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  • Suppose when you sit in a beer garden

  • beautiful weather, and you have ordered the beer,

  • the beer comes, you put it in a table, and then you touch the table

  • and the table is unstable and the beer is poured out.

  • You are angry!

  • It’s a four-legged table. The table is completely stable.

  • The problem is the ground on which the table stands.

  • This is not flat and that’s why one leg is above the ground.

  • And then if you put your hands again on the table,

  • it goes down and it’s the instability of the table.

  • The moment solution is you take a sheet of paper.

  • For example this paper is under the beer glass and put it under this leg

  • and for a while, it looks okay but after a few minutes, we are angry again because of this paper

  • is compressed a little bit and instability again.

  • And we hate that.

  • Mathematicians never have unstable tables.

  • They know what to do.

  • And what you do is very very simple.

  • Turn the table and start moving the table and try to turn it

  • so that you have a quarter of a turn and on the way of your turning,

  • there will be a moment where it’s absolutely stable.

  • So youre just rotating like a rotator like rotating a disc?

  • Yes, I rotate the table like a disc and typically only a few centimeters are needed

  • and suddenly it’s stable and this is not by chance.

  • This there it’s a mathematical proof that this will always happen.

  • Youre gonna have to give me that proof now.

  • I give you that proof now.

  • Here’s the ground and here this is the position of the four legs

  • and we enumerate themthis is leg 1, this is 2

  • this is 3, this is leg 4.

  • And suppose that leg 1 is above the floor whereas these three are fixed on the ground.

  • Now, of course, if we put pressure on 1, then we still the instability.

  • And now, we do the following:

  • We measure the height of leg 1.

  • Remember, we always measure the height of leg 1.

  • So if you do that in time, then we get associate it to time T,

  • we associate height of leg 1.

  • So time is zero, we get some T=0,

  • we get some number say X>0.

  • Now nothing is happening, now let’s start moving

  • and we do that obviously in time and at each time, we measure the height of leg 1.

  • And we turn it in this way.

  • All we turn it so that we try to bring leg 1 to the position of leg 2.

  • At each time, we measure the height of leg 1.

  • So this gives the function f (t).

  • For each time, T we measure.

  • Here’s something important.

  • It can happen that if we fix It all 2, 3 and 4,

  • this is all you remember, we fix 2, 3 and 4 and now it could happen that the height of leg 1 is negative.

  • Yeah because this will happen, if we now put leg 1 into position of leg 2,

  • leg 2 to the position of leg 3, leg 3 to the position of leg 4

  • and leg 4 is at the position of leg 1.

  • But now, we remember that we fix the position of 2, 3 and 4.

  • I fixed them on the ground.

  • I keep them on the ground.

  • And since we did it here, at this position, this was above the ground,

  • and now we force these three to be on the ground.

  • That means this position has to be under the ground.

  • You see that before you fix these three, now you force these to go down,

  • and this is suddenly under the ground.

  • So it’s time 1, let’s suppose take time into 1 until at this position,

  • so it’s t=1, this height is negative.

  • So now, we draw if I can get you a sheet of paper.

  • Now we draw this curve so this is time = 0, this is time=1

  • Here we draw the height and at times zero, the height was something positive.

  • And at time =1, the height was negative.

  • So this is f (0), this is f (1).

  • And now, at each time T, we get the position of the height and you see we get a curve

  • and it might go even up and down but in the very end, it has to end here.

  • And now comes the famous theorem of Mathematics, the Intermediate Value Theorem

  • which just says that if you have a continuous function which is positive here and negative here,

  • they match the opposition here where it is 0.

  • It could be multiple ~It could beyou can have fun with it, you turn your table further

  • and it might be in the 2nd position of this table.

  • You don’t need that but it’s fun to try that out.

  • And if you are in the beer garden,

  • and if you do it the next time in the beer garden, you will easily fix the table.

  • And you will be pleased and you can taste it even better.

  • I’ll do it all the time whenever I’m in the beer garden or even in a restaurant,

  • often the ground is not flat, and I sit there with my friends

  • and they are saying, “Ahh, let’s put this under.”

  • I said, “Don’t do it!”

  • I move it just a little bit and they are obviously very surprised.

  • And we do not change and it’s fix for the whole evening.

  • What if the tables are all lined-up or with special shape or something?

  • Oh, that’s of courseMathematics is always theoretical so if you cannot move the table,

Suppose when you sit in a beer garden

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A2 初級

グラグラするテーブルを固定する(数学で (Fix a Wobbly Table (with Math))

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    Go Tutor   に公開 2021 年 01 月 14 日
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