Placeholder Image

字幕列表 影片播放

由 AI 自動生成
  • these incredibly pretty geometric patterns are penrose tiling 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 power 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 stylings, a Penta grid and it's quite possibly the best way to understand penrose stylings, at least.


  • It's what finally helped me feel like I understood them.


  • Here's how you find the Penta grid start with a single tile and highlight neighboring tiles whose edges are parallel and their neighbors, and you end up with a wobbly rib 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 to 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 their ori, you see a whole mess of ribbons jump out at you.


  • These ribbons are the key to understanding penrose tiling because the ribbons form a pentagram and what exactly is a Penta grid.


  • 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 five sets of lines evenly rotated from each other and intersecting at either 36 or 72 degrees.


  • You get a Penta grid.


  • Penta grids are made up of five sets of parallel lines, and penrose tiling 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 pen to grid 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 line up in the pen to grid, 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 Penta grid in disguise, of course you don't have to use this particular Penta grid.


  • 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 tiling.


  • And we're not limited to a Penta grid, Here's a hep to grid and its corresponding penrose like tiling and here's an Octa grid, a non a grid, a deck, a grid and so on.


  • And that beautiful ribbon pattern we showed before was from a grid with 17 different sets of lines.


  • My friend Otis 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 coloring is 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 the 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 Penta grids 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 white 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 Penta grid, the white 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 one over the sine of 36 degrees.


  • And the spacing between 72 degree lines is we One over the sine of 72°. So the ratio of wide tiles too 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 thin tiles would have to be rational.


  • Which the golden ratio isn't.


  • Of course this just proves the Thailand can't repeat in one direction.


  • The whole proof a little bit more than we want to get into here.


  • The Penta grid 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 wide 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 there are 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 hep to grid or deck a grid or whatever the ribbons.


  • Still don't repeat because the ratio of the spacings of 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.


  • Alright, go play with the beautiful penrose tile patterns over at dot com slash pattern collider and send the prettiest ones to me on Patreon at metaphysics.

    好了,去玩美麗的彭羅斯瓷磚圖案吧,在dot com slash pattern collider上,把最漂亮的圖案發給我,在metaphysics的Patreon上。

  • And speaking of beautiful geometric patterns head over to brilliant.


  • This video's sponsor for their interactive course on beautiful geometry.


  • You'll explore how to make test relations fractals, infinite tile ing's and more brilliant has dozens of courses covering broad swaths of math and science and there's something for everyone from entertaining puzzles, too clever, problem solving strategies for high school math competitions, two black holes.


  • Actually all of those subjects are for me, you can choose your own by signing up for free at brilliant dot org slash minute.


  • Physics.


  • The 1st 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 dot org slash minute physics.


these incredibly pretty geometric patterns are penrose tiling and if you've heard anything about them, it's probably that they never repeat themselves.


由 AI 自動生成

單字即點即查 點擊單字可以查詢單字解釋

B2 中高級 中文

為什麼彭羅斯瓷磚從不重複 (Why Penrose Tiles Never Repeat)

  • 14 1
    林宜悉 發佈於 2022 年 12 月 06 日