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From a recent trip to Japan, my home country, I brought back a little present.
It's a traditional toy. A Japanese toy called "Kamifusen", which means paper balloon.
[robot pumping air]
And as you can see, it's a balloon made of paper, and there's a hole in there, yeah?
So the way you inflate this paper balloon, when you squash it, is to blow into this hole...
[Tadashi blowing up balloon]
And make it rounder. Okay, but there is another way to make it rounder.
[balloon crinkles]
You can squash it, and then you can start tossing it around.
[Tadashi hitting balloon]
You hit it from all sides at random... oops!
Now, when you hit something, don't you have the impression that you are denting it and therefore making
it less and less round and you're denting it, right, from both sides?
But look at what's happening as I toss it and hit it from all sides. I'm doing it completely randomly.
You can hit it quite hard.
[Tadashi hitting balloon]
What was shriveled to a little crumple
is now getting rounder and rounder
and becoming more and more spherical.
And look! I ended up with a round sphere.
So, this is a really strange way of making something round by denting it repeatedly from all sides.
How does this trick work?
I think the key to this behavior,
which all Japanese children know instinctively, is really in the following behavior of the balloon. When I
shrivel it to a little crumple,
watch what happens when I release it.
You see, it starts fighting its way back...
...at least initially. It doesn't inflate all the way back of course, but it has some memory.
And it wants to come back to where it started. It fights its way back as I said.
[Brady] Why is it even doing that? What's causing that?
That's... that's an interesting question. Um, I have to think about this.
[laughter] Okay.
So let's imagine the following toy model. Let's say that you have a state of this balloon, which is
crumpled on both sides: on the left as well as on the right,
and I'm going to strike this balloon at this moment.
Pafff!
From the left, let's say. What happens,
to a good approximation,
I think, is that the side that you hit gets over crumpled because I just hit it.
You know, you introduce a lot of crumples.
But the other side gets puffed out literally because there is this force which
Pushes it rightward. But, you see, then something interesting happens.
That "fighting back" effect means that the left side which was over crumpled,
starts fighting back, so it starts expanding to roughly the
degree of wrinkliness that it had in the beginning, whereas the right hand side stays puffed out.
So, per this cycle, you see, what happened is that
I increase the overall smoothness a little bit. The other side which got puffed out, whereas this side that I over wrinkled
fought back by itself. So, if you keep hitting it from all sides gradually you are making the thing rounder and rounder.
So let's plot the evolution of the volume against time, and let's say in time
I'm hitting the balloon at various regular intervals. It doesn't have to be regular intervals.
Okay. So the volume of the balloon starts like this and
as soon as I hit it, of course,
instantaneously, I suddenly decrease the volume. I just shrivel it. So, a hit
gives you this dip. But then, the thing fights back.
Yeah? This is the fight back. And then the next hit will decrease the volume.
And then it will fight back, fight back and so on, and gradually it asymptotes,
or it approaches this spherical shape
from below like this. So it never becomes completely spherical, but it grows like that. So I think there is a process of
"grow, shrink, grow, shrink, grow, shrink, zig, zag, zig, zag" and then goes back then. That's why you can do this strange thing of
hitting it, or denting it, repeatedly, and yet the result outcome is that it becomes rounder and rounder.
Okay, this will probably take a long time.
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An unexpected way to inflate a balloon - Numberphile

23 分類 收藏
林宜悉 發佈於 2020 年 3 月 28 日
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