字幕列表 影片播放 列印英文字幕 If you ask real experts in probability they'll say this is not random because it's far too uniform, you know. Really, Poisson... That kind of thing would have more gaps and so forth. But it's random enough and what I'm about to show you works with any sufficiently random dots. OK. Now, I photocopied this on a transparency. The exact copy, I put one on top of the other. The result is this. Interestingly boring, shall I say, really amazingly boring. But now, let's shift the transparency against the bottom sheet a little bit and what you see is this. You see those concentric circles around a common center, and the reason, actually in retrospect, is not too surprising because whatever I'm doing to the transparency is a Euclidean motion and we know that a Euclidean motion in 2D is always a rotation. Actually it could be a translation, but a translation is a rotation with a center at infinity. So it's always a rotation. So you see those traces of rotation, so to speak, under your eyes. What's happening is that the original random dots are displaced slightly and images are close enough to the original dots. And when you have a situation like that, human eyes cannot help connecting the dots. The dots are connected and you see those small segments in your eyes and then you see those trajectories of the rotation that are concentric circles. So, if you rotate them too far, the images and the originals get too far and they lose all correlation and you don't see anything. Now, let's suppose that for example this common center of all the concentric circle is too high and you don't want it to be too high, you want to bring it back to the center. So when I first tried this I naturally shifted the transparency in the direction that I want to move the center, but watch what happens when I do this. I'm going to move it up and down in this direction. The center moves sideways, horizontally! And when I move the transparency sideways, it goes up and down. So the motion of the center is always perpendicular to the direction in which I'm moving the transparency, that's really strange. Now, once you notice this you can actually figure out what's happening by calculation. So let's say this is a line and that's another line, and one has been rotated into another, so that's the center of rotation. That's where they intersect, so to speak. Now, let's say that I move the image sideways and you can see that the point of intersection actually goes up and down like that. So, although I'm moving the knife horizontally, the intersection point is going up and down vertically. So that's more or less why if you move the transparency horizontally the center goes vertically and if you move it vertically it goes horizontally. Next, I'm going to show you another example. This is the same picture as the previous one, but only shrunk a little bit. I went to the photocopier and hit the shrink button. It shouldn't be shrunk too much, maybe this is 96% of the original because if you shrink it too much, again, you lose the correlation and you don't see anything. But this time, what you see is this. You see those spirals. And of course you see those spirals, because in addition to the rotation that's present from the previous one, you rescaled the whole thing so you have this radial shrinking and if I now adjust the rotation so it gets rid of the rotational component and make it pure translation then you get this radial pattern. OK, so that was the random end of the experiment. Let's try the other end, which is the regular end of the spectrum. And we'll begin by seeing something that is very classical. This kind of thing makes you seasick. Square lattice, and I took an exact photocopy of this on the transparency, and if I put one on top of the other, you get something completely black. Now, nothing interesting is happening. Let's start rotating this. When I rotate this: some ghost-like sorts of shades appear and resolve into, resolving to, this beautiful pattern of square lattice but at 45 degrees to the original one. And, as I keep rotating, it resolves into a smaller and smaller pattern and at 45 degrees it becomes the smallest, and around that point you see there's some kind of wave that comes in and out, that's quite curious. and then when I keep spinning, eventually, this pattern becomes larger and larger, comes back out at us, again, 45 degrees and then the scale diverges and it becomes a blur and keeps going and so on, and I have rotated by 90 degrees, because that's the period, if you like. This is called a moiré pattern, which is quite well known. You can do the same thing with another kind of lattice; this is lots and lots of equilateral triangles that have been put together and colored in an automated fashion black and white. And if I put the photocopy of this on top, again you see nothing. Now, let's start rotating this. You again see the moiré pattern: a ghost comes in and then resolves into this kind of thing. I think at this stage, most people see black walls partitioning the plane into white triangles. Black walls and white triangles. OK. Let's keep rotating. These triangles become smaller and smaller, and they kind of go away and resolve into the smallest scale, and then, if I keep rotating, these come back. But when they come back at us, what comes back is not what went out! Many people at this stage see hexagons, but you can probably be convinced that these are white walls dividing the plane into black triangles, I don't know if you can see this. And so what comes back is dual to what went out. What went out was black walls dividing the plane into white triangles but what came back is white walls dividing the plane into black triangles. And then it becomes a blur. And if I keep rotating, you get back what has just disappeared into infinity, and then if you keep rotating further you again get triangles and hexagons and smaller and smaller scale and eventually the pattern starts coming back. But what comes back is what had gone out in the first place, that is, black walls with white triangles, and then it becomes a blur and by the time I have completed the cycle I have rotated it by 120 degrees, which is the angle's symmetry. But within this single period it turns out that you have two sub-periods and those two sub-periods are dual to each other: what goes out and what comes back are sort of complimentary to each other so that's quite curious. Numberphile is brought to you by the Mathematical Science Research Institute, MSRI. That's the building behind me there. This is a place where many of the world's top mathematicians come together for sometimes a semester at a time cracking some of the hardest problems in mathematics. If you'd like to find out more, I've put some links in the description under the video.

B1 中級 詭異的圓點圖案 - Numberphile (Freaky Dot Patterns - Numberphile) 1 0 林宜悉 發佈於 2021 年 01 月 14 日 更多分享 分享 收藏 回報 影片單字