Attack on the pentagon results in discovery of new mathematical tile (2015)(theguardian.com)
theguardian.com
Attack on the pentagon results in discovery of new mathematical tile (2015)
https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile
15 comments
Nicely done. I envy your etched windows.
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... :)
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.
I think there is a lot of latent amateur mathematical talent going underutilized.
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.
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.
It didn't need a subject domain expert- just computer engineers who understood the problem and how to test counterexamples efficiently.
I think most of us has unused talents in several other fields we could have gone into.
What matters is that we do good work in the field we're in.
What matters is that we do good work in the field we're in.
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].
[0]: https://www.scottaaronson.com/blog/?p=304
[0]: https://www.scottaaronson.com/blog/?p=304
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.
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.
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?
> attack on the pentagon
Wow that definitely sounds like a clickbait title.
Wow that definitely sounds like a clickbait title.
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.
Are we seriously not doing "phrasing"?
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!
You might be interested in these https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_m...
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!
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!
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.
[1] https://en.wikipedia.org/wiki/Pentagonal_tiling
[1] https://en.wikipedia.org/wiki/Pentagonal_tiling
It's an open problem to find a single aperiodic tile. It's called the Einstein Problem.
https://en.m.wikipedia.org/wiki/Einstein_problem
https://en.m.wikipedia.org/wiki/Einstein_problem
The einstein problem (lower-case E).
Not to be confused with the physicist.
Not to be confused with the physicist.
TIL ein stein means one tile.
It's an open problem for non-convex tiles (and seems to have a solution for non-continuous tiles).
Previous discussion:
https://news.ycombinator.com/item?id=10045297
Man those 3 years went quick!
https://news.ycombinator.com/item?id=10045297
Man those 3 years went quick!
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.
Do a polished concrete floor but stamped with the tiling pattern cut out through a CNC.
Done. Super cheap.
Done. Super cheap.
Penrose tilings are aperiodic. The stamp (or combined area of all CNC-ed stamps) would be as large as the floor.
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.
Nice.
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.
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.
You could use a programmable laser cutter, I suppose. Or less expensively, but slower and with more waste, just score and use a tile breaker.
That would be a very boring process. And you would probably lose some of them.
Cutting tiles by hand is not difficult, it's just annoying. Very annoying. Especially in weird shapes.
Cutting tiles by hand is not difficult, it's just annoying. Very annoying. Especially in weird shapes.
Is it really feasible to cut ceramics with a laser? I'd say these only do (thin) wood pieces?
This paper [0] seems to contain information you would want to do it yourself with a common medium power CO2 lasercutter.
[0]: https://www.sciencedirect.com/science/article/pii/S003039929...
[0]: https://www.sciencedirect.com/science/article/pii/S003039929...
Yes. UV laser cutters should be able to cut at least certain ceramics. Not certain about IR.
Not cheap, but https://www.aliexpress.com/w/wholesale-ceramic-tile-laser-cu...
And perhaps they're bullshitting.
And perhaps they're bullshitting.
The article says that "tiling the plane" is:
"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?
"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?
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.
Afaik the tiling should be able to fill a finite plane of arbitrary size. Thus, the same tiling should also be able to go on forever.
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.)
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.)
"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.
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.
Border is irrelevant, for what it's worth it can be whatever you want
Very dishonest title
> “We discovered the tile using using a computer to exhaustively search through a large but finite set of possibilities,”
Brute force. That's my kind of math.
Brute force. That's my kind of math.
> 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?
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?
Hint: The interior angles of any four sided shape add up to 360°.
An example of when Title Case would be most misleading.
I want to build boats that are shaped like this, then you can tile them together and create a new country.
Yeah the headline is a bit misleading, it sounded like a cyberattack on a US government institution
But interesting nonetheless
But interesting nonetheless
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.
It's neither capitalized nor near a capital, so I can't object the headline. It's about pentagons.
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.
It's clickbait, but it's at least a step cleverer than most.
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Yea it's clickbait, but I could easily see mathematicians working on the problem making this joke themselves. I found it very charming.
Startup idea: ceramic tiles for geeks.
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. :)
[0] https://i.guim.co.uk/img/static/sys-images/Guardian/Pix/pict...
[0] https://i.guim.co.uk/img/static/sys-images/Guardian/Pix/pict...
Is acrylic ok?
https://acryliccommunity.com/en/the-xtx-project/
https://acryliccommunity.com/en/the-xtx-project/
This.
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
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
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!