Why Penrose Tiles Never Repeat

[Music] these incredibly pretty geometric patterns are Penrose tilings and if you've heard anything about them it's probably that they never repeat themselves I mean they look pretty similar all over and there are patches that are perfect matches but if you slide the whole thing over and around it will never completely line up with itself again patterns like this that go on forever and feel like they should repeat but don't are called quasi-periodic but I never really felt like I understood these patterns like how do you make them how do we know they don't ever repeat I just had to take somebody's word for it that they worked the way people say they do until recently when I learned there's a hidden pattern inside Penrose tilings a pentagrid and it's quite possibly the best way to understand pandro's tilings at least it's what finally helped me feel like I understood them here's how you find the pentagrid start with a single tile and highlight neighboring tiles whose edges are parallel and their neighbors and you end up with a wobbly ribbon of tiles that snakes around a bit but overall follows a straight path and if you pick another tile with the same orientation you can make a ribbon that's parallel to the first and you can keep going here's a whole set of parallel ribbons of course we could have started with the other edges of our original tile and ended up with a different ribbon of tiles and there's a whole parallel set of these ribbons too in fact jumping ahead a little bit if we make a slightly more complicated version of the Penrose tiling and color the tiles based on how they're oriented you see a whole mess of ribbons jump out at you these ribbons are the key to understanding Penrose tilings because the ribbons form a pentagrid and what exactly is a pentagrid if you take a regular array of parallel lines you can copy and rotate it so it forms a grid you're probably most familiar with a square grid where two sets of lines have been evenly rotated from one another and intersect at 90 degrees you might also have seen a triangular grid where three sets of lines have been evenly rotated and intersect at 60 degrees and if you create a grid with 5 sets of lines evenly rotated from each other and intersecting at either 36 or 72 degrees you get a pentagrid pentagrids are made up of five sets of parallel lines and Penrose tilings are made up of five sets of parallel ribbons of tiles because they're actually the same to make a Penrose tiling all you have to do is start with a pentagrid and then at every point where two lines intersect you draw a tile oriented so the sides of the tiles are perpendicular to the two lines this way at the next intersection along the line the sides of that tile will be parallel to the sides of the first tile and the same at the next intersection and so on and you can slide them all together into a ribbon And if you do the same for the next lineup in the pentagrid you get another ribbon and another and if you also do it for all the other lines in the other directions all the ribbons combined together make a Penrose tiling you can also just add a tile to every intersection and slide them all together along the grid lines either way you get a Penrose tiling every Penrose tiling is made out of five infinite sets of parallel infinitely long ribbons because every Penrose tiling is a pentagrid in Disguise of course you don't have to use this particular pentagrid we can also shift the various different sets of lines by random amounts and get a beautiful new tiling that's slightly different from Penrose tilings and and were not limited to a pentagrid here's a heptagrid and its corresponding penrose-like tiling and here's an OCTA grid a nanogrid a Deca grid and so on and that beautiful ribbony pattern we showed before was from a grid with 17 different sets of lines my friend atish made an interactive website where you can play around with all of this and make your own penrose-like patterns you can highlight the grid lines and see their counterpart tiles and vice versa you can change the colorings to bring out different aspects of the patterns you can use it to generate a bunch of other famous patterns you can save them for a phone or computer background or to print on a shirt or whatever it's really great and as you've probably noticed it's where all the visuals in this video are from but there's one more thing remember how I said that pentagrids helped me see why these patterns never repeat themselves this isn't a proof but it at least gives you a flavor of the non-repetition so start with a single ribbon if the ribbon ever did repeat itself then after a certain point in time you'd have the same pattern of thin and wide tiles over again and again and again so the ratio of thin to wide tiles would be a rational number the number of thin tiles in a given chunk divided by the number of wide tiles in this example there are six wide tiles for every four thin ones in an actual Penrose tiling we can directly calculate the ratio of thin tiles to wide tiles since the ribbons of tiles correspond to the intersections along the line of the pentagrid the wide tiles are from the intersections with the 72 Degree lines and the thin tiles from the 36 degree lines some basic trigonometry shows that the spacing between 36 degree lines is 1 over the sine of 36 degrees and the spacing between 72 Degree lines is 1 over the sine of 72 degrees so the ratio of wide tiles to thin tiles is the ratio of these which happens to be the golden ratio which is irrational so there's no way the pattern could ever repeat if it did the Golden Ratio would have to be rational remember if the pattern did repeat the ratio of wide to thin tiles would have to be rational which the golden ratio isn't of course this just proves the tiling can't repeat in One Direction the hole proof is a little bit more than we want to get into here the pentagrid allows us to directly calculate that as you go out along any ribbon in a Penrose tiling for every 10 thin tiles you see there are on average 16.18 y tiles a golden ratio worth and because the golden ratio is irrational sometimes there are slightly more wide tiles for every 10 thin ones and sometimes they're slightly fewer in a way that is perfectly predicted by the value of the golden ratio but never repeats and the more tiles you look at the more closely their ratio matches the golden ratio of course there's nothing special about the golden ratio here it happens to show up a lot when you have five-sided things for the heptagrid or Deca Grid or whatever the ribbons still don't repeat because the ratio of the spacings with the grids and the ratios of the numbers of types of tiles is some other irrational number all these patterns are quasi-periodic they may never repeat but they also aren't just a random jumble of tiles all right go play with the beautiful Penrose tile patterns over at autishbead.com pattern collider and send the prettiest ones to me on patreon at minutephysics and speaking of beautiful geometric patterns head over to brilliant this video sponsor for their interactive course on beautiful geometry you'll explore how to make tessellations fractals infinite tilings and more brilliant has dozens of courses covering broad swaths of math and science and there's something for everyone from entertaining puzzles to clever problem solving strategies for high school math competitions to black holes actually all of those subjects are for me you can choose your own by signing up for free at brilliant.org minut physics the first 200 people get 20 off an annual premium subscription with full access to all of Brilliance courses and puzzles and more exclusive content added monthly or you can give a brilliant subscription to somebody as a gift again that's brilliant.org minutephysics and thanks to brilliant for their support foreign

Channel: minutephysics

Views: 1,180,208

Keywords: physics, minutephysics, science

Id: -eqdj63nEr4

Length: 6min 36sec (396 seconds)

Published: Thu Dec 01 2022