Name: 為什麼彭羅斯瓷磚從不重複 (Why Penrose Tiles Never Repeat)
Uploaded: 2022-12-06T18:40:19.000Z
Duration: 6 min 37 s

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

這些令人難以置信的漂亮的幾何圖案是彭羅斯瓷磚，如果你聽說過它們，那可能是它們從不重複。

關係子句

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.

我的意思是，它們看起來都很相似，而且有一些補丁是完全匹配的，但如果你把整個東西滑過去，它永遠不會與自己完全對齊。

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.

同樣，像這樣永遠持續下去，感覺應該重複，但又不重複的模式被稱為準週期性，但我從來沒有真正感覺到我理解這些力量模式。

我們怎麼知道他們永遠不會重複？

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.

我只是不得不相信別人的話，認為他們是按照人們所說的方式工作的，直到最近我瞭解到，在彭羅斯風格里面有一個隱藏的模式，即彭塔網格，這很可能是理解彭羅斯風格的最好方法，至少。

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

這是最終幫助我覺得我理解他們的原因。

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.

以下是你如何找到Penta網格的方法，從一個單一的瓷磚開始，突出邊緣平行的相鄰瓷磚和它們的鄰居，最後你會得到一個搖擺不定的瓷磚肋骨，它有點蜿蜒，但總體上遵循一條直線。

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.

而如果你挑選另一塊方向相同的瓷磚，你可以做一個與第一塊平行的絲帶，你可以繼續下去。

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.

當然，我們可以從我們原來的瓷磚的其他邊緣開始，最後得到一個不同的瓷磚帶，而且有一整套平行的這些瓷磚帶，事實上，如果我們做一個稍微複雜一點的彭羅斯瓷磚版本，並根據它們的口碑給瓷磚著色，你會看到一大堆的瓷磚帶跳出來。

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

這些絲帶是理解彭羅斯瓷磚的關鍵，因為這些絲帶形成了一個五角星，而究竟什麼是彭塔格。

If you take a regular array of parallel lines, you can copy and rotate it.

如果你取一個有規律的平行線陣列，你可以複製和旋轉它。

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.

是以，它形成了一個網格，你可能最熟悉的是方形網格，其中兩組線已經從彼此均勻旋轉，並相交於90度。

You might also have seen a triangular grid where three sets of lines have been evenly rotated and intersect at 60 degrees.

你可能也見過一個三角形的網格，其中三組線被均勻地旋轉並相交於60度。

And if you create a grid with five sets of lines evenly rotated from each other and intersecting at either 36 or 72 degrees.

而如果你創建一個有五組線的網格，彼此均勻旋轉，相交於36度或72度。

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.

Penta網格是由五組平行線組成的，而penrose瓦片是由五組平行的瓦片帶組成的，因為它們實際上是一樣的。

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.

你所要做的就是用筆開始畫格子，然後在每一個兩條線相交的地方，畫出一個有方向的瓦片，這樣瓦片的邊就與兩條線垂直了。

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.

沿著這條線，那塊瓷磚的邊會與第一塊瓷磚的邊平行，在下一個交叉點也是如此，以此類推，你可以把它們都滑到一起，成為一條絲帶，如果你對筆到格的下一行做同樣的事，你會得到另一條絲帶和另一條。

And if you also do it for all the other lines, in the other directions, all the ribbons combined together.

如果你也對所有其他線，在其他方向，所有的絲帶結合在一起做。

You can also just add a tile to every intersection and slide them all together along the grid lines.

你也可以只在每個交叉點添加一塊瓷磚，然後沿著網格線把它們滑到一起。

無論哪種方式，你都會得到一個筆直的瓷磚。

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.

每個彭羅斯瓦片都是由五組無限長的平行帶組成的，因為每個彭羅斯瓦片都是一個變相的彭塔網格，當然你不一定要用這個特定的彭塔網格。

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.

我們還可以將各種不同的線組隨機移位，得到一個美麗的新瓦片，與彭羅斯瓦片略有不同。

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.

我們並不侷限於五邊形網格，這裡有一個七邊形網格及其相應的五邊形網格，這裡有一個八邊形網格，一個非八邊形網格，一個甲板，一個網格等等。

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

而我們之前展示的那個美麗的絲帶圖案是來自一個有17組不同線條的網格。

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.

我的朋友奧蒂斯做了一個互動網站，在那裡你可以玩弄所有這些，並製作你自己的類似彭羅斯的圖案，你可以突出網格線並看到它們的對應瓷磚，反之亦然。

You can change the coloring is to bring out different aspects of the patterns.

你可以改變著色是，以帶出圖案的不同方面。

You can use it to generate a bunch of other famous patterns.

你可以用它來生成一堆其他的著名圖案。

You can save them for a phone or computer background or to print on the shirt or whatever.

你可以把它們保存為手機或電腦背景，或打印在襯衫上或其他地方。

And as you've probably noticed it's where all the visuals in this video are from.

你可能已經注意到，這段視頻中的所有視覺效果都來自這裡。

But there's one more thing, remember how I said that Penta grids helped me see why these patterns never repeat themselves.

但還有一件事，記得我說過五角星網格幫助我看到為什麼這些模式永遠不會重複。

This isn't a proof, but it at least gives you a flavor of the non repetition.

這不是一個證明，但它至少給你一個不重複的味道。

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.

如果絲帶真的重複了，那麼在某個時間點之後，你就會有同樣的薄而寬的瓦片圖案，一遍又一遍地重複。

So the ratio of thin to wide tiles would be a rational number.

是以，薄瓦和寬瓦的比例將是一個有理數。

The number of thin tiles in a given chunk divided by the number of white tiles.

某一大塊中的薄瓦片數量除以白瓦片數量。

In this example, there are six wide tiles for every four thin ones.

在這個例子中，每四塊薄磚有六塊寬磚。

We can directly calculate the ratio of thin tiles to wide tiles.

我們可以直接計算出薄瓦片與寬瓦片的比例。

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.

由於瓦片帶對應於沿五角星網格線的交叉點，白色瓦片來自與72度線的交叉點，薄瓦片來自36度線。

Some basic trigonometry shows that the spacing between 36 degree lines is one over the sine of 36 degrees.

一些基本的三角函數表明，36度線的間距是36度正弦的1倍。

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.

而72度線之間的間距是我們超過72度正弦的1。所以寬瓦片與薄瓦片的比例是這些的比例，而這恰好是黃金比例，是無理的。

So there's no way the pattern could ever repeat.

If it did, the golden ratio would have to be rational.

如果是這樣，黃金比例就必須是合理的。

Remember if the pattern did repeat, the ratio of wide thin tiles would have to be rational.

請記住，如果圖案確實重複，那麼寬的薄的瓷磚的比例就必須是合理的。

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

當然這只是證明了泰國不能在一個方向上重複。

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

整個證明有點超過我們想在這裡討論的範圍。

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.

Penta網格允許我們直接計算出，當你沿著penrose瓦片中的任何一條帶子出去時，每10個薄瓦片就可以計算出。

You see there are on average 16.18 wide tiles, a golden ratio worth.

你看，平均有16.18塊寬的瓷磚，這是一個黃金比例的價值。

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.

而且由於黃金比例是無理的，有時每10塊薄磚會有稍多的寬磚，有時會有稍少的寬磚，這種方式完全可以用黃金比例的值來預測，但絕不會重複。

And the more tiles you look at the more closely their ratio matches the golden ratio.

而且你看的瓷磚越多，它們的比例就越接近黃金比例。

Of course there's nothing special about the golden ratio here.

當然，這裡的黃金比例並沒有什麼特別之處。

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.

當你有五面的東西給hep打格子或甲板打格子或什麼絲帶時，它就會經常出現。

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.

還是不要重複，因為網格的間距和瓷磚類型的數量之比是其他無理數。

All these patterns are quasi periodic, they may never repeat, but they also aren't just a random jumble of tiles.

所有這些圖案都是準週期性的，它們可能永遠不會重複，但它們也不是隨機的雜亂的瓷磚。

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.

說到美麗的幾何圖案，請到輝煌。

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

這個視頻的贊助商是他們的美麗幾何互動課程。

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.

你將探索如何使測試關係分形，無限的瓷磚ING的和更多的輝煌有幾十個課程覆蓋廣泛的數學和科學的領域，有每個人的東西，從娛樂的難題，太聰明瞭，解決問題的策略，為高中數學比賽，兩個黑洞。

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

實際上所有這些主題都是為我準備的，你可以通過在輝煌點org斜線分鐘免費註冊來選擇自己的主題。

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.

最初的200人可以獲得20%的年度高級訂閱，可以完全訪問所有的輝煌課程和謎題，以及每月增加的更多獨家內容。

 Or you can give a brilliant subscription to somebody as a gift again, that's brilliant dot org slash minute physics. 

或者你可以再給別人一個輝煌的訂閱作為禮物，這就是輝煌的點陣式斜線分鐘物理學。