Darts in Higher Dimensions (with 3blue1brown) - Numberphile
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Date: 2022-04-09
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Comments and reviews: 9
Adam
-Because why not- is the definition of why math is so fun. The rules are clear, the reasoning rock solid, but the freedom to play with the assumptions is unmatched. No physics to restrain you. No -right- way to approach a problem.
Another fun one is -without loss of generality-. I can't count how many times these words stopped me for quite a few minutes while I tried to understand why and make sure it really is. The revelation was usually fun. Not for obvious cases of course, but some authors can make the word -clearly- work VERY hard.
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-Because why not- is the definition of why math is so fun. The rules are clear, the reasoning rock solid, but the freedom to play with the assumptions is unmatched. No physics to restrain you. No -right- way to approach a problem.
Another fun one is -without loss of generality-. I can't count how many times these words stopped me for quite a few minutes while I tried to understand why and make sure it really is. The revelation was usually fun. Not for obvious cases of course, but some authors can make the word -clearly- work VERY hard.
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Гаврил
You can actually get 2. 16 (which is only about 0. 03 away from the true answer) not thinkig of the high-dimensional shapes, just assuming that the bulleye radius gets sqrt(2) times smaller every move (there is 1/2 possibility that it gets more than sqrt(2) times smaller and 1/2 possibility for less than sqrt(2. The video is absolutely beatiful, I just find it interesting that for practical applications (if 1. 4% error is not a problem) such utterly rough methods can actually work really well.
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You can actually get 2. 16 (which is only about 0. 03 away from the true answer) not thinkig of the high-dimensional shapes, just assuming that the bulleye radius gets sqrt(2) times smaller every move (there is 1/2 possibility that it gets more than sqrt(2) times smaller and 1/2 possibility for less than sqrt(2. The video is absolutely beatiful, I just find it interesting that for practical applications (if 1. 4% error is not a problem) such utterly rough methods can actually work really well.
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eekee
I'm watching the first part; the shrinking bullseye, and thinking this would make a wild 2-player game! With 2 players getting alternate shots, what's the best strategy? Do you want to aim for the outer part of the circle every time, only on your first shot, never, or something else? Does the best strategy for the player who gets the first shot differ from the best strategy for the other player?
Anyway, fascinating video! Using higher dimensions for a 2D puzzle blew my mind. :D
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I'm watching the first part; the shrinking bullseye, and thinking this would make a wild 2-player game! With 2 players getting alternate shots, what's the best strategy? Do you want to aim for the outer part of the circle every time, only on your first shot, never, or something else? Does the best strategy for the player who gets the first shot differ from the best strategy for the other player?
Anyway, fascinating video! Using higher dimensions for a 2D puzzle blew my mind. :D
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gtapanda
How can a perfect bullseye be probability 0? If that's the case because it's an infinitesimally small point, then couldn't you say the same about every single point on the dartboard? But the dart will definitely hit one of those points, so how can a dart hit a point that has a probability of 0 of being hit by the dart? Doesn't the fact that the dart hit that point disprove that the point had a 0 probability of being hit?
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How can a perfect bullseye be probability 0? If that's the case because it's an infinitesimally small point, then couldn't you say the same about every single point on the dartboard? But the dart will definitely hit one of those points, so how can a dart hit a point that has a probability of 0 of being hit by the dart? Doesn't the fact that the dart hit that point disprove that the point had a 0 probability of being hit?
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Groovy
Please retitle the video; based on the only -scientific equation- NOT mentioned in the video, but clearly referred to by all the information provided: ie. probability, the 4th and successive even steps and the equivalent odds of finding successively higher dimensions
TIME IS NOT YOUR FRIEND
PS. Also shows splitting a pie with more than 3 other people only creates wasteful cubes
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Please retitle the video; based on the only -scientific equation- NOT mentioned in the video, but clearly referred to by all the information provided: ie. probability, the 4th and successive even steps and the equivalent odds of finding successively higher dimensions
TIME IS NOT YOUR FRIEND
PS. Also shows splitting a pie with more than 3 other people only creates wasteful cubes
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Calculated
Grant is most certainly the best person in the world at taking topics in mathematics and explaining them in a way that's easy to understand. I've taken calculus classes, and have seen approaches to topics that would have made them far easier to learn in class. For a single example, he made e-(i-pi) seem intuitive instead of just -remember this fact-.
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Grant is most certainly the best person in the world at taking topics in mathematics and explaining them in a way that's easy to understand. I've taken calculus classes, and have seen approaches to topics that would have made them far easier to learn in class. For a single example, he made e-(i-pi) seem intuitive instead of just -remember this fact-.
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Poly
In this Game a score of Zero is possible, only if on the first throw you miss the target. The chances of getting zero increases as the size of our first target gets smaller. If you are in the center of a sphere (meaning completely surrounded by the target, a score of zero is impossible leaving an infinite number of new possibilities.
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In this Game a score of Zero is possible, only if on the first throw you miss the target. The chances of getting zero increases as the size of our first target gets smaller. If you are in the center of a sphere (meaning completely surrounded by the target, a score of zero is impossible leaving an infinite number of new possibilities.
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BrutusJV
If I understand correctly, to calculate the prob of S > 2 we just take for random numbers between -1 and 1.
This doesn't take into account whether the first shot would even hit, does it? Because the sum of the squares of four random numbers could be < 1 while two of these (what would be the first shot) could be > 1 right?
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If I understand correctly, to calculate the prob of S > 2 we just take for random numbers between -1 and 1.
This doesn't take into account whether the first shot would even hit, does it? Because the sum of the squares of four random numbers could be < 1 while two of these (what would be the first shot) could be > 1 right?
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Erhannis
4: 54 Well, probably not rotationally symmetric; probably more elliptical than circular. You have to counter gravity in one plane and not the other, and the mechanics of your arm probably affect the result. But a rotationally symmetric gaussian distribution would likely indeed be more correct than a uniform distribution, yes.
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4: 54 Well, probably not rotationally symmetric; probably more elliptical than circular. You have to counter gravity in one plane and not the other, and the mechanics of your arm probably affect the result. But a rotationally symmetric gaussian distribution would likely indeed be more correct than a uniform distribution, yes.
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