Why Penrose Tiles Never Repeat
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Published 2022-12-01
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 That Never Repeats
Music algorithmically generated, algorithm designed by Henry Reich
N.G. de Bruijn’s paper introducing the pentagrid/Penrose idea: www.math.brown.edu/reschwar/M272/pentagrid.pdf
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/feature-column/fcarc-ri…
The Empire Problem in Penrose Tilings by Laura Effinger-Dean: www.cs.williams.edu/~bailey/06le.pdf
Pentagrids and Penrose Tilings by Stacy Mowry & Shriya Shukla: web.williams.edu/Mathematics/sjmiller/public_html/…
Penrose Tiling by Andrejs Treibergs: www.math.utah.edu/~treiberg/PenroseSlides.pdf
Algebraic Theory of Penrose's Non-Periodic Tilings of the Plane by N. G. de Bruijn: www.math.brown.edu/reschwar/M272/pentagrid.pdf
Particularly good and helpful, and (we think) an undergrad thesis which is impressive!: www.cs.williams.edu/~bailey/06le.pdf
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-kind-of-impossible-t…
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Created by Henry Reich
All Comments (21)
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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.
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you know it’s a good day when minutephysics drops some obscure math problems
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The golden ratio shows up in nature a lot because it is the main part of an efficient packing algorithm. Thanks Numberphiles!
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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.
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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.
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Finally i understand why it never repeats, veritasium made an interesting showcase but i never understood why it never repeats
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Great explanation Henry!
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I love how the music is algorithmically generated. Really fits the video!
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this video makes me blame my old geometry teacher for not making class this fun
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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
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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.
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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!
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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 ❤❤❤
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That was simple and intuitive. And my respect for Penrose only increases the more I know about his work.
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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!
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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.
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Please do one on the newly discovered a periodic monotile
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4:49 This was a cool proof! Pretty much the highlight of the video. Also nice to see minutephysics drop.
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This is the perfect kind of math/science video we need. Thank you. I wish other channels were as good as yours.
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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.
