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Discovery of a new irregular pentagon that can cover the plane
- devindotcom 11y agoThis is cool, but I'm not sure why it's a hard problem to solve for a computer running through a ton of 5-sided polygons with some basic rules and attempting to tessellate them? Is there a reason that approach doesn't work, other than being rather un-romantic as far as discovering cool new things like this goes?
- pavpanchekha 11y agoYeah, great reasons why it doesn't work—what 5-sided polygons are you going to run through? The hard bit is the side lengths; there are good reasons to think that the angles cannot be too weird. But for example, this pentagon has one side of side length 1 / (sqrt(2) (sqrt(3) - 1)). How long is your exhaustive enumeration going to go until it finds that one? Anyway, it sounds like clever enumeration is exactly what these authors did—but you do have to be clever to find something at all.
- Pyxl101 11y ago> 1 / (sqrt(2) (sqrt(3) - 1)) I don't mean in any way to demean their research, but I think your exhaustive search could build up formulas in exactly that form and iterate on them. It starts with a valid pentagon. Then it permutes that pentagon with an evolutionary method by modifying the formula. For example, maybe the lengths it tries are: 1 / 2 1 / sqrt(2) 1 / sqrt(2) * 3 1 / sqrt(2) - 1 1 / sqrt(2) * (3) 1 / sqrt(2) * sqrt(3) It would try both these and many others along the way. 1 / exp(2) ... 2 / sqrt(2) ... Again, not to trivialize, but there are only so formulas made up of a fixed number of terms and operators, and as long as it's easy to check whether a shape is a valid pentagon, and whether it tiles, then I think you could check a considerable number of them. It looks like a number of the pentagon formulas have a few sides with complex lengths, while the others are simple or equal to each other, so you could bias the algorithm to search for those. I'm sure there's way more complexity I'm overlooking, but that's how one might get started.
- ntoronto 11y ago"I'm sure there's way more complexity I'm overlooking, but that's how one might get started." That's how I'd start. There's probably a way to make the search a lot smarter. Here's the part you're overlooking, though: "... there are only so [many] formulas made up of a fixed number of terms and operators..." "So many" = "countably infinite." Paring it down to finitely many would require understanding tantamount to having solved the problem in the first place. [Edit: I can words.]
- rtpg 11y agoThis could work for approximations. But if you want the exact value, this might not be super effective, as there are values without closed forms (that is to say, that are not expressible simply with the *-+/ and sqrt/log and other usual operations). So your iteration procedure will probably get to a good approximation, but might not find the "core" expression.
- gizmo686 11y agoIn this case, (with the hindsight of knowing the answer), it would probably be easier to enumerate the angles, as they were all rational multiples of pi (with the biggest denominator being 12). Still, that gives a naive 5^144 combinations of angles to test. If you reduce this to proper, reduced fractions this comes down to 5^47 [1]. You can probably improve on this further by requiring that the angles form a (convex) polygon, but I suspect that still leaves you with to many to be practical. [1] http://oeis.org/A005728 http://oeis.org/A005728
- tlarkworthy 11y agoThe angles don't seem so weird though
- JoshTriplett 11y agoI'd be even more curious about the computation used to determine if a given set of sides and angles tessellates, and how long that computation takes. Approaches to enumerate possible expressions don't seem that far-fetched, if the resulting possibilities can be evaluated quickly.
- foota 11y agomind elaborating on why the angles can't be too weird? I thought it was odd that the angles given were all integers.
- gizmo686 11y agoThe one thing that the unit of "degrees" has going for it is that it tends to give integer answers (because 360 is so divisible). If you convert them into rotations around a circle, they angles become (1/6, 3/8, 7/24, 1/4, 5/12). (Multyply those by τ=2π to get the angle in radians). EDIT: To your question as to why angles can't be to weird, consider vertices, where the corners of the pentagons meet. Each vertex is composed of three angles, one from each of the three neighboring vertices (although it is not a-priori obvious that a vertex contains only three angles). The sum of these angles (in rotations) must be 1. This means that, heuritstically, you would want as many permutations of the angles as possible to add up to one.
- DanBC 11y agoThat approach does work. From the article: "We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey.
- bradjohnson 11y agoRight? I love when people ask "Why don't they do X?" without even bothering to read the damn article. Does the parent think that people just randomly stumbled upon this shape?
- nikanj 11y agoJust going to leave this here https://xkcd.com/793/ https://xkcd.com/793/
- nether 11y agomath dept is getting shitfaced tonight...
- cft 11y ago"That same year an unlikely mathematical pioneer entered the fray: Marjorie Rice, a San Diego housewife in her 50s, who had read about James’ discovery in Scientific American. An amateur mathematician, Rice developed her own notation and method and over the next few years discovered another four types of pentagon that tile the plane. " https://en.wikipedia.org/wiki/Marjorie_Rice https://en.wikipedia.org/wiki/Marjorie_Rice
- shoo 11y agoRice's pentagons are illustrated here [1]: Type 9, Type 11, Type 12 and Type 13. [1] http://www.mathpuzzle.com/tilepent.html http://www.mathpuzzle.com/tilepent.html
- morcheeba 11y ago>Types 6-8 were found by R. B. Kershner in 1968. >Type 10 was found R. James in 1975. >Types 9, 11-13 were found by M. Rice in 1976-1977. Dammit, R. James! You can't just skip numbers like that and mess up Rice's streak!!
- 1arity 11y agothat's awesome. Outsider math and outsider art deserve as much respect as "outsider" programming ( startups / garage hackers ) garner. Creating semi-closed communities of insiders, united by what are essentially arcane handshakes, and dismissive of value emerging outside of themselves, is really just both a narrow-mindedness that doesn't behoove innovation and also an admission of an unnecessary insecurity about the lack of substantive reasons for cohesiveness in that community resulting in the desire to fabricate arbitrary insubstantial reasons for cohesiveness. At the same time, you could say that narrow-mindedness was focus and restraint, things which __do__ work for innovation. Another way of looking at the cohesiveness issue, is that if your community is so substantially strong, why do you need to exclude outsiders ? The best democracies include difference, revealing their strength through diversity. The worst extremist fascist states put it to death, revealing their dread of diversity. Let the ad-hoc groups of intellectual collaborators be like democratic meritocracies not frightened facist states.
- Vexs 11y agoI think it's kinda funny that most of the tessellations are just using pentagons to make other shapes that tessellate naturally. I suppose the same could be said of most tessellations though, but it's still interesting.
- Someone 11y agoThat's because there are only 17 wallpaper groups (https://en.m.wikipedia.org/wiki/Wallpaper_group https://en.m.wikipedia.org/wiki/Wallpaper_group). Any _repeating_ pattern must match one of them (non-repeating patterns by definition do not)
- cLeEOGPw 11y agoCan you NOT link to mobile version? Mobile phones switch to mobile automatically, desktop version does not.
- sp332 11y agoThe mobile version is very readable on a desktop browser, and it's probably easier for you to edit the URL on your PC than for the poster to edit it using their phone.
- snaily 11y ago...all effort being equal. And assuming, of course, that comments are read (on average) once.
- sp332 11y agoAssuming that the total effort expended by the few who really dislike the mobile interface is less than fiddling with a URL on a phone, plus gratitude toward the person for taking the effort to make the post at all.
- chillingeffect 11y ago
- jagermo 11y agovery well explained article. even I (with my little grasp on math) can see why its a big thing. Cudos to the author
- nickysielicki 11y agoKudos for not using the same clickbait title The Guardian did. Journalistic integrity is dead.
- matthewmacleod 11y agoIt's a pun, not clickbait. This is not the death of "journalistic integrity".
- teddyh 11y agoNo, the original title is clearly clickbait – meant to be misinterpreted as something other than what the article is about. Here is what a real pun in a newspaper headline looks like: http://threepanelsoul.com/2010/03/22/on-astronomy-minors/ http://threepanelsoul.com/2010/03/22/on-astronomy-minors/
- bbcbasic 11y agoIt's a pun PUN: Attack on the pentagon results in discovery of new mathematical tile CB: Breaking News: Pentagon attack
- nickysielicki 11y agoYeah, it's a pun. So what? The title being a pun doesn't make it any less misleading. Puns and clickbait are not mutually exclusive. My real gripe is that there's a time and place for everything and a newspaper ought to know and respect that line. You don't mislead someone skimming your headlines into thinking the pentagon has been attacked for the sake of a joke. I like clever headlines as much as the next guy, and I understand that for the sake of a joke you might leave some ambiguity in the air and mislead your reader slightly, but this goes far past that. I think his editor should have caught this.
- bbcbasic 11y agoI think using the word 'mathematical tile' indicates that it is a maths/geek article rather than serious news, and takes this into the safe zone. Just. It's a fine line though.
- 1arity 11y agoPerhaps the more impressive number is that they found 7 quintillion new irregular pentagons that can't tile the plane.
- oldstrangers 11y agoI've read your comment a few different ways. I'm concluding that you find it impressive that it took 7 quintillion tries to find a new pentagon that can tile the plane.
- wingerlang 11y agoI think he just meant that the number 7 quintillion is an impressive number, more than 15.
- laverick 11y ago"Attack on the pentagon" ...aaaand the guardian has lowered themselves to buzzfeed's standards.
- mahouse 11y agoAs you can see, it's some kind of column.
- Dylan16807 11y agoIsn't that every newspaper that has ever existed?
- matt4077 11y agoNot at all. Witty headline have a long history which was unfortunately cut short by google which, at least for a while, only rewarded the most boring, literal drivel. It's good to see the tradition making a comeback. This headline is also completely unlike the buzzfeed-clickbait headlines. Buzzfeed tries to exploit psychological weaknesses (Mathematicians attack the pentagon – you won't believe what happened next). Headlines like the Guardian's are much better in that they're still entertaining after you've read the story. Sometimes I think they're written more for the amusement of the editor than anything else. Other guardian headline I saved: "Blight in Italy leaves pine nut nuts pining for more"
- unfamiliar 11y agoIts a joke, I found it pretty funny.
- CPLX 11y agoI think it's more that they have raised themselves to The Economist standards.
- est 11y agoMeta question: where can I find list of simple unsolved/undiscovered problems like these in math? It does not appear in https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...
- jbrooksuk 11y agoI don't think that tiling a plane is exactly a "simple" problem.
- est 11y agoI mean something simple enough to understand, and could be played by average programmer in a IPython Notebook. I didn't mean simple to solve, just lower barrier of entry.
- leni536 11y agoI like this one: http://www2.stetson.edu/~efriedma/squinsqu/ http://www2.stetson.edu/~efriedma/squinsqu/ Can be easily generalized to other shapes and more dimensions too.
- pavel_lishin 11y agoDoes the lack of n = 16 mean that there's no current proof?
- leni536 11y ago> For the n not pictured, the trivial packing (with no tilted squares) is the best known packing.
- Dylan16807 11y agoWhy not? It seems simple to me. Simple doesn't mean easy. Factoring a 2048 bit number is very simple and very hard.
- deleted 11y ago[deleted]
- ab 11y agoIt's interesting how your eye naturally groups the pentagons into the larger primitive unit, like the pinwheels of type 5. Wolfram Alpha also has some things about tiling: http://www.wolframalpha.com/input/?i=pentagon+tiling http://www.wolframalpha.com/input/?i=pentagon+tiling http://www.wolframalpha.com/input/?i=pentagon+type+5+tiling http://www.wolframalpha.com/input/?i=pentagon+type+5+tiling
- infinity0 11y agoMany of these types are basically combining two pentagons into an octagon (or even hexagon) then tiling it across the plane. For some reason, intuitively those seem more easy to me (2 * 3, 2 * 4), so that you could just generate a bunch of them, and split them in two to create tessellating pentagons? Even the example in the article can be viewed as a regularly tessellating nonagon. I don't see what's "irregular" about it? The article doesn't mention that word, but the HN title does.
- raverbashing 11y agoirregular - with sides of different sizes https://en.wikipedia.org/wiki/Regular_polygon https://en.wikipedia.org/wiki/Regular_polygon
- ColinWright 11y agoSides and/or angles. It's possible to have all the sides the same length, and yet still not be regular. Similarly, it's possible to have all the angles the same, and yet still not be regular. It is instructive to construct examples of both types of failure.
- raverbashing 11y agoAh true. A lozenge has all sides with the same size but not the same angles
- Someone 11y ago"then tiling it across the plane": see https://news.ycombinator.com/item?id=10046349 https://news.ycombinator.com/item?id=10046349. "just generate a bunch of them, and split them in two" Try splitting a 'random' polygon into a set of identical convex (regular or irregular) pentagons. "what's "irregular" about it?": this has been answered, but for completeness: the sides and angles of the pentagons aren't all identical.
- huuu 11y agoBesides tiling your bathroom is there any use case for this? Or is this pure for fun and gaining knowledge? Edit: The article is talking about building structures but isn't a triangle the most rigid form? And triangles are already used in building.
- fractalb 11y agoShould it be discounted if it were done for pure fun?
- matt4077 11y agoSure there are. There are many processes where you want to "tile" a surface with identical smaller parts. I'd assume it's something that happens with solar panels. Possibly also on the molecular level with coatings etc. Having more options for the shape of the constituent parts may be an advantage. see also https://en.wikipedia.org/wiki/Buckminsterfullerene https://en.wikipedia.org/wiki/Buckminsterfullerene, a molecule inspired by architecture.
- chronic40 11y agoAnd rectangles aren't good enough.
- akent 11y agoCan I actually tile my bathroom with tiles like this? Do any tile manufacturers make them that shape? EDIT: I found these, at least http://www.ballerhouse.com/2010/08/19/precision-concrete-tiles/ http://www.ballerhouse.com/2010/08/19/precision-concrete-til...
- brunnsbe 11y agoFor those interested in the subject here is an interesting video about using penrose tiling for street tiling in Helsinki, Finland: https://www.youtube.com/watch?v=yxlEojkVJ0c https://www.youtube.com/watch?v=yxlEojkVJ0c
- zimpenfish 11y agoThere's a lovely building at North Greenwich which is Penrose'd. http://wordpress.mrreid.org/2011/02/26/penrose-tiling/ http://wordpress.mrreid.org/2011/02/26/penrose-tiling/ http://www.bdonline.co.uk/foa%E2%80%99s-peninsula-patterns-for-ravensbourne-college/3144928.article http://www.bdonline.co.uk/foa%E2%80%99s-peninsula-patterns-f...
- jessaustin 11y agoI'd never heard that Penrose sued Kleenex. Not his finest hour.
- unfamiliar 11y agoI wish I had a bathroom to tile - I reckon this could be considered "in vogue" for the next 30 years or so, until they find a newer pentagon. Does anyone know if this can be coloured with 3 colours? Obviously 4 is possible due to the 4 colour theorem and 2 will not work due to to three faces sharing a corner.
- rlpb 11y ago> Obviously 4 is possible due to the 4 colour theorem... Unless your bathroom includes a loop, such as all four walls (even with holes for windows and doors) or over the ceiling. Then the coloured area is no longer a plane, and so the 4 colour theorem does not apply.
- T-hawk 11y agoWouldn't all four walls still function as a plane, topologically? Think about looking into a cube (box) with one side open, so that you see five faces. The projection of those faces onto your retina or a photograph is a direct mapping to a plane. Four walls plus the ceiling are equivalent to that open box. Four walls minus the ceiling are equivalent to a plane with a hole. You have to include both the ceiling and the floor to break out of plane topology. And what you get is a sphere.
- Chinjut 11y agoA sphere is also four-colorable, functioning equivalently to a plane for such purposes; consider stereographically projecting the sphere (minus a point) onto a plane.
- jordigh 11y agoWhat's the point group for the first tiling on that webpage? It's a periodic tiling (unlike a Penrose tiling, which is only quasiperiodic), so the crystallographic restriction for two dimensions says the rotation subgroup of the point group must be one of C_2, C_3, C_4 (not C_5!) or C_6: http://mathworld.wolfram.com/CrystallographyRestriction.html http://mathworld.wolfram.com/CrystallographyRestriction.html I can't tell by eye-balling it what the symmetry is for the first one, but its periodicity says it must be one of those. Quasicrystals with 5-fold symmetry are not exactly periodic. There are only 17 wallpaper groups. Since this is a wallpaper, what is its group? https://en.wikipedia.org/wiki/Wallpaper_group https://en.wikipedia.org/wiki/Wallpaper_group
- dooglius 11y agoFrom the table on the wikipedia page, it looks like it's p1
- ai_ja_nai 11y agowe can shape building blocks like those, now
- zkhalique 11y agoTake that, Penrose! LOL
- deleted 11y ago[deleted]
- hinkley 11y agoI don't know why, but this was the first question that popped into my head after seeing the diagrams: Does anyone make bricks in these shapes? Those would make an awesome paver pattern.
- cammil 11y ago"Every triangle can tile the plane. Every four-sided shape can also tile the plane." Can someone point me to a proof of this?
- deleted 11y ago[deleted]
- function_seven 11y agoNot a formal proof, but close enough I suspect. See comments as well: http://www.quora.com/Why-do-all-quadrilaterals-tessellate http://www.quora.com/Why-do-all-quadrilaterals-tessellate
- thyrsus 11y agoThe triangle is pretty easy: take two of the (same size) triangles, with vertices ABC and A'B'C'. Rotate and translate the second triangle to fit the matching side of the first triangle, e.g. AB to B'A'. You now have a quadrilateral with two pairs of equal sides (sides (AB')C and (A'B)C'). The angle on the corners comprised of the two triangles (e.g., C(AB')C') will add together to be 180 degrees minus the angle of the adjacent corners, due to the three interior angles of every triangle summing to 180 degrees. Duplicate that quadrangle and fit the second to a matching side. The angles put together will form 180 degrees, i.e., a straight line. Now you have indefinitely extensible strip. Place the strips next to each other and you've tiled the plane.
- sandworm101 11y agoWait a second. Aren't there actually two different pentagon shapes in use here? Look at the yellow and blue in the OP. They are actually mirror images of each other. Maybe a mathematician would say they are the same, but certainly not someone cutting tile for a bathroom floor. And if these were proteins trying to form a cell wall, that mirroring would be a serious hurdle.
- cammil 11y agoAll depends how you define "the same". Mathematicians use "congruency" which allows mirror images.
- sandworm101 11y agoAh. You learn something every day. I guess the guardian was therefore incorrect in their definition and should have said "congruent copies" ... but "copies" is also probably wrong. Congruent congruencies? >"If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps, then that shape is said to tile the plane."
- ngneer 11y ago"Identical copies" is informal, as is "copies" for that matter. You can think of the information that is defining the shape as what is being copied, hence mirroring and other variations are allowed. A possibly related term to look into would be "isomorphism".
- sandworm101 11y agoIs mirroring also allowed when 'filling' a 3d space?
- goldenkey 11y agoMirroring is allowed because in a higher dimension space, it is simply rotating the lower dimensional object by 180 degrees upon its own axis :-) Ie. an ant can't flip the hexagon you lay flat, but you can. Similarly, you cannot "flip" the 3d object you have in terms of chirality, but a 4d being can :-)
- Sniffnoy 11y agoDoes anyone have a link to the actual paper?
- XaspR8d 11y agoI'm not certain there is an actual paper yet; I suspect from some of the phrasings that it was merely an announcement that their program succeeded, but a formal paper will take a while. I did find this reddit post by Dr. Mann [1] where he says: > We were just in the process of debugging and optimizing the code when our new example was found. Because we are in the early stages of the computational experiments, we were surprised to find this example so quickly. We are hopeful of finding more new examples as we proceed. [1] https://www.reddit.com/r/math/comments/3fe347/15th_pentagon/cto75m0 https://www.reddit.com/r/math/comments/3fe347/15th_pentagon/...
- infogulch 11y agoThe first of the image of other tilings at the end of the article is almost obvious: it's just a hexagonal tiling where the hexagons are bisected.
- ben174 11y agoWith only five points and < 180 degrees at each intersection, couldn't this problem have been solved by a computer via brute force in seconds?
- jjxw 11y agoIt kind of was: >“We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey. “We were of course very excited and a bit surprised to find the new type of pentagon.
- tribe 11y agoFrom the article: “We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,” said Casey. “We were of course very excited and a bit surprised to find the new type of pentagon. I would expect it to take longer than seconds, since there are many ways that these shapes can fit together, and there are many possible edge lengths.
- deleted 11y ago[deleted]