Why Penrose Tiles Never Repeat
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- Опубликовано: 30 ноя 2022
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This video is about a better way to understand Penrose tilings (the famous tilings invented by Roger Penrose that never repeat themselves but still have some kind of order/pattern).
This project was a collaboration with Aatish Bhatia (aatishb.com).
REFERENCES
Explore Penrose and Penrose-like patterns: aatishb.com/patterncollider
Video by Derek Muller/Veritasium about Penrose Patterns: • The Infinite Pattern T...
Music algorithmically generated, algorithm designed by Henry Reich
N.G. de Bruijn’s paper introducing the pentagrid/Penrose idea: www.math.brown.edu/reschwar/M...
De Bruijn, N.G., 1981. Algebraic theory of Penrose’s non-periodic tilings of the plane. Kon. Nederl. Akad. Wetensch. Proc. Ser. A, 43(84), pp.1-7.
Here are some excellent in-depth references on how to construct Penrose Tiles Using the Pentagrid Method:
Penrose Tilings Tied up in Ribbons by David Austin: www.ams.org/publicoutreach/fea...
The Empire Problem in Penrose Tilings by Laura Effinger-Dean: www.cs.williams.edu/~bailey/06...
Pentagrids and Penrose Tilings by Stacy Mowry & Shriya Shukla: web.williams.edu/Mathematics/...
Penrose Tiling by Andrejs Treibergs: www.math.utah.edu/~treiberg/Pe...
Algebraic Theory of Penrose's Non-Periodic Tilings of the Plane by N. G. de Bruijn: www.math.brown.edu/reschwar/M...
Particularly good and helpful, and (we think) an undergrad thesis which is impressive!: www.cs.williams.edu/~bailey/06...
An interesting popular science read on the discovery on quasicrystals and their connection to Penrose Tilings:
The Second Kind of Impossible by Paul Steinhardt: bookshop.org/books/the-second...
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4:25 Wow, the proof of why it never repeats is pretty elegant! It also makes sense why a "tri-grid" (triangular tiling) DOES repeat, because sin(120)/sin(60) = sqrt(3)/2/sqrt(3)/2 = 1/1 = 1, which is rational. That explains why, when you take a ribbon of a triangular tiling, you see the same number of upside-down triangles and rightside-up triangles: it's a 1:1 ratio.
Maybe on a curved surface, the ratio can be changed to become rational and a pentagonal tiling does repeat
woah nice catch cary :)
That’s pretty interesting!
@TimesByTwo you just did that
Oh hey cary
you know it’s a good day when minutephysics drops some obscure math problems
scientists make me sad
@@rgw5991 why
@@anonymousfish2456 crippling depression IDK?
@@rgw5991 1a
@@rgw5991 why do scientists make you sad?
I saw a bus seat pattern just a couple of weeks ago and it drove me nuts that the pattern seemed like it should repeat but every time I thought I figured it out there were one or two elements that were off.
Thank you for reassuring me that I'm not crazy! And educating me in an entertaining way at the Same time.
I saw that pattern in the back of bus seats too. Just awful the things some kids carve in there!
The golden ratio shows up in nature a lot because it is the main part of an efficient packing algorithm. Thanks Numberphiles!
do you remember which numberphile video that was? i just checked and they have a bunch of videos on the golden ratio
@@noshiko5398 the 'most irrational' number
@@maxthomas-bland4842 thank you!!!
Finally i understand why it never repeats, veritasium made an interesting showcase but i never understood why it never repeats
I still don't get why it doesn't repeat. Could someone help?
edit: oh my God lol, I thought the video ended at 3:15 when he mentioned the friend's website. Too used to clicking away from sponsors :p
@@iwanttwoscoops I don't have an exact proof, but know the general gist of how it works. With the square and triangular grids, notice how all the intersections of lines all meet at a point, and that the spokes radiating out of that point are all regular and form a neat tiling pattern. Then compare that with the pentagrid, where only some lines meet each other, and you get groups of "near misses" where several lines almost (but not quite) meet. -- It's that almost but not quite meeting that makes the pattern non-repeating.
The number of spokes S is 2 times the number of parallel line sets L, so S=4 for square (L=2) grids, S=6 for triangualar (L=3) grids, and S=10 for penta (L=5) grids. The angle between two closest parallel line sets is 360/S (90 for square, 60 for triangular, 36 for pentagrids). Note how for pentagrids, Henry (in this video at 1:50) notes that the lines intersect at either 36 degrees or 72 degrees -- that is, when a line intersects at 72 degrees (2x36) there is one line missing. I suspect that this also plays a part in figuring out why the tiling can't repeat.
The only bit left really to prove (which is the part I'm not sure on) is proving that you can't make it so that everywhere in a pentagrid where at least 2 lines meet, that there is at least one of those points that does not have 10 spokes (or stated another way, has at least 1 angle between the connecting lines that is 72 degrees).
@@iwanttwoscoops Rather than clicking away just press L (forward 10 seconds) 6 times to jump forward by a minute. If the sponsorship is still going just press 3 times more, since *most* sponsors are between 60 and 90 seconds iirc. If you actually look at the video buffer rather than the recommended videos list you might sometimes see that not only is the video only halfway through, but, for some channels, they actually go through the trouble of chaptering the ads ie the video literally has the ads' beginning and end timestamped, and marked on the video bar. Lastly, there's also the video hotspots on videos which mark the most replayed portion of a video, and *sometimes* that just so happens to be after the ad. Hope this helps!
I notice that they're said to be _quasiperiodic_ and not nonperiodic. This is the thought that came to mind when you started laying out the _parallel ribbons,_ because they definitely have at least some periodic nature.
It's actually not so easy to put the difference between _quasi-periodic_ and _not at all periodic_ in rigid terms.
It's quasiperiodic because a given particular sequence of tiles along a ribbon does repeat over and over again. However, its repetitions occur at irregular intervals, and is overall still non-periodic as well.
It's a bit different than a sequence of integers in which there is no repetition at all, that wouldn't have the feature of quasiperiodic.
@@npip99 if you keep generating random integers you will find every finite sequence infinitely often, so your definition would make random numbers quasiperiodic.
@Artem Down He didn't said that the sequence of integers is random. Could simply be a strictly increasing sequence of integers; then you definitely have no repetition.
That is the difference. Quasiperiodic will not give you all possible sequence. Some sequences are guaranteed not to appear in quasi periodic sequence. Like primes. Primes are not random it is guaranteed that no primes will be divisible by 6 or 10 or 15.
I love how the music is algorithmically generated. Really fits the video!
It does sound pretty horrible tho
@@finnlyonn237 and very annoying, I couldn't focus on the content
@@Glendragon Because it kinda repeats, but never actually does *brain boom*
@@Glendragon I actually liked it, but I can definitely see how it could be annoying or distracting.
@@nahometesfay1112 , it would have been alright if it hadn't been so loud.
The Pattern Collider is fun and free and doesnt ask for any email of details or push cookies at you. Much appreciated Aatish.
The 6-Fold Stepped Plane (3:27 bottom left) looks like a marching crowd to me. To make it select 6 Fold Symmetry and slide the Disorder to the max right. Cheers Mr Henry
Used to have that pattern on a rug a long time ago, it always mesmerized me into checking how quickly i can switch between seeing a pattern of stairs going "up" in one direction vs in another... or seeing the "inner" bits as concave vs convex. :-)
PS Had no idea those were the terms i'll eventually use to describe the options, for that kiddo-aged me it was just "bulgy" vs "holey". ^^
Great explanation Henry!
Omg veritasium
ITS VERITASIUM HIMSELF
What only 3 reply 43 like c'mon
my favourite youtuber here
Bruh only 88 likes that explains why henry doesn't have boring guys in comment sections
Art and Math are best friends. By themselves a lot of people are intimidated by them, yet they can help explain each other and they both in turn become approachable for everyone ❤❤❤
For sure! Apparently a lot of visual art employs the golden ratio, a mathematical constant
this video makes me blame my old geometry teacher for not making class this fun
I would’ve killed myself if my 8th grade geometry teacher busted out a grid with 5 axises like I wasn’t already struggling with two lmao
Largely it's because teachers are paid $50k a year to cover a completely new topic every day on top of crowd control, documentation, assignment creation, grading for up to 120 students every other night, and assessments.
A content creator maybe needs to make a video every couple weeks at least, can have a team, and can devote most of their time for just that one project.
@@spiderplant yes obviously and yeah teachers should be paid more although i don’t know that everything you said is quite true
@@jhawkingsgrey As a former educator of 10 years, there's actually more i didn't bother to mention, such as meetings, frequent trainings, conferences, procuring supplies, writing emails, etc.
@@spiderplant oh maybe consider moving to new jersey i have friends who are teachers and my parents are teachers and i know they don’t have to create their own assignments because that’s normally supplied by curriculum director, they don’t have grading that often, etc etc
The reason why the tiling is aperiodic can be seen more readily when observing the cut-projection method for constructing it.
The Penrose tiling can be seen as a projection of the 5D integer lattice, Z^5, to a specially chosen 2D subspace -- the squares closest to this plane project onto the plane as rhombuses.
The a-periodicity comes from the fact that Z^5 is a regular lattice and the 2D plane lies at irrational angles to the Z^5 lattice root vectors.
Hey it's Dugan Hammock!👋I was just watching your QGR presentation on this very subject a few days ago. I agree, I prefer the cut-projection method for quasicrystal construction but it's neat to see the multi-grid method mentioned here. Quasi-order is so fascinating, especially when investigating physical uses. The fact that quasicrystals can inherit symmetries from their higher-dimensional parent crystals (as in Fibonacci) is intriguing. There was a great paper earlier this year about using a Fibonacci-based quasi-periodic drive system to stabilize a quantum computer against several error modes via emergent dynamics (DOI 10.48550/arXiv.2107.09676 for preprint). I think I'm quasi-obsessed but I'm still trying to wrap my head around some of the QGR stuff you work on. 🤯
@@StackCanary Thank you! 👋
I should note that the multi-grid method allows for a much wider variety of tilings than the cut-project method.
Only certain special arrangements of multi-grids can be re-contextualized into a cut-project scheme from a regular lattice.
Also there are cut-project schemes which can not be re-contextualized as multi-grid constructions.
It is also possible to a cut-project of an arbitrary honeycomb or well-behaved tiling -- it's is possible to take a cut-project of a quasicrystal tiling to get a more different quasicrystal tiling in a smaller dimension.
My brain exploded trying to read this lol. Sounds awesome but I have no idea where I would even begin with something like that.
@VJDugan I am not a mathematician, but what you wrote gave me an intuition to why there are no solutions in radicals to the quintic (or higher order) equation (i.e. Abel's impossibility theorem).
@@VJDugan ΩΩΩΩ
Are you familiar with quasicrystals? They are similar to normal crystals, but instead of having a normal repeating unit cell their atoms are-you guessed it-penrose tiled More or less). They were long predicted and made in the lab, but only recently have been found in nature. Could make an interesting video!
I will check it out. [DiowE]
They got the Nobel Prize for quasicrystals
check out the book 'the second kind of impossible'
Hey! That is part of the video where I first heard about this (Veritasium's, 2 years ago) Personally, I thought this had a more elegant mathematical proof but touched on fewer outside implications(Not really a fault of minutephysics, though. Just different styles)
Wow! I'm really into this now. Could you perhaps share any resources on this?
I would love to see how far the research has come on this subject...
That was simple and intuitive. And my respect for Penrose only increases the more I know about his work.
I love how you acknowledged your explanation doesn't meet the requirements of a proof, but still gives us enough baseline information to follow why without needing a math degree to follow along.
Go back to the beginning, with the green & blue tiles. If you cross your eyes, like it's a stereoscopic image, you can see very well defined straight lines following the pentagrid. Line up two areas with identical patterns, and the pentagrid pops out like it's floating above the Penrose tiles.
That is so rad!
I watched Veritasium's video about Penrose tiles 2 years ago and I couldn't understand why it's never repeating, but your video made it very clear! Thank you!
I suppose - as I remember - that the aim of that video was not to prove this attribute.
This is the perfect kind of math/science video we need. Thank you. I wish other channels were as good as yours.
4:49 This was a cool proof! Pretty much the highlight of the video. Also nice to see minutephysics drop.
I opened the Pattern Collider and, for some reason, my first experiment was to play with 3-fold symmetry. Then I shifted the pattern variable down to 0 and got a very nice result that CGP Grey would like.
Hexagons are the Bestagons.
Moorish tilework, that should be pointed out for anyone wanting to learn more. There is also sacred geometry involved beyond the flower of life/golden ratio. In real life the patterns continue across multiple planes(walls and ceilings). It’s incredibly breaktaking to witness in real life. I believe there were other cultures who knew how to create irregular patterns, but the Moors made massive rooms with this stuff.
Please do one on the newly discovered a periodic monotile
Interesting things happen to the (4:29) ratio as the grid goes from a 3-grid, to a 4-grid, to a 5-grid, and so on.
To see it graphed out, paste this text string
3gaxkag510
into the desmos calculator address bar.
Penrose is one of my favorites. I only learned about all this when Derek over at Veritasium did his video on this. BUT! I never knew you could scale it up with additional sets! This is absolutely GOD-TIER because I'm planning to tile my living room with penrose tiles, and you just opened up a whole plethora of new tile designs for me? I made my own based on the pentagon, like Penrose did. Now I have to EXPERIMENT!! THANK YOU!! 😻😍💖👍
Sidenote: previously I knew of penrose via his diagrams related to space-time. So many reasons to be in awe of the dude!
Incredible vue work by Mr. Aatish. I will be reading the source of this!
Thanks for the video Henry!
I like the background audio, it sounds fitting to the topic of something that never repeats itself
Thank you! Super cool. I've been interested in Penrose tilings for some time but never knew the underlying structure. I want to use them for marquetry patterns. Now the Pattern Collider gives me a lot more options.
not only the video but the references!!!! well done!
I just read yesterday Penrose’s Wikipedia page and I wondered what that pattern is, but skipped because I was interested in other things. Hugely interesting!
For quite some time, I have had this hypothesis that maybe a Penrose tiling does repeat, but you have to go so far to find it that it appears that it never repeats. Now, you have shown me why it is impossible for it to repeat.
I already knew about this, but I enjoyed the way this video presented it.
Wow! I was thinking about these patterns the other day! thank you for these amazing videos!
Super fascinating! I really like this video! It reminds me of my days studying computer graphics!
The premise of this video exactly aligned with my experience. I believed this conclusion because sources I trusted said so, but it was deeply unsatisfying, because their arguments never truly made me understand WHY we KNEW the pattern couldn't repeat. THIS video finally scratched that itch. From unrelated concepts, I eventually absorbed how different rational and irrational are, and new neurons have formed in my brain to link Penrose to my brain's continent of math knowledge.
Fascinating stuff! I quite liked the marimba music playing in the background! :)
The binary numbers at the start are also non-periodic. If you count from zero up and put all of the numbers in a single row
0110111001011101111000...
You can get an infinite number of repeating segments of any size, but since each number is larger than the previous, the pattern never repeats
Likewise if you do it with decimal numbers you'll eventually hit 123456789, which is a repeat of the first nine numbers, but it's not periodic as the next number doesn't start with 10..., its 123456790
No shit. That was the point of showing it as an example
@@B3Band you don't have to shoot someone down for sharing their thoughts, we're all here to contemplate these things
Im not sure if i understand what you mean by 123456789...do you mean for irrational numbers? Bc you can definitely have 123456789 repeat in an infinite decimal, and that *would* be periodic
@@Konchok_Dawa no, he means natural numbers (including zero) expressed in decimal form
if you write out all decimal numbers in decimal form, you can always add a pattern, that never occured before
for example
0123456789 has no repeated pattern
you can co on
01231467891011121314151317181920
if you pick a random digit from form the list, it might occur elsewhere - lets say 1, which occurs multiple times locally (like for example stars in the pentrose pattern)
you can then extend this pattern by another random digit (before or after) and you are less likely to find this pattern - lets say 12 - we can find den squence 12 multiple times in our list, 2 times to be exact
now add another digit, 121 - this is there exactly one time
obviously we can extend this sequence by adding all numbers with 3 digits to have a list von 012345678910111213 ... 999 to find, that 121 occurs multiple times now at least at the edge of 12 to 13 like before and obviously when adding 121
so when we add another random digit number to the list, we might not find it in our existing list - like the penrose pattern, when you select your pattern to search big enought, it will be unique
I asked some people about generating noise by stacking waves together at different angles and they said it would end up repeating. I think this really proves that even with very regular angles, frequencies, and amplitudes, that definitely doesn't happen.
Not sure what you mean by angles, but if you want to generate noise, you could do it by overlaying two repeating sound snippets, one with duration 1s and one with sqrt(2)s. This will never repeat because sqrt(2) is irrational. Of course sqrt(2) can't be computed to infinite precision, so it will repeat at some point. But you can delay that point by taking multiple sound snippets where for all durations t_i it is true that t_i/t_j is an irrational number. So for example, 1 s, sqrt(2) s, phi s, pi s etc.
@@pirmelephant probably meant different phases
Noise is supposed to have uniform frequency distribution so even if it is not periodic sound it can still have non-uniform frequencies.
Thanks you so much for creating this super informative video!!❤❤
Wow the visuals were amazing!
Thank you, you made me understand better something that I always wondered about.
It’s a wonderful day when minute physics makes a 5+ minute video!
*Keanu voice* whoa. Thanks for this and the link. Going to look into ways of adapting these geometries into rhythms. Similar, yet endlessly changing patterns is the feeling I want to put in my sounds.
Nice video minutephysics!
I didn't even know about Penrose tiles, but this video explains it beautifully! Thank you 🙂
I like shirts with patterns, and I think these penrose patterns would look pretty dope.
This is great, love the channel
This is helpful! Thank you very much! Six minutes of my life well spent!
The background music also Quasi-Periodic. Awesome little detail! It's so funny to listen to when you actually pay attention to it haha
I thought the music was going wild on this one, then saw it was algorithmically generated, fun stuff.
@Artem Down Well, the result to me is definitely musical, but just wonky enough to grab my attention
This is one of the best videos I have ever seen. Brilliant!
If you liked this, you'll love veritasium's video on the Penrose tiles.
Hey, was the background music also quasi periodic?? Nice touch! I love it!!
I'm redecorating my bathroom soon, and I feel inspired
Damn, I have been playing with minesweeper on voronoi tilings, but using penrose tiling might actually be much better :) I need to try to generate some now
Where can I try that? It sounds so cool!
@@nahometesfay1112 I tried linking once or twice, but I think youtube removed it. Anyway I've put it on itch, its TileGame by JohannGambolputty, may not appear in search right now though...
I like the darts and kites better. But, the way I have always understood it, given any tiling, you can break the pieces into smaller pieces to create a new tiling or you can build the pieces into larger pieces to make a new tiling. (Well, the tiling might not be new. Some build up into copies of themselves.) But the ratio of the pieces can still be shown to be the golden ratio.
Fascinated by the algorithmically generated music, because it bears some resemblances to pieces I have made - (see "Notes From The Analytical Engine" by Beat Frequency on Bandcamp) - please can you post some details about the algorithm.
Excellent presentation.
Oh my god, content like this is what makes the internet great!
great video, subscribed!
Loved the music in this one
What would the resulting tiling look like if you only used 3 of the 5 sets of parallel lines in the pentagrid?
Gaps in your tiling?
Typing something to find the answer
Then it wouldn't be a pentagrid anymore, wouldn't it? It would be a grid made of 3 sets of parallel lines, like a triangular grid. But how would the tiling look like?
Perfect illustrative music there ;)
Dunno if it's just me, but the music in the background is just a little too loud for me to properly hear what you're saying without trying too hard.
I do however get that it's essentially an example of a non-repeating pattern which is very similar, but I don't know if the video would come across a little better if there was a bit larger difference in how loud your voice and the music is.
Though I suppose people might be less likely to notice the music being different if it was the same, but oh well.
Great video nonetheless!
Edit: Spaced out the statement a little to make it easier to read.
Agreed. The music is a little to loud here.
I agree too, the music was really obnoxious in this one. Still a super cool video, just... that music isn't a good fit.
I didnt even notice there was music in the background. I guess different people have different perception.
It was fine for me; I didn't really notice it much.
...And hmm, I'll have to go back and listen again, to check if the music is quasi-periodic itself. 🙂
I'm a musician and tend to fixate on musical elements... and I barely noticed the music. Maybe the balance got changed in the 16 hours since your comment got posted?
Amazing! Thank you!!
Brilliant video!
Good stuff! Thank you!
Such patterns are not only a mathematical conception but exist in nature, in the materials called quasicrystals, with atoms that never repeat. This discovery awarded a Nobel price to itz finder.
This video was REALLY GOOD
Excellent video!
interesting vidéo, thanks for your work ^^
4:33 the fact that it happened to be the Golden Ratio blew me away. It's awesome that mathematics and science go down some path of research and in the end find something within that is known/discovered.
BRB gonna draw up a Penrose-crawl for my next D&D session
Wow...
By far, the best and most brain-melting video I've seen in ages... !!
Lovely Video!
More of this plz!
I think this is yet another time to remind people that Hexagon are the bestagons!
Thanks insightful
This channel is educating me
Who's smol 10 years using mamas account
And it's insane so thank you!
Nice! To further understand aperiodic tilings, how about a video on the cut-and-project method? Also it might be a good idea to mention inflation (in the context of tilings).
I love your friend Aatish!
I have almost no idea what's going on, but this still has to be one of the best ads I've ever seen.
You had me at "patches that are perfect matches".
Here's my interpretation on why they never repeat
1) Start with 5 wide tiles connected by a corner.
2) Surround the shape with narrow tiles, by filling every 216 angle with 144 angles, making a decagons
3) Surround it completely with wide tiles, alternating between filling 144 angles with two 108 angles, and three 72 angles.
4) Repeat step 2
5) Repeat step 3, filling 252 angles with 2 72 angles, and filling the sets of three 144 angles by putting 3 72 angles in the middle ones.
6) Repeat steps 4 and 5 ad inf.
Since Each band of wide tiles is surrounded both inside and out with narrow tiles, the only time when 5 wide tiles get together is in the center.
But... the bands don't need to be complete... As can be seen at 1:10, there are plenty of "5 wide tiles connected by a corner" shapes in there, it's not just a single one in the whole plane. 🤔
I will absolutely accept more minute-physics-math videos.
This helped me understand the grid in dreamscaper and why it always seemed to never make sense it's actually a pentagrid
if I start with the regular square tessellation tiling, then replace one square with an arbitrary asymmetrical design, I also get a tiling that doesn't have global translational symmetry.
I have a question I was hoping you could answer. Due to special relativity, if I were to somehow escape the effects of the movement of the galaxy and everything in it, reducing my velocity and the effect of gravity on me to zero, how would I perceive time? Would it stop? Would it travel only slightly slower? If I were to travel to a planet that moved slower relative to Earth, would I experience time differently, and by how much?
I was following along pretty well and all was good. Then, out of nowhere, phi appeared. Suddenly, the world made sense.
there is no minutephysics video without a sponsor
The rationality of this blows my mind!
Please do a video on hyperbolic geometry and horocycles!!
This would make a cool floor tiling!
...and the reason a square tiling repeats is because their tiling would be sine of 90°, which is 1.
That makes sense!
Thanks, minute physics!
minute... maths?
Love it.
I love that the soundtrack is also quasi-periodic
That site is way too fun. I wanna take black and white ones to play in a coloring app
So interesting !
Love this!!
Penrose tiling touches on so many fundamental questions of life, beauty, and meaning, that it's kind of incredible.
Great stuff. But I especially appreciate it when I discover one of the diminishing number of people who know the difference between fewer and less.
Best kaleidoscope ever!