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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

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))

• 250 26
Go Tutor に公開 2021 年 01 月 14 日