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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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# 詭異的圓點圖案 - Numberphile (Freaky Dot Patterns - Numberphile)

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林宜悉 發佈於 2021 年 01 月 14 日