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  • 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.

If you ask real experts in probability

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気まぐれなドットパターン - Numberphile (Freaky Dot Patterns - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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