A Mathematical Fable - Numberphile
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Date: 2022-04-08
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Comments and reviews: 9
River
Perhaps another way to gain some intuition on this is to ask, -how would things behave if this were NOT true? -. Imagine we trace the perimeter of all four shapes (the triangle and three squares. Now imagine that we proportionally grow the combined shape by 2x - if the fastest-growing of the four shapes gains in area slightly faster than 2x, it will make up a larger percentage of the combined shape than it did before it was enlarged. Now grow the combined shape by 2x again and again and again. You might need to zoom out to the edge of the universe to see the new combined shape. Then ask yourself, does it make any sense at all that this fastest-growing inner-shape (be that the triangle or one of the three blobs) now seems to occupy close to 100% of the total area while all of the other shapes have virtually disappeared? No. The only way for us to have something that still -looks right- as we grow larger and larger is to grow all shapes proportionally.
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Perhaps another way to gain some intuition on this is to ask, -how would things behave if this were NOT true? -. Imagine we trace the perimeter of all four shapes (the triangle and three squares. Now imagine that we proportionally grow the combined shape by 2x - if the fastest-growing of the four shapes gains in area slightly faster than 2x, it will make up a larger percentage of the combined shape than it did before it was enlarged. Now grow the combined shape by 2x again and again and again. You might need to zoom out to the edge of the universe to see the new combined shape. Then ask yourself, does it make any sense at all that this fastest-growing inner-shape (be that the triangle or one of the three blobs) now seems to occupy close to 100% of the total area while all of the other shapes have virtually disappeared? No. The only way for us to have something that still -looks right- as we grow larger and larger is to grow all shapes proportionally.
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Lee
I found another mathematical constant related to pythagorus, (sqrt(x2 + x2)/x) always equals 1. 4142135624 (didn't check for more digits, used my phone to calculate it) - of which I'm dubbing TI, the T comes from Triangle and the I comes from PI so it could TI with PI as a pun: ) - anyways further to that when x is a power of 2 (only tried with powers of 2 since I was lazy) 2 - TI multiplied by 2 n times will also equal sqrt(x2+x2, I have not done an in depth trial an error on the formula derived from using TI. Since the name TI is built from half of PI I tried seeing if there was any upside down PI symbol, didn't find any so I'm proposing that as TI's symbol as well, I'm too lazy to write and submit a paper on TI so just dump this entire comment in as a quote if you do so yourself (in case the date is lost it is 20/12/2020)
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I found another mathematical constant related to pythagorus, (sqrt(x2 + x2)/x) always equals 1. 4142135624 (didn't check for more digits, used my phone to calculate it) - of which I'm dubbing TI, the T comes from Triangle and the I comes from PI so it could TI with PI as a pun: ) - anyways further to that when x is a power of 2 (only tried with powers of 2 since I was lazy) 2 - TI multiplied by 2 n times will also equal sqrt(x2+x2, I have not done an in depth trial an error on the formula derived from using TI. Since the name TI is built from half of PI I tried seeing if there was any upside down PI symbol, didn't find any so I'm proposing that as TI's symbol as well, I'm too lazy to write and submit a paper on TI so just dump this entire comment in as a quote if you do so yourself (in case the date is lost it is 20/12/2020)
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CAbbott71
I still find this one hard to see, but once I take on board that the reveal is not a random act, then it makes sense that the final probability is not as simple as 1/2.
As a proof to myself, I ran a simulation in Excel:
- Pick a random door for the car
- Pick a random door
- Randomly pick a reveal that was not the chosen door or the car.
Then test the outcome of both strategies, Switch or Stay
Over 1, 000 iterations: 671 wins to the Switch, only 329 to the Stay.
Over 100, 000 iterations it was 66571 to 33429.
It's pretty clear that the result is closer to 66. 6% than 50%!
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I still find this one hard to see, but once I take on board that the reveal is not a random act, then it makes sense that the final probability is not as simple as 1/2.
As a proof to myself, I ran a simulation in Excel:
- Pick a random door for the car
- Pick a random door
- Randomly pick a reveal that was not the chosen door or the car.
Then test the outcome of both strategies, Switch or Stay
Over 1, 000 iterations: 671 wins to the Switch, only 329 to the Stay.
Over 100, 000 iterations it was 66571 to 33429.
It's pretty clear that the result is closer to 66. 6% than 50%!
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Zafoshin
I don't get it. He proved it the 2nd day. 3d day was just a special case with an unnecessary different proof. Why does he state if you prove it for one type of blob you prove it for all? In general, It's prove for all cases or disprove for one. If you prove for one, that's no guarantee for the rest.
Furthermore, he uses square-cube law without proving it. Why does the king not ask proof of that? Is there any proof of this besides expressing each blob as limit of similar squares?
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I don't get it. He proved it the 2nd day. 3d day was just a special case with an unnecessary different proof. Why does he state if you prove it for one type of blob you prove it for all? In general, It's prove for all cases or disprove for one. If you prove for one, that's no guarantee for the rest.
Furthermore, he uses square-cube law without proving it. Why does the king not ask proof of that? Is there any proof of this besides expressing each blob as limit of similar squares?
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Montana
But Counselor, my land isn't on a flat plane, one side rises up steadily in elevation, while the other two remain primarily flat! Also the middle sized blob contains the river, and most of the farmland!
And the councelor said 'Do not worry. None of this is real. This was just a math lesson, and you're stepping off script. All the land is worth exactly the value of the lesson learned. '
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But Counselor, my land isn't on a flat plane, one side rises up steadily in elevation, while the other two remain primarily flat! Also the middle sized blob contains the river, and most of the farmland!
And the councelor said 'Do not worry. None of this is real. This was just a math lesson, and you're stepping off script. All the land is worth exactly the value of the lesson learned. '
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Thomas
The king comes back and says -I forgot about the queen. I need to divide everything by 3 instead-
The counsellor says -No problem. Just create a 1/3 scale blob on the straight border inside each of the original blobs. The queen gets all 3 of those smaller blobs and this gives everyone an equal area. -
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The king comes back and says -I forgot about the queen. I need to divide everything by 3 instead-
The counsellor says -No problem. Just create a 1/3 scale blob on the straight border inside each of the original blobs. The queen gets all 3 of those smaller blobs and this gives everyone an equal area. -
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Timotej
Very entertaining video but I'm not sure I entirely follow the line of reasoning that one blob implies all blobs, if I chose a blob of area 0, the relation would be satisfied in that one case and, by the logic used here, all cases even though thats clearly not true from that example alone
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Very entertaining video but I'm not sure I entirely follow the line of reasoning that one blob implies all blobs, if I chose a blob of area 0, the relation would be satisfied in that one case and, by the logic used here, all cases even though thats clearly not true from that example alone
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Martin
I think the proof is incomplete. He proved that if you know the blob Pythagorean theorem for one blob you know it for all similar blobs, but he didn't generalize to differently shaped blobs. But that's easy to demonstrate. This is definitely my new favorite Pythagorean theorem proof.
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I think the proof is incomplete. He proved that if you know the blob Pythagorean theorem for one blob you know it for all similar blobs, but he didn't generalize to differently shaped blobs. But that's easy to demonstrate. This is definitely my new favorite Pythagorean theorem proof.
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JanickGers0
I've been teaching the Pythagorean Theorem in high school for a couple of years now. I can't believe I just discovered this video. I'm definitely going to use this in my class. Now I just have to find a translation for 'blob' in spanish. That is the tricky part!
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I've been teaching the Pythagorean Theorem in high school for a couple of years now. I can't believe I just discovered this video. I'm definitely going to use this in my class. Now I just have to find a translation for 'blob' in spanish. That is the tricky part!
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