One thread in that linked discussion was asking if one could buy ceramic tiles to tile a floor/wall, but no links provided. I'd really love to use one of these spectre tiles to redo a bathroom or backsplash. If anyone finds or knows of a manufacturer making these tiles please provide a link!
Yes, the one with reflections was found months ago, and got a lot of coverage and discussion here. Then the one without reflections was found more recently (and also had a few discussions here). This article is about the first of these, so rather old-hat now, not even old-turtle.
I still maintain this naive belief that all of math is elegant, and from that perspective 13 sides sounds and looks oddly specific and feels contrived to me, at least without knowing specificities of the field.
Where do 13 sides come from? Is it related to a number of transformations?
This is just the first tile we've found with this property. There might be a tile with the same property and a smaller number of sides. Perhaps the such tile with the smallest number of sides will be especially elegant.
Well, from one perspective, since the requirement is that a tile must be able to tile the plane only irregularly;
- All three-sided shapes can tile the plane regularly,
- All four-sided shapes can tile the plane regularly,
So we know the minimum number of sides is at least five.
Assuming 13 sides is established as the minimum, I suspect there is no 'nice' reason for it; its just that this may be the minimum number to give you sufficient degrees of freedom.
It's based on a tiling of kites. You combine multiple kites together to create the aperiodic monotile. It just happens that the perimeter of the tile has 13 sides.
Lol it repeats alright... blah blah blah "it doesn't technically repeat"... To the brain it does, it's like white noise. If I showed you an evolving animation of this shit for 3 hours it would be torture. Someone should measure something like the multiscale entropy of this pattern.
Edit: I will say the coolest thing about this is the cross-disciplinary connection hints at a metapattern: https://youtu.be/48sCx-wBs34?t=1007
It needn't be multiscale entropy precisely, but anything similar to it.
The way I intuit it, though I could be wrong, is that as you have a "ground level" or "close" measurement of white noise, it appears complex and varied. As you go up and up in the scale, you get the "eagle eye's view" of white noise, you realize "oh shit, it's all kinda the same!"
So it wouldn't necessarily prove anything, but if the entropy dropped as you scaled, it would show that the pattern isn't really that complex, and there's a lot of hidden redundancy. I mean it is called a "pattern" after all to be fair.
The plane is infinite, so the special property that mathematicians want is that the tiling doesn't contain arbitrarily large repeating patches. Any tiling that necessarily repeats itself will contain arbitrary large repeated (translated) chunks when the whole infinite plane is considered. The tricky thing is to create a set of one or more tiles that can only tile the plane without repetitions.
It's a simplified explanation of the geometric concept of "tiling the plane" where shapes are placed, like flooring tiles, onto a 2D surface, a plane. In this case I think pattern is just the layman's terms being used for simplicity's sake because, when you look at the assembled tiles, it's the same shape repeated (but not the same repeated configuration of several of these shapes, which is what makes this novel).
This is not mentioned until half way through, but I find fascinating..
> David Smith, a retired printing technician and nonprofessional mathematician, was the first to come up with the shape that could be a solution to the long-standing “einstein problem.” He shared his ideas with scientists who took on the challenge of trying to mathematically prove his conjecture
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[ 3.6 ms ] story [ 55.3 ms ] threadEdit - a visual explanation of this journey: https://www.youtube.com/watch?v=IfVwelta1fE
edit: and a live stream chat with the article's authors, worth a watch for the quirky tilings by Yoshiaki Araki alone: https://www.youtube.com/watch?v=OImGgciDZ_A
https://news.ycombinator.com/item?id=35274773
Where do 13 sides come from? Is it related to a number of transformations?
Anyway, I agree that people like to focus on the elegant parts. Math popularization materials have too much kawaii math.
It has been proven using a computer. The problem was first reduced to a few hundred cases, then a brute force algorithm was used to solve each case.
So we know the minimum number of sides is at least five.
Assuming 13 sides is established as the minimum, I suspect there is no 'nice' reason for it; its just that this may be the minimum number to give you sufficient degrees of freedom.
The monster: https://youtu.be/mH0oCDa74tE
Ideal packing of squares (some are elegant, 17 is not) https://kingbird.myphotos.cc/packing/squares_in_squares.html
The 4 color theorem was solved by reducing it down to 633 cases and just using a computer to find a coloring for each case: https://en.m.wikipedia.org/wiki/Four_color_theorem
Mathematicians discover shape that can tile a wall and never repeat (newscientist.com) - 488 points, 3 months ago, 160 comments
https://news.ycombinator.com/item?id=35242458 - 104 points, 3 months ago, 22 comments
https://news.ycombinator.com/item?id=35264965 - 222 points, 3 months ago, 52 comments
https://news.ycombinator.com/item?id=35265569 - 3 points, 3 months ago, 1 comment
The irony is palpable.
Edit: I will say the coolest thing about this is the cross-disciplinary connection hints at a metapattern: https://youtu.be/48sCx-wBs34?t=1007
The way I intuit it, though I could be wrong, is that as you have a "ground level" or "close" measurement of white noise, it appears complex and varied. As you go up and up in the scale, you get the "eagle eye's view" of white noise, you realize "oh shit, it's all kinda the same!"
So it wouldn't necessarily prove anything, but if the entropy dropped as you scaled, it would show that the pattern isn't really that complex, and there's a lot of hidden redundancy. I mean it is called a "pattern" after all to be fair.
https://en.m.wikipedia.org/wiki/Aperiodic_tiling
Most definitions I could find (I am not a mathematician) seems to imply one of: repetition, growth or shirking.
This shape appears to have none of these properties but they still call it a pattern.
https://en.wikipedia.org/wiki/Tessellation
> David Smith, a retired printing technician and nonprofessional mathematician, was the first to come up with the shape that could be a solution to the long-standing “einstein problem.” He shared his ideas with scientists who took on the challenge of trying to mathematically prove his conjecture