title

Darts in Higher Dimensions (with 3blue1brown) - Numberphile

description

Grant Sanderson from 3Blue1Brown joins us to discuss an intriguing puzzle with a shrinking bullseye.
More links & stuff in full description below ↓↓↓
3Blue1Brown: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw
Grant Sanderson on the Numberphile podcast: https://youtu.be/A0RH93XvSyU
Greg Egan's tweet which started it all: https://twitter.com/gregegansf/status/1160461092973211648
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): http://bit.ly/MSRINumberphile
We are also supported by Science Sandbox, a Simons Foundation initiative dedicated to engaging everyone with the process of science. https://www.simonsfoundation.org/outreach/science-sandbox/
And support from Math For America - https://www.mathforamerica.org/
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Videos by Brady Haran
Grant did the animations for this one!
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Sign up for (occasional) emails: http://eepurl.com/YdjL9
Special thanks to our friend Jeff for the accommodation and filming space.

detail

{'title': 'Darts in Higher Dimensions (with 3blue1brown) - Numberphile', 'heatmap': [{'end': 850.087, 'start': 829.861, 'weight': 0.796}, {'end': 1722.759, 'start': 1695.905, 'weight': 1}], 'summary': 'Explores dart games, probability, and geometry, covering concepts such as shrinking bullseye, punishment for misses, point systems, probability distribution, higher dimensional geometry, and the estimation of pi using random dart throws, providing a comprehensive insight into the mathematical aspects of darts and higher dimensions.', 'chapters': [{'end': 81.384, 'segs': [{'end': 45.139, 'src': 'embed', 'start': 19.599, 'weight': 0, 'content': [{'end': 25.224, 'text': "I have a proper compass, so I'm gonna go ahead and cut out a piece, just a piece of paper to indicate our enlarged bullseye.", 'start': 19.599, 'duration': 5.625}, {'end': 31.658, 'text': "It's not the greatest circle I've ever drawn.", 'start': 29.875, 'duration': 1.783}, {'end': 33.421, 'text': 'All you do is you try to hit the bullseye.', 'start': 31.999, 'duration': 1.422}, {'end': 37.609, 'text': 'Each time we hit it though, based on where our shot is, the bullseye is going to shrink.', 'start': 33.702, 'duration': 3.907}, {'end': 39.933, 'text': "So it's going to get harder and harder as we go.", 'start': 38.37, 'duration': 1.563}, {'end': 41.996, 'text': "You're not going to hit the wall, eh? I'm not that bad.", 'start': 39.953, 'duration': 2.043}, {'end': 45.139, 'text': 'Oh, okay.', 'start': 44.719, 'duration': 0.42}], 'summary': 'Using a compass, a bullseye is created on paper; hitting it shrinks the target.', 'duration': 25.54, 'max_score': 19.599, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA19599.jpg'}], 'start': 0.302, 'title': 'Dart bullseye puzzle', 'summary': 'Discusses a dart game where the bullseye shrinks with each hit, based on shot distance and chord, increasing game difficulty over time.', 'chapters': [{'end': 81.384, 'start': 0.302, 'title': 'Dart bullseye puzzle', 'summary': 'Discusses a dart game where a bullseye shrinks each time it is hit, with the shrinking determined by the distance of the shot from the center and a perpendicular chord, making it harder to hit as the game progresses.', 'duration': 81.082, 'highlights': ['The bullseye shrinks each time it is hit based on the distance of the shot from the center and a perpendicular chord, making it harder to hit as the game progresses. The length of the chord drawn from the center to where the dart hits determines the new diameter of the bullseye, making it harder to hit as the game progresses.', 'The game involves hitting as many bullseyes as possible, with the bullseye shrinking with each hit. The objective of the game is to hit as many bullseyes as possible, but with each hit, the bullseye shrinks, making it increasingly difficult to hit.', 'The bullseye starts as a giant circle that fills up the entirety of the board. Initially, the bullseye is a giant circle that fills up the entire board, providing a large target for the players.']}], 'duration': 81.082, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA302.jpg', 'highlights': ['The bullseye starts as a giant circle that fills up the entirety of the board.', 'The game involves hitting as many bullseyes as possible, with the bullseye shrinking with each hit.', 'The bullseye shrinks each time it is hit based on the distance of the shot from the center and a perpendicular chord, making it harder to hit as the game progresses.']}, {'end': 320.914, 'segs': [{'end': 118.771, 'src': 'embed', 'start': 83.371, 'weight': 1, 'content': [{'end': 88.497, 'text': 'Is the center of the new circle where your dart hit or where the original center was? Great question.', 'start': 83.371, 'duration': 5.126}, {'end': 89.838, 'text': "It's where the original center was.", 'start': 88.597, 'duration': 1.241}, {'end': 92.481, 'text': "All right, so I'll draw a new circle where the original center was.", 'start': 89.858, 'duration': 2.623}, {'end': 96.205, 'text': 'I actually wish my shot was worse now because this is kind of too close to the edge.', 'start': 92.501, 'duration': 3.704}, {'end': 99.629, 'text': "I mean, you can kind of see it's barely smaller.", 'start': 96.786, 'duration': 2.843}, {'end': 102.844, 'text': "Oh yeah, so that's just trimming that circle down a bit.", 'start': 100.103, 'duration': 2.741}, {'end': 104.565, 'text': "So I'm just going to be giving it a little trim.", 'start': 102.984, 'duration': 1.581}, {'end': 109.087, 'text': 'This is like getting your hair cut, you know, once every week or something like that.', 'start': 104.585, 'duration': 4.502}, {'end': 113.309, 'text': 'And presumably this means that the next shot is going to be similarly difficult.', 'start': 109.367, 'duration': 3.942}, {'end': 118.771, 'text': "This is actually fascinating because when I animate it, then it's like, oh, boom, the computer does it.", 'start': 115.109, 'duration': 3.662}], 'summary': 'Drawing a new circle around the original center, adjusting its size.', 'duration': 35.4, 'max_score': 83.371, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA83371.jpg'}, {'end': 212.869, 'src': 'embed', 'start': 142.856, 'weight': 0, 'content': [{'end': 145.359, 'text': 'But let me, let me try to get a worse shot.', 'start': 142.856, 'duration': 2.503}, {'end': 150.824, 'text': "Yeah, consistency is only a virtue if you're not a screw up.", 'start': 145.739, 'duration': 5.085}, {'end': 157.664, 'text': "So this one's nice and far away from the center and I'm going to be punished more because it's far away.", 'start': 153.237, 'duration': 4.427}, {'end': 159.026, 'text': "So we'll do the same game.", 'start': 157.964, 'duration': 1.062}, {'end': 159.707, 'text': "We'll take this off.", 'start': 159.066, 'duration': 0.641}, {'end': 160.568, 'text': "We're going to make some cuts.", 'start': 159.727, 'duration': 0.841}, {'end': 162.631, 'text': 'Distance is h1.', 'start': 161.77, 'duration': 0.861}, {'end': 167.238, 'text': "Okay, so it'll be a chord of the big circle perpendicular to that line.", 'start': 162.891, 'duration': 4.347}, {'end': 176.407, 'text': "The chord defines the new diameter, but it's kind of easier to think of as half the chord, as defining the new radius,", 'start': 170.981, 'duration': 5.426}, {'end': 178.249, 'text': "certainly if you're drawing with a compass.", 'start': 176.407, 'duration': 1.842}, {'end': 180.212, 'text': "So I'll set my new radius.", 'start': 178.95, 'duration': 1.262}, {'end': 185.558, 'text': 'But this time, because I was closer to the edge of my, you know, quote unquote, bullseye, that chord ends up smaller.', 'start': 180.652, 'duration': 4.906}, {'end': 190.403, 'text': "So I'm punished, which means that the new bullseye is meaningfully smaller.", 'start': 185.718, 'duration': 4.685}, {'end': 192.684, 'text': 'Smaller bullseye, play the same game.', 'start': 191.104, 'duration': 1.58}, {'end': 196.985, 'text': "I'm going to set it up so that the center is at the center of the board.", 'start': 192.904, 'duration': 4.081}, {'end': 202.267, 'text': "Now I'm going to actually try to hit closer to the center because I want more shots for a higher score.", 'start': 197.366, 'duration': 4.901}, {'end': 206.248, 'text': "And I didn't hit close to the center.", 'start': 204.347, 'duration': 1.901}, {'end': 207.288, 'text': 'Oh dear.', 'start': 206.788, 'duration': 0.5}, {'end': 209.088, 'text': 'You are going to get punished for that.', 'start': 207.808, 'duration': 1.28}, {'end': 212.869, 'text': "So at this point, maybe you can already see how much that's going to punish me.", 'start': 209.348, 'duration': 3.521}], 'summary': 'Adjusting radius based on distance from center affects accuracy and score.', 'duration': 70.013, 'max_score': 142.856, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA142856.jpg'}, {'end': 282.468, 'src': 'embed', 'start': 245.539, 'weight': 5, 'content': [{'end': 249.943, 'text': 'Based on what I want the answer to look like at the end here, you get one point just for playing.', 'start': 245.539, 'duration': 4.404}, {'end': 254.507, 'text': 'Okay? So you get one point just for playing and then one point for each bullseye that you hit.', 'start': 249.963, 'duration': 4.544}, {'end': 255.889, 'text': 'So in this case, I got a score of four.', 'start': 254.528, 'duration': 1.361}, {'end': 260.394, 'text': "You can say maybe what we're counting is how many darts you throw in total.", 'start': 256.269, 'duration': 4.125}, {'end': 263.738, 'text': 'So I threw four darts and the fourth one happened to be a miss.', 'start': 260.875, 'duration': 2.863}, {'end': 267.003, 'text': "Whatever you want to do to artificially add one, you'll see why we like that at the end.", 'start': 264.079, 'duration': 2.924}, {'end': 269.163, 'text': "All right, so I've defined a game.", 'start': 268.183, 'duration': 0.98}, {'end': 270.584, 'text': "I haven't defined the puzzle yet.", 'start': 269.183, 'duration': 1.401}, {'end': 275.146, 'text': 'So for the puzzle, let me just draw our board again, or at least draw a circle.', 'start': 270.904, 'duration': 4.242}, {'end': 277.286, 'text': "All right, so we're going to have a random darts player.", 'start': 275.166, 'duration': 2.12}, {'end': 279.547, 'text': 'So he throws at the board with some random distribution.', 'start': 277.427, 'duration': 2.12}, {'end': 280.568, 'text': 'We all do.', 'start': 280.108, 'duration': 0.46}, {'end': 282.468, 'text': "There's no skill here.", 'start': 280.948, 'duration': 1.52}], 'summary': 'Players earn one point for playing and one point for each bullseye hit. the player scored four points by hitting one bullseye and playing. the game involves a random darts player with no skill.', 'duration': 36.929, 'max_score': 245.539, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA245539.jpg'}], 'start': 83.371, 'title': 'Dartboard games', 'summary': 'Explores dartboard circle trimming, punishment for missing the bullseye, and a dart game point system. it demonstrates the impact of consistency on circle size, the concept of punishment for missing the bullseye, and the point system where players can earn four points. additionally, it introduces the distribution of throws within a square encompassing the dart board.', 'chapters': [{'end': 167.238, 'start': 83.371, 'title': 'Dartboard circle trimming', 'summary': "Explores the process of trimming a circle on a dartboard, demonstrating how shots closer to the edge result in smaller circles and the impact of consistency on the circle's size.", 'duration': 83.867, 'highlights': ['The new circle is drawn where the original center was, and shots closer to the edge result in smaller circles.', 'Trimming the circle is likened to getting a haircut, with each trim making the circle slightly smaller.', 'Consistency in hitting the same spot results in a smaller circle, while a shot further from the center leads to a larger circle.']}, {'end': 245.079, 'start': 170.981, 'title': 'Punishment for missing the bullseye', 'summary': 'Discusses the concept of getting punished for missing the bullseye in a game, resulting in a smaller bullseye and a lower score, demonstrated through the use of a compass and a board.', 'duration': 74.098, 'highlights': ['Missing the bullseye results in a smaller bullseye, leading to a lower score, demonstrated through the use of a compass and a board.', "The chord defining the new radius is smaller when it's closer to the edge, which punishes the player for missing the bullseye.", 'Scoring in the game is based on the number of hits, with a miss resulting in a lower score.']}, {'end': 320.914, 'start': 245.539, 'title': 'Dart game point system', 'summary': 'Discusses a dart game point system where players earn one point for playing and additional points for hitting bullseyes, resulting in a score of four. it also introduces the concept of a random darts player and the distribution of their throws within a square encompassing the dart board.', 'duration': 75.375, 'highlights': ['Players earn one point for playing and additional points for hitting bullseyes, resulting in a score of four.', 'Introduction of the concept of a random darts player and the distribution of their throws within a square encompassing the dart board.', 'The game point system is defined, where players earn one point for playing and additional points for hitting bullseyes.']}], 'duration': 237.543, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA83371.jpg', 'highlights': ['Consistency in hitting the same spot results in a smaller circle, while a shot further from the center leads to a larger circle.', 'The new circle is drawn where the original center was, and shots closer to the edge result in smaller circles.', 'Trimming the circle is likened to getting a haircut, with each trim making the circle slightly smaller.', 'Missing the bullseye results in a smaller bullseye, leading to a lower score, demonstrated through the use of a compass and a board.', "The chord defining the new radius is smaller when it's closer to the edge, which punishes the player for missing the bullseye.", 'Players earn one point for playing and additional points for hitting bullseyes, resulting in a score of four.', 'Introduction of the concept of a random darts player and the distribution of their throws within a square encompassing the dart board.', 'Scoring in the game is based on the number of hits, with a miss resulting in a lower score.']}, {'end': 678.097, 'segs': [{'end': 343.995, 'src': 'embed', 'start': 321.514, 'weight': 3, 'content': [{'end': 329.16, 'text': "The x coordinate for this person is always going to be something between negative 1 and 1, and it'll just be completely uniform within that.", 'start': 321.514, 'duration': 7.646}, {'end': 332.043, 'text': 'So this would be the simplest thing to model on a computer.', 'start': 329.861, 'duration': 2.182}, {'end': 338.81, 'text': 'To do it, you just choose two different numbers, each one between negative one and one, where every one of those is equally probable.', 'start': 332.643, 'duration': 6.167}, {'end': 341.432, 'text': 'So every point in this whole square is equally probable.', 'start': 339.15, 'duration': 2.282}, {'end': 343.995, 'text': "And that square doesn't shrink, it's only the circle that shrinks.", 'start': 341.753, 'duration': 2.242}], 'summary': "Modeling a person's x coordinate uniformly between -1 and 1 is simple and equally probable for every point in a square.", 'duration': 22.481, 'max_score': 321.514, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA321514.jpg'}, {'end': 414.181, 'src': 'embed', 'start': 385.355, 'weight': 4, 'content': [{'end': 387.896, 'text': "Just what's his average score for those thousand games?", 'start': 385.355, 'duration': 2.541}, {'end': 391.678, 'text': 'Another way to think about it is.', 'start': 390.037, 'duration': 1.641}, {'end': 402.482, 'text': "you say it's going to be the probability that the score is equal to one times one for that score, plus the probability because, remember,", 'start': 391.678, 'duration': 10.804}, {'end': 405.263, 'text': 'the lowest score you can get is one because you get that one point just for playing.', 'start': 402.482, 'duration': 2.781}, {'end': 414.181, 'text': 'the probability that the score is equal to 2 times 2, plus the probability that the score is equal to 3 times 3, and so on.', 'start': 405.876, 'duration': 8.305}], 'summary': 'Calculate the average score for a thousand games using probability and scores.', 'duration': 28.826, 'max_score': 385.355, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA385355.jpg'}, {'end': 553.831, 'src': 'embed', 'start': 528.358, 'weight': 0, 'content': [{'end': 535.923, 'text': "so it's 2 by 2, so it's 4, which means the probability that you hit that first bullseye, that your score will be greater than 1,", 'start': 528.358, 'duration': 7.565}, {'end': 538.025, 'text': 'because you get that 1 point free just for playing.', 'start': 535.923, 'duration': 2.102}, {'end': 543.103, 'text': 'is going to be the area of that circle divided by the area of that square.', 'start': 538.9, 'duration': 4.203}, {'end': 546.806, 'text': "Grant, what happens if the dart lands on the line? That's probability zero.", 'start': 543.303, 'duration': 3.503}, {'end': 548.267, 'text': "Don't worry about it.", 'start': 547.587, 'duration': 0.68}, {'end': 549.388, 'text': "It won't mess with the problem.", 'start': 548.407, 'duration': 0.981}, {'end': 553.831, 'text': "So now a much harder question is, OK, you've hit that first shot.", 'start': 549.468, 'duration': 4.363}], 'summary': 'Probability of scoring greater than 1 on first bullseye is area of circle divided by area of square.', 'duration': 25.473, 'max_score': 528.358, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA528358.jpg'}, {'end': 621.062, 'src': 'embed', 'start': 592.658, 'weight': 2, 'content': [{'end': 599.264, 'text': "it's the square root of x squared plus y squared, if those are the coordinates of your first shot.", 'start': 592.658, 'duration': 6.606}, {'end': 602.886, 'text': "And because we're going to have other shots, I'm also going to give these guys a subscript.", 'start': 599.924, 'duration': 2.962}, {'end': 606.609, 'text': "So now it's a geometry question to ask what's the new radius?", 'start': 603.226, 'duration': 3.383}, {'end': 614.02, 'text': "Maybe we call this our zeroth radius right? and I want to know what's the next radius after that initial shot.", 'start': 607.169, 'duration': 6.851}, {'end': 621.062, 'text': 'And if we remember our rule, we say you draw the chord, and the chord was defined to be perpendicular to that.', 'start': 614.74, 'duration': 6.322}], 'summary': 'The new radius is the square root of x squared plus y squared, with subsequent shots having subscript and a perpendicular chord.', 'duration': 28.404, 'max_score': 592.658, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA592658.jpg'}, {'end': 664.431, 'src': 'embed', 'start': 633.285, 'weight': 1, 'content': [{'end': 643.897, 'text': "So what we might do and I recognize I'll sort of be drawing over myself a bit here, Because we have a right triangle here and that's what defined it.", 'start': 633.285, 'duration': 10.612}, {'end': 645.798, 'text': "we're going to use the Pythagorean theorem.", 'start': 643.897, 'duration': 1.901}, {'end': 650.021, 'text': "Because this was our old radius, radius 1, that's the new hypotenuse.", 'start': 646.278, 'duration': 3.743}, {'end': 664.431, 'text': 'So we know that the old hypotenuse, 1 squared, is equal to one of the legs, which was h0 squared, plus the new radius squared, plus r1 squared.', 'start': 650.601, 'duration': 13.83}], 'summary': 'Using the pythagorean theorem, the old hypotenuse is equal to the sum of h0 squared and r1 squared.', 'duration': 31.146, 'max_score': 633.285, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA633285.jpg'}], 'start': 321.514, 'title': 'Dart throwing probability and geometry', 'summary': 'Discusses the uniform distribution of dart throwing within a square, the shrinking circle, and the probability of hitting the bullseye, while exploring the concept of expected score and the paradox of events with probability zero. additionally, it covers the probability of hitting the bullseye in a dart game, calculating the probability of getting a score greater than 1, and exploring the geometry involved in determining the new radius after hitting the bullseye.', 'chapters': [{'end': 510.807, 'start': 321.514, 'title': 'Dart throwing probability', 'summary': 'Discusses the uniform distribution of dart throwing within a square, the shrinking circle, and the probability of hitting the bullseye, while exploring the concept of expected score and the paradox of events with probability zero.', 'duration': 189.293, 'highlights': ["The x coordinate for this person is always going to be something between negative 1 and 1, and it'll just be completely uniform within that. Describes the uniform distribution of x coordinate within the square, emphasizing the range of -1 to 1.", 'Another way to think about the expected score: imagine this random player plays a thousand games right? Explores the concept of expected score by considering the average score over a thousand games, highlighting the approach to understanding the probability distribution.', "In principle, you could have a perfect game, right? If you hit in the center every single time and it never shrinks, you're hitting a perfect game. Raises the possibility of achieving a perfect game and emphasizes the impact of the shrinking circle on the game's outcome.", "And this a lot of people kind of try to come to philosophical terms with like how can it be possible? but probability zero? the probability would have to be slightly bigger, but it's, uh, just one of those things of math. Explores the philosophical implications of events with probability zero, emphasizing the counterintuitive nature of such events in probability theory.", 'If you hit a first shot, your score is greater than or equal to one. Or no, actually, because you do get that point for free, this is the probability that your score is greater than one. Discusses the probability of achieving a score greater than one on the first shot, considering the point awarded for participating in the game.']}, {'end': 678.097, 'start': 511.187, 'title': 'Bullseye probability and geometry', 'summary': 'Discusses the probability of hitting the bullseye in a dart game, calculating the probability of getting a score greater than 1, and exploring the geometry involved in determining the new radius after hitting the bullseye.', 'duration': 166.91, 'highlights': ['The probability of getting a score greater than 1 is the area of the circle (pi) divided by the area of the square (4). The probability of getting a score greater than 1 is calculated as the area of the circle (pi) divided by the area of the square (4), indicating a 0.785 probability of hitting a score greater than 1.', 'Determining the new radius after hitting the bullseye involves using the Pythagorean theorem and the concept of chords in a right triangle. The process of determining the new radius after hitting the bullseye involves utilizing the Pythagorean theorem and the concept of chords in a right triangle, showcasing the application of geometry in the probability problem.', 'Exploring the hit length at each point involves using coordinates and the square root of x squared plus y squared. The exploration of the hit length at each point involves utilizing coordinates and the square root of x squared plus y squared, providing a framework for analyzing the dart game scenario.']}], 'duration': 356.583, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA321514.jpg', 'highlights': ['The probability of hitting a score greater than 1 is the area of the circle (pi) divided by the area of the square (4).', 'Determining the new radius after hitting the bullseye involves using the Pythagorean theorem and the concept of chords in a right triangle.', 'Exploring the hit length at each point involves using coordinates and the square root of x squared plus y squared.', "The x coordinate for this person is always going to be something between negative 1 and 1, and it'll just be completely uniform within that.", 'Another way to think about the expected score: imagine this random player plays a thousand games right?']}, {'end': 1205.525, 'segs': [{'end': 731.646, 'src': 'embed', 'start': 696.806, 'weight': 0, 'content': [{'end': 699.107, 'text': 'because it depends on four different numbers, right?', 'start': 696.806, 'duration': 2.301}, {'end': 704.251, 'text': "So I think it's helpful to just write out what the what the actual requirement is.", 'start': 699.788, 'duration': 4.463}, {'end': 711.796, 'text': 'H1, the hit in that second shot is less than R1, and I can go ahead and rewrite that by saying H1 squared is less than R1 squared,', 'start': 704.251, 'duration': 7.545}, {'end': 715.198, 'text': 'And the reason is we have a whole bunch of squares going on.', 'start': 713.137, 'duration': 2.061}, {'end': 719.24, 'text': "Maybe that's going to be helpful for us, right? Because I can expand out what h1 squared is.", 'start': 715.218, 'duration': 4.022}, {'end': 723.682, 'text': "That's x1 squared plus y1 squared.", 'start': 720, 'duration': 3.682}, {'end': 731.646, 'text': "And that has to be less than, well, what is r1 squared? Well, that's 1 squared minus h0 squared.", 'start': 723.702, 'duration': 7.944}], 'summary': 'Requirement: h1^2 < r1^2, where h1=x1^2+y1^2 and r1=1-h0^2.', 'duration': 34.84, 'max_score': 696.806, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA696806.jpg'}, {'end': 858.789, 'src': 'heatmap', 'start': 829.861, 'weight': 0.796, 'content': [{'end': 831.862, 'text': 'You immediately knew, oh, we need to use pi.', 'start': 829.861, 'duration': 2.001}, {'end': 833.963, 'text': "We're thinking of the area of the circle right?", 'start': 831.902, 'duration': 2.061}, {'end': 835.643, 'text': "and it's actually quite subtle.", 'start': 834.603, 'duration': 1.04}, {'end': 836.263, 'text': "what's happening there?", 'start': 835.643, 'duration': 0.62}, {'end': 843.545, 'text': "you have a question about a pair of numbers and you're gaining intuition and using facts that we've discovered in math, like the area of a circle,", 'start': 836.263, 'duration': 7.282}, {'end': 849.287, 'text': 'by thinking of that not as two separate entities, but as a single point in two-dimensional space.', 'start': 843.545, 'duration': 5.742}, {'end': 850.087, 'text': 'that might sound obvious.', 'start': 849.287, 'duration': 0.8}, {'end': 854.948, 'text': "the reason I'm hampering down on it is look at what we have over here four separate numbers.", 'start': 850.087, 'duration': 4.861}, {'end': 858.789, 'text': "what you're asking for is a probabilistic property of these four numbers.", 'start': 854.948, 'duration': 3.841}], 'summary': 'Using pi to think of the area of a circle, gaining intuition in math and exploring probabilistic properties of four numbers.', 'duration': 28.928, 'max_score': 829.861, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA829861.jpg'}, {'end': 1112.305, 'src': 'embed', 'start': 1079.609, 'weight': 1, 'content': [{'end': 1091.233, 'text': "What this comes down to is asking we're going to have six different coordinates x0, y0, x1, y1, x2, y2,", 'start': 1079.609, 'duration': 11.624}, {'end': 1094.273, 'text': 'and that the sum of their squares is less than one.', 'start': 1091.233, 'duration': 3.04}, {'end': 1105.457, 'text': "I don't know how to calculate the volume of a six-dimensional bowl.", 'start': 1102.196, 'duration': 3.261}, {'end': 1108.784, 'text': 'No! Luckily, mathematicians have figured it out for us.', 'start': 1105.797, 'duration': 2.987}, {'end': 1110.485, 'text': 'And again, it could be a whole interesting story.', 'start': 1108.864, 'duration': 1.621}, {'end': 1112.305, 'text': 'But if you go to Wikipedia, you could see a chart.', 'start': 1110.545, 'duration': 1.76}], 'summary': 'Mathematicians have calculated the volume of a six-dimensional bowl, with details available on wikipedia.', 'duration': 32.696, 'max_score': 1079.609, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1079609.jpg'}, {'end': 1167.211, 'src': 'embed', 'start': 1140.401, 'weight': 2, 'content': [{'end': 1144.186, 'text': 'and then what its denominator was, which was the six, because that r ends up being one.', 'start': 1140.401, 'duration': 3.785}, {'end': 1146.148, 'text': "And now I'll tell you the more general fact.", 'start': 1144.366, 'duration': 1.782}, {'end': 1156.639, 'text': 'which is pretty mind-blowing, is that the volume of a 2n ball is equal to pi to the n, so half of the dimension, right?', 'start': 1146.909, 'duration': 9.73}, {'end': 1158.241, 'text': 'So in two dimensions you see a pi.', 'start': 1156.659, 'duration': 1.582}, {'end': 1159.883, 'text': 'in four dimensions you see a pi squared.', 'start': 1158.241, 'duration': 1.642}, {'end': 1161.044, 'text': 'in six you see pi cubed.', 'start': 1159.883, 'duration': 1.161}, {'end': 1165.269, 'text': 'So pi to half the number of dimensions divided by n factorial.', 'start': 1161.625, 'duration': 3.644}, {'end': 1167.211, 'text': 'Very clean, very beautiful.', 'start': 1165.649, 'duration': 1.562}], 'summary': 'The volume of a 2n ball is equal to pi to the n, and in six dimensions, it is pi cubed.', 'duration': 26.81, 'max_score': 1140.401, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1140401.jpg'}], 'start': 678.237, 'title': 'Probability and volume in higher dimensions', 'summary': 'Explores the probability of getting a second shot in a new coordinate system using four random numbers within the range of -1 to 1, and the volume of higher dimensional objects in 2, 4, and 6 dimensions, and concludes with the calculation of the expected score.', 'chapters': [{'end': 808.334, 'start': 678.237, 'title': 'Probability of getting second shot', 'summary': 'Discusses the probability of getting the second shot in a new coordinate system, involving the calculation of the probability based on four random numbers within the range of -1 to 1 and the requirement that the sum of their squares should be less than one.', 'duration': 130.097, 'highlights': ['The chapter discusses the requirement for the hit in the second shot to be less than R1, involving the expansion and rearrangement of the coordinates to calculate the probability based on four random numbers within the range of -1 to 1. Requirement for hit in the second shot, expansion and rearrangement of coordinates, calculation of probability based on four random numbers.', 'The discussion focuses on the challenging question of determining the probability of the sum of squares of four random numbers being less than one, considering the range of -1 to 1 for each number. Challenging question of determining probability, consideration of the range of -1 to 1 for each number.', 'An explanation is provided about the complexity of calculating the probability, considering the impact of individual numbers on the overall probability. Complexity of calculating probability, impact of individual numbers on overall probability.']}, {'end': 1205.525, 'start': 808.334, 'title': 'Volume of higher dimensional objects', 'summary': 'Discusses the probability of the sum of squares of random numbers being less than 1 in 2, 4, and 6 dimensions, revealing a pattern in the volumes of higher dimensional objects and concluding with the calculation of the expected score.', 'duration': 397.191, 'highlights': ['The probability of the sum of squares of random numbers being less than 1 in 2, 4, and 6 dimensions The discussion revolves around the geometric interpretation of the probability of the sum of squares of random numbers being less than 1 in 2, 4, and 6 dimensions, showing the connection to the volumes of higher dimensional objects.', 'Pattern in the volumes of higher dimensional objects The pattern in the volumes of higher dimensional objects is revealed, with the volume of a 2n ball being expressed as pi to the power of n divided by n factorial.', 'Calculation of the expected score The chapter concludes with the calculation of the expected score, emphasizing the collapsing chaos in a wonderful way.']}], 'duration': 527.288, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA678237.jpg', 'highlights': ['The chapter discusses the requirement for the hit in the second shot to be less than R1, involving the expansion and rearrangement of the coordinates to calculate the probability based on four random numbers within the range of -1 to 1.', 'The probability of the sum of squares of random numbers being less than 1 in 2, 4, and 6 dimensions The discussion revolves around the geometric interpretation of the probability of the sum of squares of random numbers being less than 1 in 2, 4, and 6 dimensions, showing the connection to the volumes of higher dimensional objects.', 'Pattern in the volumes of higher dimensional objects The pattern in the volumes of higher dimensional objects is revealed, with the volume of a 2n ball being expressed as pi to the power of n divided by n factorial.']}, {'end': 1518.143, 'segs': [{'end': 1256.125, 'src': 'embed', 'start': 1227.71, 'weight': 5, 'content': [{'end': 1232.961, 'text': "Because if I wanted to say, for example, what's the probability that your score is precisely equal to two?", 'start': 1227.71, 'duration': 5.251}, {'end': 1238.189, 'text': 'So this will be your probability that your score is greater than one, right?', 'start': 1234.426, 'duration': 3.763}, {'end': 1243.234, 'text': 'Minus the probability that the score is greater than two.', 'start': 1239.03, 'duration': 4.204}, {'end': 1247.177, 'text': "Because for it to equal two, it's greater than one, but it's not greater than two.", 'start': 1243.854, 'duration': 3.323}, {'end': 1250.3, 'text': 'And this entirely encompasses the possibilities, right?', 'start': 1247.717, 'duration': 2.583}, {'end': 1256.125, 'text': 'If you get a score of greater than one, that encompasses all the possibilities of getting a score greater than two.', 'start': 1250.74, 'duration': 5.385}], 'summary': 'Explaining probability using score examples and quantifiable data.', 'duration': 28.415, 'max_score': 1227.71, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1227710.jpg'}, {'end': 1299.943, 'src': 'embed', 'start': 1266.594, 'weight': 4, 'content': [{'end': 1278.425, 'text': 'So this would be 1 times the probability that the score is bigger than 0, minus the probability that the score is bigger than 1,', 'start': 1266.594, 'duration': 11.831}, {'end': 1289.913, 'text': 'plus 2 times the probability that the score is bigger than 1, minus the probability that the score is bigger than 2..', 'start': 1278.425, 'duration': 11.488}, {'end': 1291.134, 'text': 'And just one more for good measure.', 'start': 1289.913, 'duration': 1.221}, {'end': 1299.943, 'text': 'Probability that the score is bigger than 2 minus the probability that the score is bigger than 3.', 'start': 1292.235, 'duration': 7.708}], 'summary': 'Calculating probabilities for different score thresholds.', 'duration': 33.349, 'max_score': 1266.594, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1266594.jpg'}, {'end': 1366.563, 'src': 'embed', 'start': 1335.153, 'weight': 6, 'content': [{'end': 1338.654, 'text': "we're subtracting 3,, but in the next one we'll be adding 4, and so on.", 'start': 1335.153, 'duration': 3.501}, {'end': 1343.415, 'text': 'So every puzzle that we just had, every micro puzzle, we just add up all of their answers.', 'start': 1340.094, 'duration': 3.321}, {'end': 1349.513, 'text': "So the probability that your score is at least zero, that's one.", 'start': 1345.851, 'duration': 3.662}, {'end': 1352.675, 'text': 'Definitely your score is bigger than zero because you get a point just for playing.', 'start': 1350.054, 'duration': 2.621}, {'end': 1359.399, 'text': "Probability that it's bigger than one, that's the one that you answered for me first, which was saying that it's pi divided by four.", 'start': 1352.995, 'duration': 6.404}, {'end': 1360.22, 'text': 'The next one.', 'start': 1359.679, 'duration': 0.541}, {'end': 1366.563, 'text': "So I'm actually going to write this slightly differently because we always have some power of pi divided by some power of two.", 'start': 1360.44, 'duration': 6.123}], 'summary': 'Using micro puzzles, we add up answers to find probabilities, like score being at least 1 and greater than 1, where the latter is pi/4.', 'duration': 31.41, 'max_score': 1335.153, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1335153.jpg'}, {'end': 1488.856, 'src': 'embed', 'start': 1441.653, 'weight': 0, 'content': [{'end': 1443.693, 'text': 'who just see something screaming in their head right now.', 'start': 1441.653, 'duration': 2.04}, {'end': 1455.139, 'text': "There's something known as the Taylor series for e to the x, which is that it's 1 plus x, and I'll write it as x to the 1 over 1 factorial x,", 'start': 1444.014, 'duration': 11.125}, {'end': 1466.893, 'text': 'squared over 2 factorial x, cubed over 3 factorial and so on, where you evaluate it as an infinite polynomial,', 'start': 1455.139, 'duration': 11.754}, {'end': 1470.356, 'text': 'where each one of the terms is 1 over n factorial.', 'start': 1467.853, 'duration': 2.503}, {'end': 1473.699, 'text': "It's not just that e to the x happens to equal this.", 'start': 1470.776, 'duration': 2.923}, {'end': 1478.064, 'text': 'I actually think the healthier way to view the exponential function and what e to the x is.', 'start': 1473.779, 'duration': 4.285}, {'end': 1479.445, 'text': 'this defines it right?', 'start': 1478.064, 'duration': 1.381}, {'end': 1483.55, 'text': 'This is the thing that should pop into your head when you think of exponential growth in e to the x.', 'start': 1479.505, 'duration': 4.045}, {'end': 1486.393, 'text': 'is this particular infinite series, this particular polynomial?', 'start': 1483.55, 'duration': 2.843}, {'end': 1488.856, 'text': 'This is where it will come up, especially in probability.', 'start': 1486.753, 'duration': 2.103}], 'summary': 'The taylor series for e to the x is 1 + x + x^2/2! + x^3/3! + ..., defining the exponential function and its connection to exponential growth and probability.', 'duration': 47.203, 'max_score': 1441.653, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1441653.jpg'}], 'start': 1205.525, 'title': 'Probability calculation and taylor series for e', 'summary': 'Covers calculating cumulative probability in games and deriving the taylor series for e, with an emphasis on probability, calculus, and factorial relationships.', 'chapters': [{'end': 1352.675, 'start': 1205.525, 'title': 'Cumulative probability of scoring', 'summary': 'Explains how to calculate the cumulative probability of scoring in a game by summing the products of each score and its corresponding probability, and then simplifying the expression to find the final cumulative probability.', 'duration': 147.15, 'highlights': ['The chapter explains how to calculate the cumulative probability of scoring in a game by summing the products of each score and its corresponding probability, and then simplifying the expression to find the final cumulative probability.', 'The probability that your score is at least zero is 1, as you get a point just for playing.', 'The method involves subtracting the probability of a score being greater than a certain number from the probability of it being greater than the preceding number to find the probability of the score being precisely equal to that number.']}, {'end': 1518.143, 'start': 1352.995, 'title': 'Understanding the taylor series for e', 'summary': 'Discusses the derivation of the taylor series for e, demonstrating the relationship between factorials, pi, and the exponential function, and how it relates to probability and calculus.', 'duration': 165.148, 'highlights': ['The Taylor series for e to the x is derived as an infinite polynomial, demonstrating the relationship between factorials, pi, and the exponential function, offering insights into exponential growth, probability, and calculus.', 'The healthier way to view the exponential function and e to the x is through this particular infinite series, providing an easier interpretation of its derivative, extending to understand e to the power of pi i, and indicating a healthier relationship with E.', 'The relationship between factorials, pi, and the exponential function leads to an easier interpretation of why it is its own derivative, and provides insights into exponential growth, probability, and calculus.', 'The chapter explains the derivation of the Taylor series for e, emphasizing its relevance in understanding exponential growth, probability, and calculus.']}], 'duration': 312.618, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1205525.jpg', 'highlights': ['The chapter explains the derivation of the Taylor series for e, emphasizing its relevance in understanding exponential growth, probability, and calculus.', 'The Taylor series for e to the x is derived as an infinite polynomial, demonstrating the relationship between factorials, pi, and the exponential function, offering insights into exponential growth, probability, and calculus.', 'The relationship between factorials, pi, and the exponential function leads to an easier interpretation of why it is its own derivative, and provides insights into exponential growth, probability, and calculus.', 'The healthier way to view the exponential function and e to the x is through this particular infinite series, providing an easier interpretation of its derivative, extending to understand e to the power of pi i, and indicating a healthier relationship with E.', 'The chapter explains how to calculate the cumulative probability of scoring in a game by summing the products of each score and its corresponding probability, and then simplifying the expression to find the final cumulative probability.', 'The method involves subtracting the probability of a score being greater than a certain number from the probability of it being greater than the preceding number to find the probability of the score being precisely equal to that number.', 'The probability that your score is at least zero is 1, as you get a point just for playing.']}, {'end': 1924.891, 'segs': [{'end': 1571.456, 'src': 'embed', 'start': 1545.242, 'weight': 0, 'content': [{'end': 1553.985, 'text': "if you are a terrible random darts player who unrealistically hits within a square with a uniform probability which you wouldn't because it would be rotationally symmetric but whatever,", 'start': 1545.242, 'duration': 8.743}, {'end': 1558.863, 'text': "you're hitting within a square and you keep going like this and you play a thousand games.", 'start': 1553.985, 'duration': 4.878}, {'end': 1560.385, 'text': 'on average, your score would be 2.1932..', 'start': 1558.863, 'duration': 1.522}, {'end': 1564.008, 'text': 'And remember, my score was four.', 'start': 1560.385, 'duration': 3.623}, {'end': 1571.456, 'text': "So you're not quite twice as good as someone who has absolutely no skill whatsoever.", 'start': 1566.511, 'duration': 4.945}], 'summary': 'Terrible random darts player averages 2.1932 score; not twice as good as someone with no skill.', 'duration': 26.214, 'max_score': 1545.242, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1545242.jpg'}, {'end': 1628.53, 'src': 'embed', 'start': 1604.475, 'weight': 4, 'content': [{'end': 1611.499, 'text': 'What was happening is you had six numbers and you were encoding a property of those six numbers with something that we like to describe geometrically.', 'start': 1604.475, 'duration': 7.024}, {'end': 1616.843, 'text': 'Rather than saying the sum of their squares is less than one, you say it shows up inside a six-dimensional ball.', 'start': 1611.839, 'duration': 5.004}, {'end': 1618.924, 'text': 'That also means it saved you some work.', 'start': 1617.623, 'duration': 1.301}, {'end': 1626.289, 'text': 'I put out this question to some channel supporters as an early teaser of things, and one of them got back to me saying you know,', 'start': 1619.784, 'duration': 6.505}, {'end': 1628.53, 'text': "I've been working on it and it's just really hard.", 'start': 1626.289, 'duration': 2.241}], 'summary': 'Encoding six numbers geometrically to represent a property, simplifying work and posing a challenging problem.', 'duration': 24.055, 'max_score': 1604.475, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1604475.jpg'}, {'end': 1722.759, 'src': 'heatmap', 'start': 1687.215, 'weight': 2, 'content': [{'end': 1695.625, 'text': "You can play a much simpler probability game, where let's say, I'm just gonna choose numbers from zero to one with uniform probability.", 'start': 1687.215, 'duration': 8.41}, {'end': 1702.008, 'text': "And I'm gonna keep going until the sum of the numbers that I've chosen ends up being bigger than one.", 'start': 1695.905, 'duration': 6.103}, {'end': 1709.172, 'text': 'So for example, if the first number you choose is 0.3, and then the second one is 0.6, their sum is 0.9.', 'start': 1702.689, 'duration': 6.483}, {'end': 1712.194, 'text': "And if the next one you choose is 0.5, that's the point when you go over.", 'start': 1709.172, 'duration': 3.022}, {'end': 1718.497, 'text': "And so the question you can ask is what's the expected number of samples you need to take before it overblows one?", 'start': 1712.474, 'duration': 6.023}, {'end': 1722.759, 'text': 'e shows up in the answer right, and the way it shows up is actually quite similar here,', 'start': 1718.917, 'duration': 3.842}], 'summary': 'Probability game with numbers from 0 to 1, finding expected samples before sum exceeds 1.', 'duration': 21.957, 'max_score': 1687.215, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1687215.jpg'}, {'end': 1798.677, 'src': 'embed', 'start': 1763.501, 'weight': 1, 'content': [{'end': 1764.662, 'text': "Why would you do that? It's very strange.", 'start': 1763.501, 'duration': 1.161}, {'end': 1769.245, 'text': 'How do you even interpret the sum of the area of a circle to the volume of a four-dimensional sphere?', 'start': 1764.702, 'duration': 4.543}, {'end': 1774.866, 'text': 'So he specifically designed a puzzle, such that this would be the answer, which I think is beautiful and clever.', 'start': 1770.065, 'duration': 4.801}, {'end': 1777.146, 'text': "Here's a very interesting thing.", 'start': 1776.046, 'duration': 1.1}, {'end': 1780.707, 'text': "We're adding up all of these volumes, right? It converges.", 'start': 1777.506, 'duration': 3.201}, {'end': 1787.528, 'text': 'And what that means is that, higher and higher out, the volume of a high dimensional sphere is quite small, right?', 'start': 1781.067, 'duration': 6.461}, {'end': 1791.529, 'text': "In fact, we could compute it out if I said hey, what's the volume of a 100 dimensional ball?", 'start': 1787.548, 'duration': 3.981}, {'end': 1795.91, 'text': 'Well, that would be pi to the 50th.', 'start': 1793.509, 'duration': 2.401}, {'end': 1798.677, 'text': 'Divided by 50 factorial.', 'start': 1797.136, 'duration': 1.541}], 'summary': "Discussion about the volume of high-dimensional spheres and a 100-dimensional ball's volume of pi to the 50th, divided by 50 factorial.", 'duration': 35.176, 'max_score': 1763.501, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1763501.jpg'}], 'start': 1518.143, 'title': 'Estimating pi and higher dimensional geometry', 'summary': 'Discusses estimating pi using random dart throws, yielding an average score of approximately 2.1932, and explores the unexpected aspects of higher dimensional geometry, including rediscovering the volume of a four-dimensional ball and the significance of the number e in probability games. additionally, it highlights the concept of adding volumes of higher dimensional balls, converging to small values in higher dimensions, with an example of a 100-dimensional ball having a volume of 2.368 x 10^-40, challenging conventional geometric intuition.', 'chapters': [{'end': 1564.008, 'start': 1518.143, 'title': 'Estimating pi using random dart throws', 'summary': "Explains the process of estimating pi using random dart throws, where hitting within a square with uniform probability yields an average score of approximately 2.1932, compared to the value of pi, with the speaker's score being four.", 'duration': 45.865, 'highlights': ['The average score from hitting within a square with uniform probability using random dart throws is approximately 2.1932, compared to the value of pi.', "The speaker's score from playing a similar game was four."]}, {'end': 1727.2, 'start': 1566.511, 'title': 'Higher dimensional geometry and probability', 'summary': 'Discusses the unexpected aspect of higher dimensional geometry and its application in probability, with a focus on encoding properties of numbers geometrically, rediscovering the volume of a four-dimensional ball, and the significance of the number e in probability games.', 'duration': 160.689, 'highlights': ['The chapter discusses encoding properties of numbers geometrically in six dimensions, which allows for a more universal language for communication and simplifies computations.', 'It explains the discovery of the volume of a four-dimensional ball through working on integrals, providing a more universal language for discussion.', 'The significance of the number e in probability games is highlighted, showcasing its role in determining the expected number of samples needed before a sum exceeds a certain value.']}, {'end': 1924.891, 'start': 1727.2, 'title': 'Higher dimensional volumes', 'summary': 'Delves into the concept of adding volumes of higher dimensional balls, converging to small values in higher dimensions, illustrated with the example of a 100-dimensional ball having a volume of 2.368 x 10^-40, challenging conventional geometric intuition. the discussion emphasizes the importance of a balance between analytic and geometric thinking.', 'duration': 197.691, 'highlights': ['The volume of a 100-dimensional ball is around 2.368 x 10^-40, challenging conventional geometric intuition. In the discussion, it is mentioned that the volume of a 100-dimensional ball is approximately 2.368 x 10^-40, which is significantly smaller than what conventional geometric intuition might suggest.', 'The concept of adding volumes of higher dimensional balls converges to small values in higher dimensions. The discussion emphasizes the convergence of adding volumes of higher dimensional balls, illustrating that in higher dimensions, the volumes converge to small values, challenging conventional geometric intuition.', 'Emphasizing the importance of a balance between analytic and geometric thinking. The chapter highlights the significance of maintaining a balance between analytic and geometric thinking when dealing with complex mathematical concepts, showcasing the value of both approaches in understanding higher dimensional spaces and their properties.']}], 'duration': 406.748, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/6_yU9eJ0NxA/pics/6_yU9eJ0NxA1518143.jpg', 'highlights': ['The average score from hitting within a square with uniform probability using random dart throws is approximately 2.1932, compared to the value of pi.', 'The volume of a 100-dimensional ball is around 2.368 x 10^-40, challenging conventional geometric intuition.', 'The significance of the number e in probability games is highlighted, showcasing its role in determining the expected number of samples needed before a sum exceeds a certain value.', 'The concept of adding volumes of higher dimensional balls converges to small values in higher dimensions. The discussion emphasizes the convergence of adding volumes of higher dimensional balls, illustrating that in higher dimensions, the volumes converge to small values, challenging conventional geometric intuition.', 'The chapter discusses encoding properties of numbers geometrically in six dimensions, which allows for a more universal language for communication and simplifies computations.']}], 'highlights': ['The bullseye starts as a giant circle that fills up the entirety of the board.', 'The game involves hitting as many bullseyes as possible, with the bullseye shrinking with each hit.', 'The bullseye shrinks each time it is hit based on the distance of the shot from the center and a perpendicular chord, making it harder to hit as the game progresses.', 'Consistency in hitting the same spot results in a smaller circle, while a shot further from the center leads to a larger circle.', 'The new circle is drawn where the original center was, and shots closer to the edge result in smaller circles.', 'Trimming the circle is likened to getting a haircut, with each trim making the circle slightly smaller.', 'Missing the bullseye results in a smaller bullseye, leading to a lower score, demonstrated through the use of a compass and a board.', "The chord defining the new radius is smaller when it's closer to the edge, which punishes the player for missing the bullseye.", 'Players earn one point for playing and additional points for hitting bullseyes, resulting in a score of four.', 'Introduction of the concept of a random darts player and the distribution of their throws within a square encompassing the dart board.', 'Scoring in the game is based on the number of hits, with a miss resulting in a lower score.', 'The probability of hitting a score greater than 1 is the area of the circle (pi) divided by the area of the square (4).', 'Determining the new radius after hitting the bullseye involves using the Pythagorean theorem and the concept of chords in a right triangle.', 'Exploring the hit length at each point involves using coordinates and the square root of x squared plus y squared.', "The x coordinate for this person is always going to be something between negative 1 and 1, and it'll just be completely uniform within that.", 'Another way to think about the expected score: imagine this random player plays a thousand games right?', 'The chapter discusses the requirement for the hit in the second shot to be less than R1, involving the expansion and rearrangement of the coordinates to calculate the probability based on four random numbers within the range of -1 to 1.', 'The probability of the sum of squares of random numbers being less than 1 in 2, 4, and 6 dimensions The discussion revolves around the geometric interpretation of the probability of the sum of squares of random numbers being less than 1 in 2, 4, and 6 dimensions, showing the connection to the volumes of higher dimensional objects.', 'Pattern in the volumes of higher dimensional objects The pattern in the volumes of higher dimensional objects is revealed, with the volume of a 2n ball being expressed as pi to the power of n divided by n factorial.', 'The chapter explains the derivation of the Taylor series for e, emphasizing its relevance in understanding exponential growth, probability, and calculus.', 'The Taylor series for e to the x is derived as an infinite polynomial, demonstrating the relationship between factorials, pi, and the exponential function, offering insights into exponential growth, probability, and calculus.', 'The relationship between factorials, pi, and the exponential function leads to an easier interpretation of why it is its own derivative, and provides insights into exponential growth, probability, and calculus.', 'The healthier way to view the exponential function and e to the x is through this particular infinite series, providing an easier interpretation of its derivative, extending to understand e to the power of pi i, and indicating a healthier relationship with E.', 'The chapter explains how to calculate the cumulative probability of scoring in a game by summing the products of each score and its corresponding probability, and then simplifying the expression to find the final cumulative probability.', 'The method involves subtracting the probability of a score being greater than a certain number from the probability of it being greater than the preceding number to find the probability of the score being precisely equal to that number.', 'The probability that your score is at least zero is 1, as you get a point just for playing.', 'The average score from hitting within a square with uniform probability using random dart throws is approximately 2.1932, compared to the value of pi.', 'The volume of a 100-dimensional ball is around 2.368 x 10^-40, challenging conventional geometric intuition.', 'The significance of the number e in probability games is highlighted, showcasing its role in determining the expected number of samples needed before a sum exceeds a certain value.', 'The concept of adding volumes of higher dimensional balls converges to small values in higher dimensions. The discussion emphasizes the convergence of adding volumes of higher dimensional balls, illustrating that in higher dimensions, the volumes converge to small values, challenging conventional geometric intuition.', 'The chapter discusses encoding properties of numbers geometrically in six dimensions, which allows for a more universal language for communication and simplifies computations.']}