any real core of fun to it, it's just interesting
| because it's an injoke?
| janfoeh wrote:
| It is the earliest German bit of net culture to take an
| offramp from the information superhighway into regular
| culture - at least the earliest I know of. Maybe that
| gives it its staying power, humorous value aside.
| mandmandam wrote:
| [dead]
| asteroidz wrote:
| Who needs a never-ending smartphone feed if you can have a
| never-ending tile pattern to gaze upon!
| bell-cot wrote:
| If I recall (previously-submitted article -
| https://cp4space.hatsya.com/2023/03/21/aperiodic-monotile/ ),
| the shape has to be flipped upside down some fraction of the
| time. So you'd either need two tile shapes for a real bathroom,
| or your tiles would be a compromise between "both faces are
| easily cleaned" and "both faces stick firmly to the mortar".
| Psychlist wrote:
| Having a white tile and a blue tile would be fine, though.
| I'm really tempted by this, but I think I should start by
| tiling the garage floor or something rather than my whole
| living area.
| bryan0 wrote:
| > The ratio of unreflected to reflected tiles is ph^4 : 1
|
| I just love this
| kossTKR wrote:
| Cool, but for some reason i find it pretty hard to look at?
| jtms wrote:
| stop looking at it :)
| gfd wrote:
| Sounds like a pretty tough job due to the lack of pattern.
| Either you have to follow a template or you run the risk of
| randomly tiling something that can't actually be extended
| further.
| dclowd9901 wrote:
| The tiles seem like they can only join at one spot to one
| spot so I don't think it would even be possible to lay them
| incorrectly unless you straight were jamming incorrect sides
| together.
| zokier wrote:
| > First we produce a list of possible neighbours of the hat
| polykite in a tiling. There are 58 possible neighbours when
| we only require such a neighbour not to intersect the
| original polykite; these are shown in Figure B.1, with that
| original polykite shaded. The first 41 of these neighbours
| remain in consideration for the enumeration of 1-patches.
| The final 17 are immediately eliminated (in the order
| shown) because they cannot be extended to a tiling: either
| there is no possible neighbour that can contain the shaded
| kite (without resulting in an intersection, or a pair of
| tiles that were previously eliminated as possible
| neighbours), or we eliminated Y as a neighbour of X and so
| can also eliminate X as a neighbour of Y.
|
| From the preprint. It is definitely possible to lay the
| tiles so that they do not tile anymore.
| ChrisMarshallNY wrote:
| It's been a looooong time ....
|
| But I could see this making a "psychedelic" experience a bit
| more interesting.
| DoctorMckay101 wrote:
| Cowntdown until either Numberphile or Matt Parker does a video on
| this. starting now
| MontagFTB wrote:
| My kids and I geeked out over the Veritasium video on Penrose
| tiles (https://www.youtube.com/watch?v=48sCx-wBs34). It is pretty
| exciting to see approachable math like this being discovered
| before our eyes.
| litoE wrote:
| I dont't get it. In the picture, each tile is composed of 8
| identical quadrilaterals. So why don't these quadrilaterals
| constitute a simpler shape that can tile a wall and never repeat?
| teawrecks wrote:
| Someone below mentioned that the important part isn't that it
| tiles aperiodically, but that it ONLY tiles aperiodically.
| There are simpler shapes that will tile both aperiodically and
| periodically.
| twanvl wrote:
| This tile forces a pattern that does not repeat. If you use the
| simpler shape you can tile a wall such that it never repeats,
| but you can also make a repeating pattern.
| plopilop wrote:
| The key point (often missed by these articles) is that
| aperiodic tilings do not have a (infinite) periodic pattern.
| This means that you cannot draw a shape on these tiles and say:
| "the tiling is based on infinite repetitions of this shape, and
| only this shape".
|
| Of course individual tiles will repeat, but never in an
| infinite periodic pattern.
|
| Edit: a novelty of this paper is that their shape is "truly"
| aperiodic, which means no matter how hard you try, you will end
| up with aperiodic tiling. Existing one-shape aperiodic tilings
| had to add constraints on how to put two shapes next to each
| other to ensure aperiodicity.
| notfed wrote:
| "Of course individual tiles will repeat"
|
| What does this mean? How can an individual tile "repeat"?
| plopilop wrote:
| I meant "a given orientation of the tile will be present
| many times (or infinitely many)".
|
| It's very probable (I did not read the paper, only their
| website page) that the tile only occupies a finite amount
| of orientations in the tiling and therefore at least (and
| probably more if not all) one orientation will also be
| present an infinite amount of times.
|
| However this does not imply periodicity of the tiling.
| mg wrote:
| They provide this demo:
|
| https://cs.uwaterloo.ca/~csk/hat/app.html
|
| I just wished one could turn off the colors.
|
| The colors really distract me from trying to see the patterns the
| shape itself creates. For me, the beauty here is that each piece
| is exactly the same. Colorizing them differently takes away from
| that.
| sjaak wrote:
| Try this:
|
| main { filter: saturate(7) grayscale(10) contrast(3); }
| gwbas1c wrote:
| > I just wished one could turn off the colors.
|
| It's not too hard to scrape the JavaScript out of the page. You
| could figure out where they set the colors and change that
| part.
|
| I also wonder if you can do that on
| https://mathigon.org/polypad/8kVqVH2Mor6JTQ
|
| There's also https://cs.uwaterloo.ca/~csk/hat/, but sadly I
| didn't see anything like a github link to the above demo.
| magicalhippo wrote:
| Ever since we went hunting for tiles for our first remodeling,
| I've been thinking about why not Wang tiles[1][2] were available.
|
| I mean obviously it'd be too much cost and hassle, since you need
| at least 5 different tiles to tile a plane, but I'm still curious
| how it would actually turn out on a real floor or wall, with an
| interesting pattern on the tiles.
|
| While you need multiple Wang tiles, at least they can be square
| rather than a rather awkward polygonal shape. So there's that...
|
| [1]: https://grahamshawcross.com/2012/10/12/wang-tiles-and-
| aperio...
|
| [2]: https://en.wikipedia.org/wiki/Wang_tile
| angry_moose wrote:
| Purely from an aesthetics standpoint, I'm guessing its because
| they have a tendency to form quasi-repeating patterns and long
| rivers which are generally unsightly (which is subjective):
|
| The long string of horizontal yellow diamonds and upside down
| blue "darts" in this image:
| https://upload.wikimedia.org/wikipedia/commons/thumb/4/40/Wa...
|
| Or the red "dominoes" in this image:
| https://grahamshawcross.files.wordpress.com/2012/10/13tiling...
| srcreigh wrote:
| seems as if this could be solved by using a different
| palette. The red sticks out cuz it's deep dark red more than
| cuz it's dominoes. Maybe.
| RC_ITR wrote:
| Maybe, I think in general though, this falls into uncanny
| valley territory.
|
| Humans like patterns or randomness, we sort of hate quasi-
| versions of either (See: our disdain for blurry pictures).
| cromulent wrote:
| They did a mall in Helsinki with Penrose tiles, I really like
| it.
|
| http://www.neverendingbooks.org/penrose-tiles-in-helsinki
| throwawaymaths wrote:
| The Salesforce center in SF looks like it's Penrose tiles,
| but I suspect they are cheating and are using a large segment
| repeated on each section of the skirt.
| RC_ITR wrote:
| Maybe, but Penrose himself was involved, so I'd be very
| bummed if that were true.
|
| SAN FRANCISCO--(BUSINESS WIRE)--The Transbay Joint Powers
| Authority (TJPA) has received approval from Dr. Roger
| Penrose, the eminent British mathematical physicist, to
| incorporate his groundbreaking geometrical pattern in the
| design of the exterior walls of the future Transbay Transit
| Center (TTC) designed by Pelli Clarke Pelli Architects
| (PCPA). Dr. Penrose and PCPA are working in tandem to
| incorporate Dr. Penrose's elegant design, known as the
| Penrose Rhombus Tiling, in the skin of the TTC. The design
| is remarkably simple but unique because it can be extended
| infinitely without repeating itself. The Penrose system is
| ideal for the perforations in the metal panels that will
| form the curved exterior of the Transit Center.
|
| https://www.businesswire.com/news/home/20130711006350/en/Ro
| g...
| iviv wrote:
| Out of curiosity, is it idiomatic to call a street next to a
| shopping center a mall? Ignorant Finn here.
| aaron695 wrote:
| [dead]
| bobbylarrybobby wrote:
| Sometimes. "Mall" alone generally refers to a big indoor
| shopping space with many distinct stores; I'd probably call
| the outdoor version an... "outdoor mall".
| cromulent wrote:
| No, probably my age and background. Street with shops and
| no cars.
| mjevans wrote:
| I think in the US that would colloquially be called an:
|
| Outdoor mall
|
| Mall alone being the 80s style indoor walkable variety. A
| 'strip mall' usually a single (often deformed to some
| degree) line of stores along a sidewalk next to a huge
| parking lot, also often with an island restaurant or
| small store that wants to stand out closer to the street
| edge of said lot.
| Sharlin wrote:
| Yeah. I wonder how the tilesetters actually did the concrete
| tile-laying part. There are "quasiperiodic" Penrose tilings
| that are composed of fairly regular "macroblocks", but this
| Keskuskatu tiling looks pretty random to me. Did they just
| have huge printouts of the pattern? Did they develop an
| intuition of which tile goes where after a while? Were they
| happy to do something different for a chance, or annoyed by
| the convoluted task? :D
| travisjungroth wrote:
| I don't think this would be too much cost and hassle at all.
| People make all sorts of fancy tiles and patterns with them.
| Four colors as triangles is so simple I'm really surprised it
| doesn't exist. Maybe a good niche online business for someone.
| swayvil wrote:
| Re : cool tiling options
|
| There's girih tiles. It's like 5 tiles that you can do a bunch
| of stuff with. Invented by some based geomystics like 1000
| years ago.
|
| And there's kisrhombille. Lots of options there.
| deelly wrote:
| I love the idea, maybe not in wiki colors but with some
| stylization. One concern thought, how can I know how many tiles
| of each type I will need?
| dist-epoch wrote:
| * for some definition of "never repeat".
|
| Most people would call that pattern obviously repeating (in the
| shape itself).
| jschveibinz wrote:
| I'm not a mathematician, but it's interesting to think about this
| as a projection onto 2d. What can be said about the multi-
| dimensional shape that creates this projection, or even if that
| is possible?
| ddingus wrote:
| That was my first thought too.
|
| Very interesting. There are specific regions that repeat, but
| the overall image does not.
| akomtu wrote:
| This tiling is hexagons painted differently, and hexagons are
| cubes projected onto a plane.
| swayvil wrote:
| This geometry is a cousin of that geometry
|
| http://www.fleen.org/generative_art_project/i0_quartersize.p...
| frankus wrote:
| Is the similarity to a Dragon Curve
| (https://en.wikipedia.org/wiki/Dragon_curve) a coincidence? This
| post (https://cs.uwaterloo.ca/~csk/hat/) mentions a substitution
| system, which makes me think there might be a connection.
| teraflop wrote:
| As far as I know, the resemblance is superficial. (For one
| thing, the standard dragon curve is based on 90deg angles, and
| this tile has a mixture of 90deg and 120deg angles, giving it a
| 6-fold pseudo-symmetry.) There are many other non-periodic or
| aperiodic tilings that are based on substitution rules, and
| many of them look totally different:
|
| https://tilings.math.uni-bielefeld.de/substitution/penrose-r...
|
| https://tilings.math.uni-bielefeld.de/substitution/fibonacci...
|
| https://tilings.math.uni-bielefeld.de/substitution/semi-deta...
|
| In addition, the dragon curve is a fractal -- mathematically,
| it's defined as the limit that the substitution process
| converges to as the details get "infinitely small", which means
| that in a sense, the true dragon curve (as opposed to the
| approximation that you can draw on a computer) has a boundary
| with no straight line segments at all. On the other hand, an
| aperiodic tiling is composed of finite, fixed-size tiles that
| extend outwards to infinity.
|
| Funnily enough, the dragon curve is a space-filling curve that
| tiles the plane _periodically_.
| youssefabdelm wrote:
| I wish it were a little more random in a sense... just enough so
| that the brain doesn't get "bored" of the evolution of the
| pattern, but not too much randomness that the randomness itself
| becomes like white noise (yet another pattern the brain can get
| "bored" of)
|
| Would be extremely curious if patterns like the one described
| exist in math.
| mckeed wrote:
| I feel Penrose P2 tiles are better in that sense. I think you
| could craft something with that kind of "fractal
| interestingness" by using color to emphasize the larger regular
| patterns that can occur in a Penrose tiling.
|
| https://en.wikipedia.org/wiki/Penrose_tiling#Kite_and_dart_t...
| MC_10 wrote:
| Previously on HN: https://news.ycombinator.com/item?id=35242458
| gniv wrote:
| Also: https://news.ycombinator.com/item?id=35264965
| sampo wrote:
| And: https://news.ycombinator.com/item?id=35265569
| mc32 wrote:
| Roughly it looks like a T-shirt. One with a jagged bottom. Jagged
| T-shirt tile.
| tromp wrote:
| Yes, this article goes more into the attire aspects:
|
| https://aperiodical.com/2023/03/an-aperiodic-monotile-exists...
| johndough wrote:
| Project website with web demo https://cs.uwaterloo.ca/~csk/hat/
|
| Direct link to PDF on ArXiv (89 pages)
| https://arxiv.org/pdf/2303.10798.pdf
| dmtroyer wrote:
| Kind of looks like overlapping t-shirts.
| iamben wrote:
| https://archive.is/iqBHP
| paulpauper wrote:
| This seems like the sort of thing Terrance Tao or a computer
| should have solved long ago.
| ReaderView wrote:
| [dead]
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