Mathematicians Have Found a Shape with a Pattern That Never Repeats

Mathematicians Have Found a Shape with a Pattern That Never Repeats(smithsonianmag.com)

86 points by rfreytag 2 years ago | 43 comments

2 months later we have the "Spectre" - an aperiodic monotile (without reflections): https://aperiodical.com/2023/05/now-thats-what-i-call-an-ape...

Edit - a visual explanation of this journey: https://www.youtube.com/watch?v=IfVwelta1fE

shrx 2 years ago | |

Also previously discussed on HN https://news.ycombinator.com/item?id=36119920

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

idbehold 2 years ago | | |

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!

jossclimb 2 years ago |

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

samwillis 2 years ago |

Significant previous discussion: https://news.ycombinator.com/item?id=35273707

Mathematicians discover shape that can tile a wall and never repeat (newscientist.com) - 488 points, 3 months ago, 160 comments

paulddraper 2 years ago | |

And before that

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.

once_inc 2 years ago |

This was found months ago, no? I saw several articles about the Einstein back then.

ape4 2 years ago | |

The shape doesn't repeat but the posts do

candiddevmike 2 years ago | | |

And in each new story, the name of the person who first discovered it is further and further down the page.

knodi123 2 years ago | | |

Even the comments do

Jarmsy 2 years ago | |

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.

thisGuyFlicks 2 years ago | |

Did you try clicking into the article? If you would have, you would've seen the published date and saved us all this unneeded comment thread.

edgyquant 2 years ago | | |

No the point is this is a repost and those are suppose to wait a year

andrethegiant 2 years ago |

Oh shit new shape just dropped

knodi123 2 years ago | |

Didn't get enough traction the first time?

https://news.ycombinator.com/item?id=35274773

softbt 2 years ago | | |

Your memory is insane

andrethegiant 2 years ago | | |

history, unlike this shape, repeats itself

Wherecombinator 2 years ago | |

Google non repeat

FireInsight 2 years ago | | |

it's everywhere....

roberthahn 2 years ago |

I would be very interested in seeing what definition of “pattern” this discovery is using.

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.

twandiz 2 years ago | |

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).

https://en.wikipedia.org/wiki/Tessellation

Maxion 2 years ago |

I guess this means that I now should re-tile my bathroom. hmm.

forkerenok 2 years ago |

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?

OscarCunningham 2 years ago | |

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.

Jarmsy 2 years ago | |

Plenty of mathematics is ugly. Just look at optimal packing. People like to focus on the elegant parts.

gus_massa 2 years ago | | |

A lot of the ugly part of math is hidden inside lemas and theorems, so the main part of the proof can be hopefully straightforward.

Anyway, I agree that people like to focus on the elegant parts. Math popularization materials have too much kawaii math.

GuB-42 2 years ago | | |

A notorious "ugly" proof is that of the 4-color theorem.

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.

kfrzcode 2 years ago |

I don't get it. The first image I can clearly see patterns of triangular arrangements. What does this news actually mean

fihfob 2 years ago | |

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.

https://en.m.wikipedia.org/wiki/Aperiodic_tiling

kfrzcode 2 years ago | | |

Fascinating, curious how it can be proven, but I'll go do some reading thanks

youssefabdelm 2 years ago |

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

JimBlackwood 2 years ago | |

What would multisce entropy prove about this pattern? I’m not even sure how you’d apply it. Iirc, multiscale entropy is specifically for timeseries.

youssefabdelm 2 years ago | | |

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.