I wish I know how to cut these shapes out of regular square tiles in large quantities. I would love to have Penrose tiling on other exotic patterns in my bathroom.
Within a fixed distance of a given type of tile, the possibilities are not unlimited. Stamp with occasional overlap, and you could get the job done with only a few different stamps.
"If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps"
So the trick, if I'm understanding this, it's that the shape must have sides that can push up against other of its sides, leaving no gaps. Obviously a circle isn't gonna do it.
Does the plane have to be a plane in the mathematical sense, i.e. infinite? Because then you're obviously not going to cover it with anything. It goes on forever.
If just a section, does the plane have to be a square or other rectangle?
The way many things relating to infinity in math are described is something like this:
You get to pick a size of tile. You can pick whatever you want, totally independently. 700x700, 1000000x1000000, whatever, it's fine.
If I can describe a mapping for that arbitrary size tile, then I'm able to map an "infinitely" large plane. That even as we approach infinity, no matter the size of the finite plane, I can make an arrangement of tiles that fully covers the plane, means I've "infinitely" mapped the plane. (In some cases where there's uncertainty you might say something more like we've shown that the limit as plane size approaches infinity is mappable.)
Making the plane infinite makes the problem easier, since you don’t need to worry about the boundary of any finite section of the plane. For example, if you choose a triangular section, you won’t be able to tile that section by squares. But it is still clear the infinite plane can be tiled by squares; just keep laying the squares edge-to-edge.
You can cover an infinite plane easily enough. You just need an infinite number of tiles. And that's no problem in math. It's no different from saying, for example, that the intervals of form [n,n+1), where n is an integer, cover the whole real number line.
"Does the plane have to be a plane in the mathematical sense, i.e. infinite? Because then you're obviously not going to cover it with anything. It goes on forever."
You certainly can tile an infinite plane. In fact, you can view this algebraically if it helps, by talking about tiling the complex plane. For example, the Gaussian Integers, which are complex numbers of the form (a+bi) where a and b are integers, can be viewed as the corners of squares that tile the complex plane.
Geometrical shapes usually don't suffer attacks, also if you wanted to specify a generic class of shapes, you would write "Attack on pentagons" as 'the pentagon' implies it's one specific pentagon.
Before I clicked into the article I assumed it was some discovery related to the aftermath of the September 11th attacks. Of course I see now that "pentagon" is not capitalized. It's just that as an American when you put a definite article before the word "pentagon" it conjures a very particular idea.
Indeed. Some of the examples [0] are very aesthetically pleasing. I imagined them as season-themed stained-glass windows, representing leaves, nuts, icicles etc., maybe in a house with indoor/outdoor rooms for each season. :)
I have been looking for a company to produce me rhombus-shaped tiles to do Penrose tiling[1]. Manual labor and gearing costs just make it prohibitively expensive for smallish batches. In the end, I'm just going to cast cement tiles myself and do it in the garden instead of my living room.
Wikipedia [1] says it has been proven in 2017 that no convex pentagons that can tile the plane other than these 15 types exist. I've been wondering if there's a single convex tile that can only tile the plane irregularly, guess not.
Sometimes I feel like there's not much left people haven't figured out, and then I see something like this: pentagons that can tile a surface aren't fully worked out mathematically. You'd think that would be something we'd have covered, but no — there's wonder left in the world!
Notably some of them like the Millennium Prize offer a reward of 1M dollars if you solve it. There's definitely still some absolute mysteries out there!
For those looking to create patterns like these in real life, I've had some success describing aperiodic tiling expansion rules in PostScript: https://github.com/steiza/postscript_fractals
From there you can send it to a laser cutter (see link for pictures), vinyl cutter, plasma cutter, CNC machine, etc.
PostScript is particularly great because it's a stack based language, and aperiodic tilings are often defined by expansion rules that are basically recursive algorithms. Just don't go too deep or you'll blow the stack!
I've long fantasized about using penrose tiling for my bathroom. Alas, my craftsmanship is terrible and I can barely manage subway style tiling.
I think someones could make some side money doing these one-offs for local builders, remodelers. Every service bureau I've ever met with their own laser (CNC, 3D printer) has spare capacity.
I've also long wanted to have client-side procedurally generated backgrounds for web pages, and user interfaces in general. Wood grain, marbling, penrose tiling, fractals... I got something kinda working using applets with Sun's HotJava browser, once upon a time.
I just saw that CSS now has a paint image hook. Made me think it's time to rescratch that itch. Unless someone beats me to it... :)
It really isn't that hard to make custom shape tiles by hand using the average Joe tools but it's a long, precise and tedious process.
I kinda miss doing it with my step father blasting Chris Rhea on the stereo and slapping my head whenever I was "unattentive" and had a 1mm difference from the model, it isn't a huge difference but add it up to all the tiles and you can get a pretty messy job in the end...I miss those days.
It's fascinating to me that Marjorie Rice, a housewife in her 50s, essentially taught herself, created her own mathematical notation, and found four pentagons on her own. (https://en.wikipedia.org/wiki/Marjorie_Rice). I'm astonished and encouraged by her work.
Makes one wonder what it take to become an "ameture" mathematician? And how does this signifier provide any credit to those who've been given this distinction?
You might not say that if you’ve seen some of the amateur “proofs” of Fermat’s Last Theorem, the Riemann Hypothesis, or other famous statements. Really, in order to contribute to mathematics, one needs, at minimum, a serious mathematical education, intelligence, and lots of persistence. Nobody can provide anyone with intelligence or grit, but the easiest way to get the required education is graduate-level study in mathematics, which is most easily done within the system itself. Beyond that, knowledge of and ability to navigate the culture of mathematics is helpful. There is a reason that not using Tex/LaTeX, and presenting standard material as if it were new are among the 10 signs Scott Aaronson lists that a paper supposedly presenting some mathematical breakthrough is wrong[0].
I said there was latent talent, not that cranks are legitimate.
I do agree that to contribute in most areas of science and math one needs a great deal of training and persistence. Not so sure about intelligence.
In my entire career, there have been a couple times when a complete outsider with little or no experience in a field came in and revolutionized it. But in those cases, generally the person was scientifically educated in an unrelated field.
There's a number of Numberphile videos I've seen where there are computer checks being done on various conjectures famous enough to on a video series like Numberphile, and even IIRC one case where a counterexample was found by an amateur with a computer. I've often thought there must be a lot of much less famous conjectures that are probably something an amateur with a computer could attack, and could well be the first to try. Or be the first one to try with gigabytes and teraflops, if the last person tried in an era where "kilo" still sounded pretty impressive.
Math is well covered, but the frontier is larger than ever; it grows every day. There's just a certain art to finding them.
Another field I've wondered about that may be relevant to HN interests is quantum algorithms. There's a lot of very smart people working the theory and the engineering and such, but I wouldn't be all that surprised there's a lot of room for even a relative amateur to learn about what quantum computers can do, and find practical speedups to existing problems, not because nobody is trying but because the field of existing problems is just so large that there almost has to be low hanging fruit still available. Theorists right now are mostly interested in solutions that change O() classes, but someone just having some fun will care if they can find a 100x speedup that is technically not in a different O() class, or something like that.
As for Conjectures: I tried one of them (http://norvig.com/beal.html) but didn't have any luck (see the part just before the Conclusion). Like Peter says, better spend your brainpower on a proof than your computerpower on finding counterexamples.
It didn't need a subject domain expert- just computer engineers who understood the problem and how to test counterexamples efficiently.
Oh geez, I literally thought that it took 14 years of research to come out with some academic research resulting from the 9/11 attacks when I read the headline. I think that's pretty terrible on both the source website and the OP's part.
Yeah, the title here is extraordinarily misleading. I literally thought that physical custom tile work had been discovered in the Pentagon 9/11 wreckage, which, unbeknownst to the craftsman, was mathematically novel.
> Every triangle can tile the plane. Every four-sided shape can also tile the plane.
The triangle case makes sense to me, intuitively, but the four-sided one makes much less sense (maybe I need a sheet of paper?). Does this also have a “trivial” proof?
66 comments
[ 2.3 ms ] story [ 207 ms ] threadAnd perhaps they're bullshitting.
[0]: https://www.sciencedirect.com/science/article/pii/S003039929...
Cutting tiles by hand is not difficult, it's just annoying. Very annoying. Especially in weird shapes.
Done. Super cheap.
Another notion I had was cutting/forming/milling the ceramic before it's glazed and kiln fired.
If you did it in sheets, it might be practical.
"If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps"
So the trick, if I'm understanding this, it's that the shape must have sides that can push up against other of its sides, leaving no gaps. Obviously a circle isn't gonna do it.
Does the plane have to be a plane in the mathematical sense, i.e. infinite? Because then you're obviously not going to cover it with anything. It goes on forever.
If just a section, does the plane have to be a square or other rectangle?
You get to pick a size of tile. You can pick whatever you want, totally independently. 700x700, 1000000x1000000, whatever, it's fine.
If I can describe a mapping for that arbitrary size tile, then I'm able to map an "infinitely" large plane. That even as we approach infinity, no matter the size of the finite plane, I can make an arrangement of tiles that fully covers the plane, means I've "infinitely" mapped the plane. (In some cases where there's uncertainty you might say something more like we've shown that the limit as plane size approaches infinity is mappable.)
You certainly can tile an infinite plane. In fact, you can view this algebraically if it helps, by talking about tiling the complex plane. For example, the Gaussian Integers, which are complex numbers of the form (a+bi) where a and b are integers, can be viewed as the corners of squares that tile the complex plane.
But interesting nonetheless
Definitely funny, definitely dishonest.
[0] https://i.guim.co.uk/img/static/sys-images/Guardian/Pix/pict...
I have been looking for a company to produce me rhombus-shaped tiles to do Penrose tiling[1]. Manual labor and gearing costs just make it prohibitively expensive for smallish batches. In the end, I'm just going to cast cement tiles myself and do it in the garden instead of my living room.
[1] https://en.wikipedia.org/wiki/Penrose_tiling
https://acryliccommunity.com/en/the-xtx-project/
https://news.ycombinator.com/item?id=10045297
Man those 3 years went quick!
[1] https://en.wikipedia.org/wiki/Pentagonal_tiling
https://en.m.wikipedia.org/wiki/Einstein_problem
Not to be confused with the physicist.
Notably some of them like the Millennium Prize offer a reward of 1M dollars if you solve it. There's definitely still some absolute mysteries out there!
From there you can send it to a laser cutter (see link for pictures), vinyl cutter, plasma cutter, CNC machine, etc.
PostScript is particularly great because it's a stack based language, and aperiodic tilings are often defined by expansion rules that are basically recursive algorithms. Just don't go too deep or you'll blow the stack!
I've long fantasized about using penrose tiling for my bathroom. Alas, my craftsmanship is terrible and I can barely manage subway style tiling.
I think someones could make some side money doing these one-offs for local builders, remodelers. Every service bureau I've ever met with their own laser (CNC, 3D printer) has spare capacity.
I've also long wanted to have client-side procedurally generated backgrounds for web pages, and user interfaces in general. Wood grain, marbling, penrose tiling, fractals... I got something kinda working using applets with Sun's HotJava browser, once upon a time.
I just saw that CSS now has a paint image hook. Made me think it's time to rescratch that itch. Unless someone beats me to it... :)
What matters is that we do good work in the field we're in.
[0]: https://www.scottaaronson.com/blog/?p=304
I do agree that to contribute in most areas of science and math one needs a great deal of training and persistence. Not so sure about intelligence.
In my entire career, there have been a couple times when a complete outsider with little or no experience in a field came in and revolutionized it. But in those cases, generally the person was scientifically educated in an unrelated field.
Math is well covered, but the frontier is larger than ever; it grows every day. There's just a certain art to finding them.
Another field I've wondered about that may be relevant to HN interests is quantum algorithms. There's a lot of very smart people working the theory and the engineering and such, but I wouldn't be all that surprised there's a lot of room for even a relative amateur to learn about what quantum computers can do, and find practical speedups to existing problems, not because nobody is trying but because the field of existing problems is just so large that there almost has to be low hanging fruit still available. Theorists right now are mostly interested in solutions that change O() classes, but someone just having some fun will care if they can find a 100x speedup that is technically not in a different O() class, or something like that.
It didn't need a subject domain expert- just computer engineers who understood the problem and how to test counterexamples efficiently.
Brute force. That's my kind of math.
Wow that definitely sounds like a clickbait title.
The triangle case makes sense to me, intuitively, but the four-sided one makes much less sense (maybe I need a sheet of paper?). Does this also have a “trivial” proof?