title

The power tower puzzle | Ep. 8 Lockdown live math

description

A fun puzzle stemming from repeated exponentiation.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks
Notes by Ngân Vũ:
https://twitter.com/ThuyNganVu/status/1261014161464516608?s=20
Play along on Desmos:
https://www.desmos.com/calculator/nul32eaaa9
Related videos.
Calculus series:
https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr
In particular look at:
https://youtu.be/CfW845LNObM
Numberphile on Grahm's constant:
https://youtu.be/XTeJ64KD5cg
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Video timeline (thanks to user "noonesperfect")
0:36 Question 1
1:13 Answer 1
1:29 Introduction to tetration
3:37 How exponentiation works in tetration
6:10 Python program for power tower iterations
8:40 Question 2
9:32 Python Program regarding question 2
10:37 Answer 2 and explanation
13:18 Power tower for infinite size converges or not? (Thumbnail question)
15:21 Question 3
16:28 Footage of Grant's setup arrangement problem due to construction-work back at home.
16:49 Answer 3 and explanation
17:40 Checking logic behind 2 different problems of power towers whose answer converges to the same value (Is it even possible?)
19:42 Checking same logic using Desmos graph tool i.e. Cobweb Graph (Desmos graph link in description)
28:12 Question 4
28:51 Questions from audience tweets
29:15 Knuth's Up Arrow Notation and Graham's Number (Check Numberphile video in description)
32:32 Answer 4 and explanation
37:29 Homework/Challenge Puzzle
39:20 Thumbnail question power tower logic
40:55 Audience questions from twitter
41:45 Power tower for complex numbers/Fractal set
45:19 Brainteaser
48:06 More questions from tweets
53:17 Notes for lock-down series in Grant's Tweeter
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The live question setup with stats on-screen is powered by Itempool.
https://itempool.com/
Curious about other animations?
https://www.3blue1brown.com/faq#manim
Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
------------------
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detail

{'title': 'The power tower puzzle | Ep. 8 Lockdown live math', 'heatmap': [{'end': 1970.527, 'start': 1937.517, 'weight': 1}], 'summary': 'Delves into construction noise impact and presents a logical puzzle, covers tetration, power towers, and exponential operations in python, explores exponential growth analysis, convergence of power towers with root 2, tangent point and tetration, solving equations and convergence, and power tower convergence with a range of complex mathematical concepts and practical implications.', 'chapters': [{'end': 100.804, 'segs': [{'end': 38.968, 'src': 'embed', 'start': 9.195, 'weight': 0, 'content': [{'end': 10.735, 'text': 'Presumably you can hear this.', 'start': 9.195, 'duration': 1.54}, {'end': 13.876, 'text': "There's some construction happening in the house next door.", 'start': 11.676, 'duration': 2.2}, {'end': 21.398, 'text': "And for the last hour or maybe two hours or so, it seems like this is the ground smashing portion of whatever construction they're doing.", 'start': 14.416, 'duration': 6.982}, {'end': 27.24, 'text': 'And just the whole house has been shaking, which you will be able to hear if they continue.', 'start': 21.838, 'duration': 5.402}, {'end': 28.52, 'text': "I think it's still going on now.", 'start': 27.38, 'duration': 1.14}, {'end': 30.521, 'text': "So if you're curious, that's what the sound is.", 'start': 28.6, 'duration': 1.921}, {'end': 35.044, 'text': 'Now, in one of the intro questions that you were just answering,', 'start': 31.641, 'duration': 3.403}, {'end': 38.968, 'text': "you are actually going to have your mind primed to think about what we're going to talk about today.", 'start': 35.044, 'duration': 3.924}], 'summary': 'Construction noise next door is disrupting the recording session.', 'duration': 29.773, 'max_score': 9.195, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod49195.jpg'}, {'end': 70.445, 'src': 'embed', 'start': 43.151, 'weight': 1, 'content': [{'end': 48.796, 'text': "It's asking if x is the number that most people are going to enter into this box, what is two to the x gonna be?", 'start': 43.151, 'duration': 5.645}, {'end': 54.358, 'text': "And it's just so mind-confuddling because you think about okay, if a lot of people are entering one, I should enter two,", 'start': 49.356, 'duration': 5.002}, {'end': 55.599, 'text': 'but everyone would think about that.', 'start': 54.358, 'duration': 1.241}, {'end': 58.2, 'text': 'so maybe I should enter four, but everyone thinks about that.', 'start': 55.599, 'duration': 2.601}, {'end': 61.881, 'text': 'so I should do two to the four, or maybe two to the that, or maybe two to the that.', 'start': 58.2, 'duration': 3.681}, {'end': 66.903, 'text': "And you know, if it was a room full of perfect logicians, you'd blow up to infinity.", 'start': 62.582, 'duration': 4.321}, {'end': 70.445, 'text': "but people aren't logicians and there is some objectively correct answer,", 'start': 66.903, 'duration': 3.542}], 'summary': "Dilemma of choosing x value for 2^x with people's choices, leading to confusion and no objectively correct answer.", 'duration': 27.294, 'max_score': 43.151, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod443151.jpg'}], 'start': 9.195, 'title': 'Construction noise and logical puzzles', 'summary': 'Discusses the impact of construction noise on recording and presents a mind-bending logical puzzle related to entering numbers into a box, illustrating its implications for the topic of the video.', 'chapters': [{'end': 100.804, 'start': 9.195, 'title': 'Construction noise and logical puzzles', 'summary': 'Discusses the impact of construction noise on the recording and then delves into a mind-bending logical puzzle about entering numbers into a box and the implications for the topic of the video.', 'duration': 91.609, 'highlights': ['Construction noise from neighboring house causes disturbance and shaking for about 1-2 hours, impacting the recording.', "The logical puzzle involves determining the exponential value of 'x' based on the number most people enter into a box, demonstrating the complexity and subjective nature of the problem."]}], 'duration': 91.609, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod49195.jpg', 'highlights': ['Construction noise from neighboring house causes disturbance and shaking for about 1-2 hours, impacting the recording.', "The logical puzzle involves determining the exponential value of 'x' based on the number most people enter into a box, demonstrating the complexity and subjective nature of the problem."]}, {'end': 493.397, 'segs': [{'end': 144.209, 'src': 'embed', 'start': 119.358, 'weight': 0, 'content': [{'end': 126.421, 'text': 'So this operation is called tetration, and a way you might think about it is you know, we all learn about addition as one of the first things,', 'start': 119.358, 'duration': 7.063}, {'end': 129.203, 'text': 'the way that we can add two numbers and multiplication.', 'start': 126.421, 'duration': 2.782}, {'end': 131.544, 'text': 'as we first see it, is repeated addition.', 'start': 129.203, 'duration': 2.341}, {'end': 135.505, 'text': 'a times b is a plus a plus a plus a, b different times.', 'start': 131.884, 'duration': 3.621}, {'end': 144.209, 'text': 'And if you say well, what happens if we repeat multiplication, you know taking a times a times, a, times a, b, different times.', 'start': 136.466, 'duration': 7.743}], 'summary': 'Tetration is a repeated multiplication, a times b is a to the power of b.', 'duration': 24.851, 'max_score': 119.358, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4119358.jpg'}, {'end': 233.966, 'src': 'embed', 'start': 212.55, 'weight': 1, 'content': [{'end': 221.56, 'text': "Now, this is actually ambiguous if we don't define our terms a little bit more clearly, because Exponentiation is not associative,", 'start': 212.55, 'duration': 9.01}, {'end': 225.542, 'text': 'meaning the order that we do these operations and start collapsing, it matters.', 'start': 221.56, 'duration': 3.982}, {'end': 230.364, 'text': "Because if I was going from left to right, let's say I was writing this as 2 squared,", 'start': 225.942, 'duration': 4.422}, {'end': 233.966, 'text': "and then I'm going to think of squaring that and then squaring the result,", 'start': 230.364, 'duration': 3.602}], 'summary': 'Exponentiation is not associative, the order of operations matters.', 'duration': 21.416, 'max_score': 212.55, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4212550.jpg'}, {'end': 308.391, 'src': 'embed', 'start': 259.033, 'weight': 2, 'content': [{'end': 260.233, 'text': 'But on the bottom.', 'start': 259.033, 'duration': 1.2}, {'end': 271.04, 'text': 'if I start by collapsing that top term, it becomes a four, And then, if I think of collapsing the current top term, that becomes two to the 16,', 'start': 260.233, 'duration': 10.807}, {'end': 272, 'text': "and that's a much bigger number.", 'start': 271.04, 'duration': 0.96}, {'end': 273.56, 'text': 'That is 65,536.', 'start': 272.16, 'duration': 1.4}, {'end': 283.544, 'text': 'And in general, this process of repeatedly exponentiating, going from the top to the bottom, explodes very quickly.', 'start': 273.561, 'duration': 9.983}, {'end': 287.108, 'text': 'And tetration in general refers to the top to the bottom part.', 'start': 284.304, 'duration': 2.804}, {'end': 289.41, 'text': 'So we start evaluating at the top and we work down.', 'start': 287.128, 'duration': 2.282}, {'end': 295.177, 'text': 'If you want to make that crystal clear, I think, instead of drawing it as a power tower,', 'start': 290.051, 'duration': 5.126}, {'end': 299.322, 'text': 'one thing that you could do is define the iterative process very exactly.', 'start': 295.177, 'duration': 4.145}, {'end': 306.61, 'text': "You might say we have some value, that we're going to start out at 1, and then each successive value is going to be 2,", 'start': 300.163, 'duration': 6.447}, {'end': 308.391, 'text': 'to the power of the previous thing.', 'start': 306.61, 'duration': 1.781}], 'summary': 'Tetration involves rapidly increasing numbers, with top to bottom evaluation process, resulting in 65,536 for 2 to the 16.', 'duration': 49.358, 'max_score': 259.033, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4259033.jpg'}, {'end': 493.397, 'src': 'embed', 'start': 453.372, 'weight': 3, 'content': [{'end': 458.135, 'text': 'And if we wanted, we could say, OK, think of that number as a string.', 'start': 453.372, 'duration': 4.763}, {'end': 462.638, 'text': "What's the length of that string? And it's telling us the number of digits that are in it.", 'start': 458.475, 'duration': 4.163}, {'end': 468.022, 'text': 'So the number we have there is a 19,000-digit expression, just monstrously huge.', 'start': 463.038, 'duration': 4.984}, {'end': 476.907, 'text': 'And if we were to try one more iteration of this, turning A into two, to the power of that 19,000 digit monstrosity,', 'start': 468.582, 'duration': 8.325}, {'end': 479.888, 'text': 'you would not be able to store the information required for that number.', 'start': 476.907, 'duration': 2.981}, {'end': 489.574, 'text': 'Any way that you were using matter to encode all the digits of whatever would come out If that was within the confines of something like the radius of the Earth.', 'start': 480.188, 'duration': 9.386}, {'end': 493.397, 'text': 'you would absolutely create a black hole in any attempt to store that kind of information.', 'start': 489.574, 'duration': 3.823}], 'summary': 'A 19,000-digit number would create a black hole when information is stored within the radius of the earth.', 'duration': 40.025, 'max_score': 453.372, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4453372.jpg'}], 'start': 100.804, 'title': 'Tetration, power towers, and exponential operations', 'summary': 'Covers the concept of tetration, the ambiguity of power towers, and demonstrates exponential power operations in python, reaching a number with 19,000 digits, highlighting the challenge of information storage within the radius of the earth.', 'chapters': [{'end': 353.17, 'start': 100.804, 'title': 'Understanding tetration and power tower', 'summary': 'Discusses the concept of tetration, the fourth stage in the process of applying repeated operations, and the ambiguity in evaluating power towers due to non-associativity of exponentiation, with a focus on the top-to-bottom evaluation and a defined iterative process.', 'duration': 252.366, 'highlights': ['The concept of tetration as the fourth stage in the process of applying repeated operations, extending beyond counting numbers, with examples involving multiplication and complex values. Tetration is explained as the next stage after exponentiation, extending the idea beyond counting numbers to include complex values and non-integer numbers.', 'Ambiguity in evaluating power towers due to non-associativity of exponentiation, with a detailed example illustrating the impact of the order of collapsing operations on the result. The non-associativity of exponentiation is showcased through a comprehensive example, highlighting the significant impact of the order of operations on the outcome of evaluating power towers.', 'The emphasis on top-to-bottom evaluation in tetration and the proposal of a defined iterative process to eliminate ambiguity in the order of operations. The chapter emphasizes the importance of top-to-bottom evaluation in tetration and proposes a clear iterative process to remove ambiguity in the order of operations when evaluating power towers.']}, {'end': 493.397, 'start': 355, 'title': 'Exponential power operations in python', 'summary': 'Explores the iterative process of exponential power operations in python, demonstrating how a small initial value can lead to a massively large number through reassignment, with the final iteration resulting in a number with 19,000 digits, posing the challenge of information storage within the radius of the earth.', 'duration': 138.397, 'highlights': ['The process of exponential power operations in Python demonstrates rapid growth, with an initial value of 1 leading to a 19,000-digit number after several iterations. 19,000-digit number', "The reassignment of values in the iterative process leads to a drastic increase in the magnitude of the number, ultimately posing a challenge for information storage within the confines of the Earth's radius. information storage challenge within the radius of the Earth", 'The concept of reassigning values using the double asterisk notation for powers in Python is explained, showcasing the unexpected growth and limitations of such operations. demonstration of unexpected growth and limitations']}], 'duration': 392.593, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4100804.jpg', 'highlights': ['Tetration extends beyond counting numbers, involving complex values.', 'Non-associativity of exponentiation impacts the result of evaluating power towers.', 'Importance of top-to-bottom evaluation in tetration to eliminate ambiguity.', 'Exponential power operations in Python lead to a 19,000-digit number.', 'Reassigning values in the iterative process leads to a drastic increase in magnitude.', 'Challenge of information storage within the radius of the Earth due to rapid growth.']}, {'end': 1002.802, 'segs': [{'end': 563.512, 'src': 'embed', 'start': 538.997, 'weight': 0, 'content': [{'end': 545.301, 'text': 'How many times do you have to repeat this operation before the value of the expression has more than 10 digits?', 'start': 538.997, 'duration': 6.304}, {'end': 550.184, 'text': 'So, with 2, we had to get to a tower of size 5 before that happened.', 'start': 546.261, 'duration': 3.923}, {'end': 553.806, 'text': 'It jumped from being a 5 digit number to a 19,000 digit number.', 'start': 550.464, 'duration': 3.342}, {'end': 559.029, 'text': 'So how many times would you have to do it, for a value of b equals 1.1?', 'start': 554.246, 'duration': 4.783}, {'end': 563.512, 'text': "And I'll give you a moment to answer that, while we listen to the soothing tones of construction, maybe slowing down.", 'start': 559.029, 'duration': 4.483}], 'summary': 'Repeated operation yields 19,000-digit number at size 5 tower, aiming for value of b equals 1.1.', 'duration': 24.515, 'max_score': 538.997, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4538997.jpg'}, {'end': 639.703, 'src': 'embed', 'start': 610.617, 'weight': 1, 'content': [{'end': 612.498, 'text': 'And you get classic exponential growth.', 'start': 610.617, 'duration': 1.881}, {'end': 617.101, 'text': 'So with 50 steps, it took us up to 117.', 'start': 612.879, 'duration': 4.222}, {'end': 618.242, 'text': "that's exponential growth.", 'start': 617.101, 'duration': 1.141}, {'end': 621.103, 'text': 'So you might wonder what is tetrational growth?', 'start': 618.602, 'duration': 2.501}, {'end': 623.664, 'text': "What happens if we're repeatedly exponentiating this thing?", 'start': 621.183, 'duration': 2.481}, {'end': 627.786, 'text': 'And before I answer that, let me see what you think.', 'start': 625.425, 'duration': 2.361}, {'end': 632.948, 'text': 'What do you think is gonna happen here if we repeatedly exponentiate this??', 'start': 628.046, 'duration': 4.902}, {'end': 639.703, 'text': 'okay, so the correct answer is that it actually never grows.', 'start': 635.06, 'duration': 4.643}], 'summary': 'Exponential growth with 50 steps led to 117, while tetrational growth results in no growth.', 'duration': 29.086, 'max_score': 610.617, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4610617.jpg'}, {'end': 697.94, 'src': 'embed', 'start': 671.401, 'weight': 4, 'content': [{'end': 677.825, 'text': 'but that somehow doing what feels like a much more powerful operation, repeatedly exponentiating, that actually stays confined.', 'start': 671.401, 'duration': 6.424}, {'end': 681.628, 'text': 'And we can see this bear out in practice.', 'start': 679.366, 'duration': 2.262}, {'end': 684.069, 'text': 'if we hop back over to our Python,', 'start': 681.628, 'duration': 2.441}, {'end': 692.306, 'text': "where now What I'm doing with each iteration and I guess I should set a to be 1 again I'm going to repeatedly turn it into 1 to the power of itself,", 'start': 684.069, 'duration': 8.237}, {'end': 697.94, 'text': 'And what we get is a little bit of initial growth, but it quickly slows down.', 'start': 693.277, 'duration': 4.663}], 'summary': 'Repeated exponentiation in python shows initial growth but quickly slows down.', 'duration': 26.539, 'max_score': 671.401, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4671401.jpg'}, {'end': 830.021, 'src': 'embed', 'start': 801.788, 'weight': 5, 'content': [{'end': 806.789, 'text': 'x, and I do not a power tower of some finite size but of an infinite size.', 'start': 801.788, 'duration': 5.001}, {'end': 812.671, 'text': "I'm going to keep going forever in the same way that evidently I can do with a base of 1.1.", 'start': 806.829, 'duration': 5.842}, {'end': 818.474, 'text': 'Can I find a value of x where this converges, for example, to 4? This is the question posed in the thumbnail of the video.', 'start': 812.671, 'duration': 5.803}, {'end': 825.518, 'text': 'And what we just saw is that we could find a value that converges to 1.111782, on and on.', 'start': 819.135, 'duration': 6.383}, {'end': 830.021, 'text': "That's evidently a value you can converge to, and the solution would be 1.1.", 'start': 825.839, 'duration': 4.182}], 'summary': 'Exploring convergence of an infinite power tower with base 1.1 to 1.111782', 'duration': 28.233, 'max_score': 801.788, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4801788.jpg'}], 'start': 493.837, 'title': 'Exponential growth analysis', 'summary': 'Explores exponential growth with a base of 1.1 and a base of 2, determining the number of iterations required for the expression to exceed 10 digits and reaching 19,000 digits, while also analyzing tetrational growth and its implications.', 'chapters': [{'end': 588.382, 'start': 493.837, 'title': 'Power tower and exponential growth', 'summary': 'Explores the exponential growth of a power tower with a base of 1.1, determining the number of iterations required for the expression to exceed 10 digits, building upon the example with a base of 2, which required a tower of size 5 before reaching 19,000 digits.', 'duration': 94.545, 'highlights': ['The exponential growth of a power tower with a base of 1.1 is explored to determine the number of iterations required for the expression to exceed 10 digits, following the example with a base of 2, which required a tower of size 5 before reaching 19,000 digits.', 'The challenge is to predict the number of iterations needed for a power tower with a base of 1.1 to exceed 10 digits, akin to the example with a base of 2 which jumped from being a 5 digit number to a 19,000 digit number after 5 iterations.', 'The process is illustrated using Python, where the value of the expression is calculated for various iterations, with the objective of determining the number of iterations required for the expression to exceed 10 digits.']}, {'end': 1002.802, 'start': 588.382, 'title': 'Exponential and tetrational growth analysis', 'summary': 'Explores the concept of exponential and tetrational growth, demonstrating how repeatedly multiplying a value by 1.1 results in exponential growth, while repeatedly exponentiating the value leads to a confined result, with detailed exploration of the solutions and implications.', 'duration': 414.42, 'highlights': ['The chapter explores the concept of exponential and tetrational growth, demonstrating how repeatedly multiplying a value by 1.1 results in exponential growth. The speaker illustrates how repeated multiplication by 1.1 results in classic exponential growth, with the example showing a value growing by 10% at each step and reaching 117 after 50 steps.', 'Repeating exponentiation of a value results in a confined result, with detailed exploration of the solutions and implications. The speaker discusses how repeatedly exponentiating a value with 1.1 as the base leads to a confined result, unlike the exponential growth, and explores the solutions and implications, highlighting the difference from the scenario with 2 as the base.', 'The speaker presents the intriguing question of finding a value where a power tower converges to a specific number, providing an example of finding a value that converges to 1.111782. The speaker poses the question of finding a value where a power tower converges to a specific number, and demonstrates finding a value that converges to 1.111782, providing insight into solving such problems and showcasing the surprising convergence of certain values.']}], 'duration': 508.965, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod4493837.jpg', 'highlights': ['The exponential growth of a power tower with a base of 1.1 is explored to determine the number of iterations required for the expression to exceed 10 digits, akin to the example with a base of 2 which required a tower of size 5 before reaching 19,000 digits.', 'The chapter explores the concept of exponential and tetrational growth, demonstrating how repeatedly multiplying a value by 1.1 results in exponential growth, with the example showing a value growing by 10% at each step and reaching 117 after 50 steps.', 'The challenge is to predict the number of iterations needed for a power tower with a base of 1.1 to exceed 10 digits, similar to the example with a base of 2 which jumped from being a 5 digit number to a 19,000 digit number after 5 iterations.', 'The process is illustrated using Python, where the value of the expression is calculated for various iterations, with the objective of determining the number of iterations required for the expression to exceed 10 digits.', 'Repeating exponentiation of a value results in a confined result, with detailed exploration of the solutions and implications, highlighting the difference from the scenario with 2 as the base.', 'The speaker presents the intriguing question of finding a value where a power tower converges to a specific number, providing an example of finding a value that converges to 1.111782.']}, {'end': 1654.729, 'segs': [{'end': 1057.571, 'src': 'embed', 'start': 1020.177, 'weight': 0, 'content': [{'end': 1027.143, 'text': "where we have this infinite power tower and we're assuming that it equals 2 and we recognize the self-similarity, They're like ah, yes,", 'start': 1020.177, 'duration': 6.966}, {'end': 1029.165, 'text': 'the power tower copy of it in itself.', 'start': 1027.143, 'duration': 2.022}, {'end': 1031.887, 'text': 'So that should mean x squared equals 2.', 'start': 1029.746, 'duration': 2.141}, {'end': 1033.69, 'text': 'That means x equals the square root of 2.', 'start': 1031.887, 'duration': 1.803}, {'end': 1035.792, 'text': 'Well, hang on a second.', 'start': 1033.69, 'duration': 2.102}, {'end': 1036.493, 'text': "This can't be right.", 'start': 1035.813, 'duration': 0.68}, {'end': 1044.041, 'text': 'Because on the one hand, this seems to be suggesting that an infinite power tower converges to 2 when the base is root 2.', 'start': 1036.914, 'duration': 7.127}, {'end': 1046.904, 'text': 'But on the other hand, it converges to 4 when the base is root 2.', 'start': 1044.041, 'duration': 2.863}, {'end': 1047.824, 'text': "It can't be both.", 'start': 1046.904, 'duration': 0.92}, {'end': 1051.187, 'text': "We've got a very deterministic process up here for what it converges.", 'start': 1048.204, 'duration': 2.983}, {'end': 1053.328, 'text': "So it's got to be just one of them, if any.", 'start': 1051.467, 'duration': 1.861}, {'end': 1055.549, 'text': 'Maybe the entire logic of the situation is false.', 'start': 1053.468, 'duration': 2.081}, {'end': 1057.571, 'text': "So let's get empirical.", 'start': 1056.27, 'duration': 1.301}], 'summary': 'Infinite power tower seemingly converges to conflicting values, prompting empirical investigation.', 'duration': 37.394, 'max_score': 1020.177, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41020177.jpg'}, {'end': 1148.474, 'src': 'embed', 'start': 1120.53, 'weight': 3, 'content': [{'end': 1122.872, 'text': "So that's going to be just some simple numerical error.", 'start': 1120.53, 'duration': 2.342}, {'end': 1129.296, 'text': 'I guess, however, the math square root library is implementing square roots.', 'start': 1124.093, 'duration': 5.203}, {'end': 1131.698, 'text': "there's always going to be a little bit of numerical error with floating points.", 'start': 1129.296, 'duration': 2.402}, {'end': 1134.179, 'text': "So when we square it, we don't exactly get 2 back.", 'start': 1131.998, 'duration': 2.181}, {'end': 1134.88, 'text': "That's fine.", 'start': 1134.48, 'duration': 0.4}, {'end': 1135.56, 'text': 'Not a problem.', 'start': 1135.04, 'duration': 0.52}, {'end': 1145.447, 'text': 'But it does seem to suggest that the correct answer to our question of what happens when we have a power tower with root 2 is that it equals 2.', 'start': 1136.081, 'duration': 9.366}, {'end': 1148.474, 'text': 'or that the sequence of numbers that this represents approaches to.', 'start': 1145.447, 'duration': 3.027}], 'summary': 'Math square root library has numerical errors in the power tower with root 2, but approaches 2.', 'duration': 27.944, 'max_score': 1120.53, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41120530.jpg'}, {'end': 1184.791, 'src': 'embed', 'start': 1159.071, 'weight': 4, 'content': [{'end': 1163.835, 'text': 'First is to represent this thing a little bit more visually, a little bit graphically,', 'start': 1159.071, 'duration': 4.764}, {'end': 1167.878, 'text': 'so we can try to understand what is going on with this iterative process.', 'start': 1163.835, 'duration': 4.043}, {'end': 1173.663, 'text': 'And then from there, understand which values will converge and which values will blow up.', 'start': 1168.339, 'duration': 5.324}, {'end': 1180.468, 'text': "And by answering that, we can get back in the direction of what's wrong with the logic associated with having this thing approach for.", 'start': 1174.083, 'duration': 6.385}, {'end': 1184.791, 'text': "So for that, let's take a look over at our good friend Desmos.", 'start': 1181.449, 'duration': 3.342}], 'summary': 'Visualize iterative process to identify converging and diverging values.', 'duration': 25.72, 'max_score': 1159.071, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41159071.jpg'}], 'start': 1002.802, 'title': 'Convergence of power towers with root 2', 'summary': 'Explores the convergence of a power tower with root 2, affirming its behavior through empirical testing and programming. the first 20 iterations slow down as it approaches 2, and further iterations continue to slow down significantly. it also discusses numerical errors associated with floating points and the graphical representation of the iterative process using desmos.', 'chapters': [{'end': 1099.909, 'start': 1002.802, 'title': 'Convergence of infinite power tower', 'summary': 'Explores the convergence of an infinite power tower to the square root of 2, affirming its behavior through empirical testing and programming, where the first 20 iterations slow down as it approaches 2 and further iterations continue to slow down significantly.', 'duration': 97.107, 'highlights': ['The infinite power tower converges to the square root of 2. The chapter reveals the convergence of an infinite power tower to the square root of 2, highlighting the surprising result derived from the logic applied.', 'Empirical testing through programming shows the infinite power tower slowing down as it approaches 2. The empirical testing through programming demonstrates the slowing down of the infinite power tower as it approaches 2, providing quantifiable evidence of its behavior.']}, {'end': 1654.729, 'start': 1100.35, 'title': 'Convergence of power towers with root 2', 'summary': 'Discusses the convergence of a power tower with root 2, showing how the value approaches 2, while highlighting the numerical errors associated with floating points and the graphical representation of the iterative process using desmos.', 'duration': 554.379, 'highlights': ['The iterative process of a power tower with root 2 approaches the value of 2, indicating the convergence of the sequence of numbers. (Relevance: 5)', 'Numerical errors occur in floating point calculations when using the math square root library to implement square roots. (Relevance: 4)', 'Graphical representation of the iterative process using Desmos demonstrates how the values converge and which values blow up, providing a visual understanding of the iterative process. (Relevance: 3)', 'The convergence of the iterative process is influenced by the slope of the graph of the function, where a slope less than 1 ensures stability. (Relevance: 2)', 'When the initial value of the iterative process is changed, it can lead to either convergence or blowing up of the values, highlighting the sensitivity of the process to the initial value. (Relevance: 1)']}], 'duration': 651.927, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41002802.jpg', 'highlights': ['The infinite power tower converges to the square root of 2.', 'Empirical testing through programming shows the infinite power tower slowing down as it approaches 2.', 'The iterative process of a power tower with root 2 approaches the value of 2, indicating the convergence of the sequence of numbers.', 'Numerical errors occur in floating point calculations when using the math square root library to implement square roots.', 'Graphical representation of the iterative process using Desmos demonstrates how the values converge and which values blow up, providing a visual understanding of the iterative process.']}, {'end': 1950.87, 'segs': [{'end': 1707.137, 'src': 'embed', 'start': 1678.118, 'weight': 0, 'content': [{'end': 1682.18, 'text': "And rather than me setting up the equations that we need to solve for that, I'm going to have you do the same.", 'start': 1678.118, 'duration': 4.062}, {'end': 1691.544, 'text': "I'm going to have you take a look at this condition of looking for a value of b Actually, let's just read what the question asks us more specifically.", 'start': 1682.541, 'duration': 9.003}, {'end': 1701.051, 'text': 'We want a value of b such that the graph of y equals b to the x sits tangent to the graph y equals x.', 'start': 1693.425, 'duration': 7.626}, {'end': 1707.137, 'text': 'Which of the following represents the pair of equations that we need to solve? Okay, so take a moment to think about this.', 'start': 1701.051, 'duration': 6.086}], 'summary': 'Solve for the value of b where y=b^x is tangent to y=x.', 'duration': 29.019, 'max_score': 1678.118, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41678118.jpg'}, {'end': 1767.344, 'src': 'embed', 'start': 1740.661, 'weight': 3, 'content': [{'end': 1746.687, 'text': "Yeah, there's this whole notation that talks about the idea of repeating these processes.", 'start': 1740.661, 'duration': 6.026}, {'end': 1750.991, 'text': "It's called Knuth-Arrow notation, and you can kind of have as many as you want.", 'start': 1747.588, 'duration': 3.403}, {'end': 1761.039, 'text': "so the way this works is that if I write something like a arrow b, that's the same thing as a to the power b,", 'start': 1751.491, 'duration': 9.548}, {'end': 1767.344, 'text': 'but then a with two arrows is the repetition of that process.', 'start': 1761.039, 'duration': 6.305}], 'summary': 'Knuth-arrow notation allows for repeating processes, with as many arrows as desired.', 'duration': 26.683, 'max_score': 1740.661, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41740661.jpg'}, {'end': 1853.485, 'src': 'embed', 'start': 1824.027, 'weight': 1, 'content': [{'end': 1828.089, 'text': "awesome. what's the deal with that second solution?", 'start': 1824.027, 'duration': 4.062}, {'end': 1830.791, 'text': 'we can take a look exactly in the graph that we have.', 'start': 1828.089, 'duration': 2.702}, {'end': 1842.678, 'text': "so if we take b and we make it 1.1, okay, It's what I was talking about how, at the place where it intersects with a slope greater than 1, yeah,", 'start': 1830.791, 'duration': 11.887}, {'end': 1847.021, 'text': "it's a fixed point for our iterative process, but it's not a stable fixed point.", 'start': 1842.678, 'duration': 4.343}, {'end': 1853.485, 'text': "So up here is that 38 value that you were just referencing, and because it intersects at a slope that's bigger than 1,", 'start': 1847.461, 'duration': 6.024}], 'summary': 'Analyzing a graph for a fixed point with slope greater than 1, intersecting at 38 value.', 'duration': 29.458, 'max_score': 1824.027, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41824027.jpg'}, {'end': 1963.201, 'src': 'embed', 'start': 1931.129, 'weight': 2, 'content': [{'end': 1934.413, 'text': 'you might do that involve more practical iterative processes.', 'start': 1931.129, 'duration': 3.284}, {'end': 1937.176, 'text': 'And iterative processes absolutely do come up all throughout science.', 'start': 1934.533, 'duration': 2.643}, {'end': 1939.299, 'text': 'I mean, chaos theory is like the big one there.', 'start': 1937.517, 'duration': 1.782}, {'end': 1947.587, 'text': "so, with all of that as the uh answering people's question side, and i'll take more through twitter as we go and probably do more of that at the end.", 'start': 1940.72, 'duration': 6.867}, {'end': 1950.87, 'text': "let's see how you've done on our problem.", 'start': 1947.587, 'duration': 3.283}, {'end': 1957.016, 'text': "so we're looking for a system of equations here and the correct answer which 1777 of you got.", 'start': 1950.87, 'duration': 6.146}, {'end': 1963.201, 'text': "i think that's not quite french revolution territory, but american revolution territory of you.", 'start': 1957.016, 'duration': 6.185}], 'summary': 'Iterative processes in science, chaos theory, and solving system of equations were discussed, with 1777 correct answers received.', 'duration': 32.072, 'max_score': 1931.129, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41931129.jpg'}], 'start': 1654.969, 'title': 'Tangent point and tetration', 'summary': 'Covers finding the tangent point of exponential and linear functions by solving a pair of equations, and explores the mind-warping knuth-arrow notation, practical implications of tetration and its fixed points with an example of 1.1 to the power x yielding two real solutions, one of which is a stable fixed point, and its practical use cases in problem-solving tactics with relevance to chaos theory.', 'chapters': [{'end': 1740.641, 'start': 1654.969, 'title': 'Finding tangent point of exponential and linear functions', 'summary': 'Delves into finding the value of b such that the graph of y equals b to the x is tangent to the graph y equals x, involving the pair of equations that need to be solved for this condition.', 'duration': 85.672, 'highlights': ['The chapter discusses finding the value of b such that the graph of y equals b to the x is tangent to the graph y equals x. It involves solving a pair of equations to determine the value of b, which represents the point where the two graphs are tangent.', 'The speaker emphasizes the need to identify the pair of equations that need to be solved for the given condition. The audience is prompted to think about the pair of equations that need to be solved to find the value of b, which represents the point of tangency between the two graphs.', 'The idea of repeated titration and pentation is mentioned, sparking curiosity about its usefulness in this context. The concept of repeated titration and pentation is briefly pondered upon, raising the question of its relevance and practicality in the given mathematical context.']}, {'end': 1950.87, 'start': 1740.661, 'title': 'Tetration and knuth-arrow notation', 'summary': 'Discusses the mind-warping knuth-arrow notation, including the concept of repeating processes through arrows, and delves into the practical implications of tetration and its fixed points, with an example of 1.1 to the power x yielding two real solutions, one of which is a stable fixed point, and explores the practical use cases of tetration in problem-solving tactics with relevance to chaos theory.', 'duration': 210.209, 'highlights': ['The chapter delves into the concept of repeating processes through Knuth-Arrow notation, emphasizing the mind-warping nature of the notation for even numbers like two and three. Knuth-Arrow notation, mind-warping nature of notation', 'Exploration of the practical implications of tetration, with an example of 1.1 to the power x yielding two real solutions, one of which is a stable fixed point. 1.1 to the power x yielding two real solutions, stable fixed point', 'Discussion on the practical use cases of tetration in problem-solving tactics with relevance to chaos theory, emphasizing the relevance of iterative processes in science. practical use cases of tetration, problem-solving tactics, relevance to chaos theory']}], 'duration': 295.901, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41654969.jpg', 'highlights': ['The chapter discusses finding the value of b such that the graph of y equals b to the x is tangent to the graph y equals x. It involves solving a pair of equations to determine the value of b, which represents the point where the two graphs are tangent.', 'Exploration of the practical implications of tetration, with an example of 1.1 to the power x yielding two real solutions, one of which is a stable fixed point. 1.1 to the power x yielding two real solutions, stable fixed point', 'Discussion on the practical use cases of tetration in problem-solving tactics with relevance to chaos theory, emphasizing the relevance of iterative processes in science. practical use cases of tetration, problem-solving tactics, relevance to chaos theory', 'The chapter delves into the concept of repeating processes through Knuth-Arrow notation, emphasizing the mind-warping nature of the notation for even numbers like two and three. Knuth-Arrow notation, mind-warping nature of notation', 'The speaker emphasizes the need to identify the pair of equations that need to be solved for the given condition. The audience is prompted to think about the pair of equations that need to be solved to find the value of b, which represents the point of tangency between the two graphs.']}, {'end': 2289.953, 'segs': [{'end': 1997.482, 'src': 'embed', 'start': 1971.668, 'weight': 3, 'content': [{'end': 1978.213, 'text': 'And the derivative of b to the x is itself, but scaled by the natural log of b, and we want that slope to equal 1,', 'start': 1971.668, 'duration': 6.545}, {'end': 1980.856, 'text': "because it's got to be the same as the slope of the graph we're just looking at.", 'start': 1978.213, 'duration': 2.643}, {'end': 1989.579, 'text': "And if ever you don't remember what the derivatives of your exponential functions are, if you do remember that e to the x is its own derivative,", 'start': 1981.696, 'duration': 7.883}, {'end': 1997.482, 'text': "which, if there's any one thing you remember in calculus with respect to e to the x, it should be that e to the x is its own derivative.", 'start': 1989.579, 'duration': 7.903}], 'summary': 'The derivative of b to the x is b^x scaled by natural log of b, aiming for a slope of 1.', 'duration': 25.814, 'max_score': 1971.668, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41971668.jpg'}, {'end': 2047.989, 'src': 'embed', 'start': 2011.651, 'weight': 2, 'content': [{'end': 2015.932, 'text': 'So then, if you wanna take the derivative of this, take its derivative.', 'start': 2011.651, 'duration': 4.281}, {'end': 2021.716, 'text': 'On the one hand, it should be the derivative of this, which, by the chain rule,', 'start': 2017.273, 'duration': 4.443}, {'end': 2030.521, 'text': 'is that constant sitting in the exponent natural log of b times itself e to the ln, b times x.', 'start': 2021.716, 'duration': 8.805}, {'end': 2037.085, 'text': "So what we're looking at is, okay, it should be itself, but scaled by something, and that something was the natural log of b.", 'start': 2030.521, 'duration': 6.564}, {'end': 2047.989, 'text': 'Now, for our puzzle of understanding when is it that our graph is going to nicely lie tangent to y equals x.', 'start': 2037.085, 'duration': 10.904}], 'summary': 'Derivative involves natural log of b, to find tangent to y equals x', 'duration': 36.338, 'max_score': 2011.651, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42011651.jpg'}, {'end': 2222.114, 'src': 'embed', 'start': 2142.157, 'weight': 0, 'content': [{'end': 2147.359, 'text': "that's an unlisted link and just say like as soon as 100 of you hop on here, I'm deleting the tweet and we're just going to do a dry run.", 'start': 2142.157, 'duration': 5.202}, {'end': 2153.801, 'text': 'And when I was solving this, for some reason, I was just confuddled for like 10 minutes where I got to this point.', 'start': 2148.019, 'duration': 5.782}, {'end': 2158.543, 'text': "I'm like, that can't be right because we're looking for a value that's between 1.44 and 1.45.", 'start': 2153.841, 'duration': 4.702}, {'end': 2159.683, 'text': "So x can't be e.", 'start': 2158.543, 'duration': 1.14}, {'end': 2160.324, 'text': "That's way too big.", 'start': 2159.683, 'duration': 0.641}, {'end': 2161.844, 'text': "That's like 2.718.", 'start': 2160.364, 'duration': 1.48}, {'end': 2164.826, 'text': 'So I went back and I was trying to think through like what have I done wrong?', 'start': 2161.844, 'duration': 2.982}, {'end': 2166.006, 'text': 'What on earth has gone wrong here?', 'start': 2164.866, 'duration': 1.14}, {'end': 2169.008, 'text': 'It was mildly embarrassing because it took so long.', 'start': 2166.026, 'duration': 2.982}, {'end': 2172.61, 'text': 'And then I ultimately realized, no, you idiot.', 'start': 2169.628, 'duration': 2.982}, {'end': 2180.595, 'text': "It's not that we're looking for a value of x that's between 1.44 and 1.45.", 'start': 2173.23, 'duration': 7.365}, {'end': 2184.837, 'text': 'That was the condition for b, for the base of our exponential, when we were playing around over here.', 'start': 2180.595, 'duration': 4.242}, {'end': 2187.937, 'text': 'x is just wherever they intersect.', 'start': 2186.517, 'duration': 1.42}, {'end': 2189.998, 'text': "so it's fine for that to be around e.", 'start': 2187.937, 'duration': 2.061}, {'end': 2191.438, 'text': 'in fact that looks consistent.', 'start': 2189.998, 'duration': 1.44}, {'end': 2196.46, 'text': 'that around where this point of tangency is, is e, which, incidentally, is also the output.', 'start': 2191.438, 'duration': 5.022}, {'end': 2199.061, 'text': 'because this is happening with the line y equals x.', 'start': 2196.46, 'duration': 2.601}, {'end': 2206.403, 'text': 'so the point of tangency has coordinates e, comma, e, evidently, and what does that mean for the value of b itself?', 'start': 2199.061, 'duration': 7.342}, {'end': 2206.843, 'text': 'once we solve?', 'start': 2206.403, 'duration': 0.44}, {'end': 2212.306, 'text': 'Well, we have an exact expression for x, we have an expression for b in terms of x.', 'start': 2207.563, 'duration': 4.743}, {'end': 2222.114, 'text': 'so this would seem to imply that b is e to the power of 1 divided by e, which is such a delightfully bizarre answer e to the power of 1 divided by e.', 'start': 2212.306, 'duration': 9.808}], 'summary': 'After a realization, the value of b is e^(1/e), consistent with the point of tangency at (e, e).', 'duration': 79.957, 'max_score': 2142.157, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42142157.jpg'}, {'end': 2279.143, 'src': 'embed', 'start': 2249.501, 'weight': 6, 'content': [{'end': 2257.125, 'text': 'As a challenge puzzle for you, if you want a little bit of homework today, I want you to find a lower bound on where this converges.', 'start': 2249.501, 'duration': 7.624}, {'end': 2259.046, 'text': 'So we found the highest value.', 'start': 2257.545, 'duration': 1.501}, {'end': 2265.11, 'text': "that'll make this converge, but we can also start playing with values of b that are less than one, and if we do that,", 'start': 2259.046, 'duration': 6.064}, {'end': 2267.913, 'text': "I'll just kind of get rid of our cobweb here.", 'start': 2266.311, 'duration': 1.602}, {'end': 2270.575, 'text': 'If we do that there will be some value.', 'start': 2268.813, 'duration': 1.762}, {'end': 2279.143, 'text': "I guess I should keep the cobwebbing where it's no longer going to zero in on some exact value and it instead becomes an unstable point.", 'start': 2270.575, 'duration': 8.568}], 'summary': 'Find lower bound for convergence, exploring values of b less than one.', 'duration': 29.642, 'max_score': 2249.501, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42249501.jpg'}], 'start': 1950.87, 'title': 'Solving equations and convergence', 'summary': 'Discusses solving systems of equations involving exponential functions resulting in x=e, a dry run of solving an equation, and explores the convergence of exponential functions to find the value of b as e to the power of 1 divided by e, between 1.44 and 1.45.', 'chapters': [{'end': 2142.157, 'start': 1950.87, 'title': 'System of equations and derivatives', 'summary': 'Discusses solving a system of equations involving exponential functions, finding the value of x at the point of intersection, and using derivatives to determine the slope, resulting in x=e.', 'duration': 191.287, 'highlights': ['The derivative of b to the x is itself, scaled by the natural log of b, and the slope needs to equal 1 at the point of intersection.', 'Expressing b as e to the natural log of b, then taking the derivative using the chain rule yields the constant ln(b) times e^(ln(b)x).', 'Solving the system of equations leads to the value of x at the point of intersection being x=e.']}, {'end': 2180.595, 'start': 2142.157, 'title': 'Solving equation dry run', 'summary': 'Describes a dry run of solving an equation, encountering confusion but ultimately realizing the correct approach, with a focus on finding a value between 1.44 and 1.45.', 'duration': 38.438, 'highlights': ['Realizing the correct approach after initial confusion, highlighting the process of identifying the mistake and correcting it.', 'Encountering confusion and spending 10 minutes in confuddled state, emphasizing the initial challenge faced during the dry run.', 'Engaging in a dry run to solve an equation with the aim of finding a value between 1.44 and 1.45, providing the specific focus of the task.', 'Setting up an unlisted link for a dry run with an audience target of 100, indicating the preparation for the exercise.']}, {'end': 2289.953, 'start': 2180.595, 'title': 'Convergence of exponential functions', 'summary': 'Explores the convergence of exponential functions and finds the value of b as e to the power of 1 divided by e, which is between 1.44 and 1.45, indicating the point when the function goes from converging to exploding.', 'duration': 109.358, 'highlights': ['The point of tangency has coordinates e, e, and the value of b is e to the power of 1 divided by e, between 1.44 and 1.45, indicating the convergence of the function.', 'The challenge is to find a lower bound on where the function converges, which involves exploring values of b that are less than one.']}], 'duration': 339.083, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod41950870.jpg', 'highlights': ['Solving the system of equations leads to the value of x at the point of intersection being x=e.', 'The point of tangency has coordinates e, e, and the value of b is e to the power of 1 divided by e, between 1.44 and 1.45, indicating the convergence of the function.', 'Expressing b as e to the natural log of b, then taking the derivative using the chain rule yields the constant ln(b) times e^(ln(b)x).', 'The derivative of b to the x is itself, scaled by the natural log of b, and the slope needs to equal 1 at the point of intersection.', 'Realizing the correct approach after initial confusion, highlighting the process of identifying the mistake and correcting it.', 'Engaging in a dry run to solve an equation with the aim of finding a value between 1.44 and 1.45, providing the specific focus of the task.', 'The challenge is to find a lower bound on where the function converges, which involves exploring values of b that are less than one.', 'Encountering confusion and spending 10 minutes in confuddled state, emphasizing the initial challenge faced during the dry run.', 'Setting up an unlisted link for a dry run with an audience target of 100, indicating the preparation for the exercise.']}, {'end': 3222.647, 'segs': [{'end': 2333.263, 'src': 'embed', 'start': 2307.764, 'weight': 0, 'content': [{'end': 2313.467, 'text': "And that'll give you kind of a range of convergence, for what values could we have as the base in our tetration,", 'start': 2307.764, 'duration': 5.703}, {'end': 2315.668, 'text': 'such that the power tower actually converges to something?', 'start': 2313.467, 'duration': 2.201}, {'end': 2323.855, 'text': "And one thing that's noteworthy here, is that it goes from converging to a value of e to just blowing up to infinity.", 'start': 2316.449, 'duration': 7.406}, {'end': 2330.021, 'text': 'It never converges to values between e and infinity, which maybe runs against intuition,', 'start': 2323.975, 'duration': 6.046}, {'end': 2333.263, 'text': 'because you would think that it somehow smoothly blows up that rather than going.', 'start': 2330.021, 'duration': 3.242}], 'summary': 'Exploring tetration convergence with base values and noting it diverges to infinity instead of converging to values between e and infinity.', 'duration': 25.499, 'max_score': 2307.764, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42307764.jpg'}, {'end': 2650.623, 'src': 'embed', 'start': 2624.98, 'weight': 1, 'content': [{'end': 2632.465, 'text': 'and you get this totally intricate pattern very analogous to the Mandelbrot set, which is also defined in terms of a certain repeated operation.', 'start': 2624.98, 'duration': 7.485}, {'end': 2640.719, 'text': "And what this indicates is there's actually a lot more intricacy than you might expect associated with these power towers and this tetration operation.", 'start': 2633.108, 'duration': 7.611}, {'end': 2646.141, 'text': 'And you get this a lot where repeated applications of something can yield chaos,', 'start': 2641.459, 'duration': 4.682}, {'end': 2650.623, 'text': 'and that chaos is often pictorially reflected in the fact that a fractal emerges.', 'start': 2646.141, 'duration': 4.482}], 'summary': 'Repeated operations yield intricate fractal patterns, demonstrating chaos and complexity.', 'duration': 25.643, 'max_score': 2624.98, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42624980.jpg'}, {'end': 2687.701, 'src': 'embed', 'start': 2664.428, 'weight': 2, 'content': [{'end': 2673.674, 'text': "what no one in the world actually knows? is if you can ever take a power tower with pi's, so pi to the power, pi to the power, pi to the power.", 'start': 2664.428, 'duration': 9.246}, {'end': 2676.74, 'text': 'pi. you do that at any point?', 'start': 2673.674, 'duration': 3.066}, {'end': 2678.324, 'text': 'is this ever equal to an integer??', 'start': 2676.74, 'duration': 1.584}, {'end': 2680.375, 'text': 'Almost certainly not.', 'start': 2679.454, 'duration': 0.921}, {'end': 2685.139, 'text': "it would seem highly unlikely, because it's an irrational number and if you're thinking of it kind of probabilistically.", 'start': 2680.375, 'duration': 4.764}, {'end': 2687.701, 'text': "But it's not probabilistic, it's a deterministic process.", 'start': 2685.359, 'duration': 2.342}], 'summary': 'The probability of a power tower with pi resulting in an integer is highly unlikely due to its irrational nature and deterministic process.', 'duration': 23.273, 'max_score': 2664.428, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42664428.jpg'}, {'end': 2805.986, 'src': 'embed', 'start': 2777.173, 'weight': 3, 'content': [{'end': 2779.554, 'text': 'six magically drop into the next cup beside it.', 'start': 2777.173, 'duration': 2.381}, {'end': 2782.635, 'text': 'And if that was the only ability that you had,', 'start': 2780.294, 'duration': 2.341}, {'end': 2788.658, 'text': 'kind of a warm-up teaser is to understand that the maximum number of coins you can have ends up being 63..', 'start': 2782.635, 'duration': 6.023}, {'end': 2796.521, 'text': 'If you just sort of keep doing this process of taking out coins, doubling it, but shifted one over, you can get up to 63.', 'start': 2788.658, 'duration': 7.863}, {'end': 2799.523, 'text': 'Now, if I introduce the second magical operation,', 'start': 2796.521, 'duration': 3.002}, {'end': 2805.986, 'text': 'which is that I can eliminate one of the coins in a cup and instead of magically dropping two into the next one,', 'start': 2799.523, 'duration': 6.463}], 'summary': 'Magical coin operations can double and shift, with max 63 coins.', 'duration': 28.813, 'max_score': 2777.173, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42777173.jpg'}, {'end': 2929.635, 'src': 'embed', 'start': 2899.164, 'weight': 4, 'content': [{'end': 2901.346, 'text': 'Awesome, yeah, yeah, solution to x squared equals two.', 'start': 2899.164, 'duration': 2.182}, {'end': 2904.789, 'text': "Can I explain why? Ah, yeah, that's absolutely outstanding.", 'start': 2901.386, 'duration': 3.403}, {'end': 2907.531, 'text': 'So if we think about this graphically again,', 'start': 2905.51, 'duration': 2.021}, {'end': 2915.278, 'text': "where that helps us be a little bit more concrete when it comes to what's going on with the iterative process, when I said b equal to around 1.41,", 'start': 2907.531, 'duration': 7.747}, {'end': 2923.213, 'text': "okay, it seems like nothing's really happening around the negative square root of two.", 'start': 2915.278, 'duration': 7.935}, {'end': 2926.374, 'text': "So let's try to understand why that's happening.", 'start': 2924.213, 'duration': 2.161}, {'end': 2929.635, 'text': "There's not even necessarily an intersection at that point.", 'start': 2926.394, 'duration': 3.241}], 'summary': 'Iterative process for x^2=2 converges around b=1.41 with no intersection at the negative square root of two.', 'duration': 30.471, 'max_score': 2899.164, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42899164.jpg'}, {'end': 3215.078, 'src': 'embed', 'start': 3188.262, 'weight': 5, 'content': [{'end': 3193.744, 'text': "yeah, i'll think on that excellent question and i think that's a great place to end it.", 'start': 3188.262, 'duration': 5.482}, {'end': 3198.765, 'text': "also, i couldn't help but notice the the twitter handle, for who was asking that?", 'start': 3193.744, 'duration': 5.021}, {'end': 3205.468, 'text': "because this is someone who's created these super beautiful notes for all of the lockdown series so far,", 'start': 3198.765, 'duration': 6.703}, {'end': 3209.872, 'text': "which I've been throwing in the video descriptions just because I'm like, yeah, this is.", 'start': 3205.468, 'duration': 4.404}, {'end': 3211.814, 'text': 'Boy, do I wish my handwriting looked anything like that.', 'start': 3209.872, 'duration': 1.942}, {'end': 3215.078, 'text': "So if anyone wants to check them out, she's done a beautiful job.", 'start': 3211.854, 'duration': 3.224}], 'summary': "Acknowledges a viewer's exceptional note-taking skills and encourages others to check it out.", 'duration': 26.816, 'max_score': 3188.262, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod43188262.jpg'}], 'start': 2290.914, 'title': 'Power tower convergence', 'summary': 'Delves into the behavior of power tower convergences, covering the range of convergence for the base in tetration, non-convergence to values between e and infinity, existence of convergence for specific values, the intricacy of the fractal set of complex numbers, and the uncertainty surrounding power towers with pi. it also discusses the iterative process of solving x^x^x = 2 and invites deeper thoughts and appreciation for viewer contributions.', 'chapters': [{'end': 2664.428, 'start': 2290.914, 'title': 'Puzzle of power tower convergence', 'summary': 'Explores the behavior of power tower convergences, highlighting the range of convergence for the base in tetration, the non-convergence to values between e and infinity, the existence of convergence for specific values, and the intricacy of the fractal set of complex numbers for which the power tower converges.', 'duration': 373.514, 'highlights': ['The range of convergence for the base in tetration is from converging to a value of e to blowing up to infinity, with non-convergence to values between e and infinity.', 'The process of determining the existence of convergence for specific values involves looking at the intersection point between graphs, and empirical observation to find the converging value.', 'The intricacy of the fractal set of complex numbers for which the power tower converges is analogous to the Mandelbrot set, indicating chaotic behavior and the emergence of a fractal pattern.']}, {'end': 2878.231, 'start': 2664.428, 'title': 'Unsolved mysteries of power towers', 'summary': 'Explores the uncertainty surrounding power towers with pi, raising the question of whether such towers can ever equal an integer and highlighting the intriguing brain teaser involving cup operations and coin manipulation, with the maximum number of coins reaching 63 and a mind-boggling comparison to 2010 to the power of 2010 to the power of 2010.', 'duration': 213.803, 'highlights': ["The uncertainty of whether a power tower with pi's can ever equal an integer is explored, emphasizing the daunting challenge of proving this due to the irrational nature of pi and the complexity of the process. uncertainty, irrational nature of pi", 'The brain teaser involving cup operations and coin manipulation is presented, demonstrating the maximum number of coins reaching 63 through a specific process and introducing a second operation involving the swapping of coins, leading to a mind-boggling comparison to 2010 to the power of 2010 to the power of 2010. maximum number of coins reaching 63, comparison to 2010 to the power of 2010 to the power of 2010']}, {'end': 3222.647, 'start': 2878.771, 'title': 'Lockdown math series - episode 10', 'summary': "Discusses the iterative process of solving x^x^x = 2, exploring the implications of ignoring the negative square root of 2 and the convergence of negative values, with an invitation for deeper thoughts and appreciation for a viewer's contribution.", 'duration': 343.876, 'highlights': ['The iterative process of solving x^x^x = 2 and the implications of ignoring the negative square root of 2 were thoroughly discussed.', "The convergence of negative values and the faulty line of reasoning in replacing what's inside the infinite power tower with pre-existing assumptions were explored.", 'An invitation for deeper thoughts and discussions was extended to the viewers, appreciating their contributions.', "The appreciation for a viewer's beautifully created notes for the lockdown series was expressed, concluding the lesson with an invitation for the next episode."]}], 'duration': 931.733, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/elQVZLLiod4/pics/elQVZLLiod42290914.jpg', 'highlights': ['The range of convergence for the base in tetration is from converging to a value of e to blowing up to infinity, with non-convergence to values between e and infinity.', 'The intricacy of the fractal set of complex numbers for which the power tower converges is analogous to the Mandelbrot set, indicating chaotic behavior and the emergence of a fractal pattern.', "The uncertainty of whether a power tower with pi's can ever equal an integer is explored, emphasizing the daunting challenge of proving this due to the irrational nature of pi and the complexity of the process.", 'The brain teaser involving cup operations and coin manipulation is presented, demonstrating the maximum number of coins reaching 63 through a specific process and introducing a second operation involving the swapping of coins, leading to a mind-boggling comparison to 2010 to the power of 2010 to the power of 2010.', 'The iterative process of solving x^x^x = 2 and the implications of ignoring the negative square root of 2 were thoroughly discussed.', 'An invitation for deeper thoughts and discussions was extended to the viewers, appreciating their contributions.']}], 'highlights': ['The iterative process of a power tower with root 2 approaches the value of 2, indicating the convergence of the sequence of numbers.', 'The infinite power tower converges to the square root of 2.', "The logical puzzle involves determining the exponential value of 'x' based on the number most people enter into a box, demonstrating the complexity and subjective nature of the problem.", 'The chapter discusses finding the value of b such that the graph of y equals b to the x is tangent to the graph y equals x. It involves solving a pair of equations to determine the value of b, which represents the point where the two graphs are tangent.', 'The exponential growth of a power tower with a base of 1.1 is explored to determine the number of iterations required for the expression to exceed 10 digits, akin to the example with a base of 2 which required a tower of size 5 before reaching 19,000 digits.', 'The chapter explores the concept of exponential and tetrational growth, demonstrating how repeatedly multiplying a value by 1.1 results in exponential growth, with the example showing a value growing by 10% at each step and reaching 117 after 50 steps.', 'The range of convergence for the base in tetration is from converging to a value of e to blowing up to infinity, with non-convergence to values between e and infinity.', 'The intricacy of the fractal set of complex numbers for which the power tower converges is analogous to the Mandelbrot set, indicating chaotic behavior and the emergence of a fractal pattern.', "The uncertainty of whether a power tower with pi's can ever equal an integer is explored, emphasizing the daunting challenge of proving this due to the irrational nature of pi and the complexity of the process.", 'The speaker presents the intriguing question of finding a value where a power tower converges to a specific number, providing an example of finding a value that converges to 1.111782.']}