title

From Newton’s method to Newton’s fractal (which Newton knew nothing about)

description

Who knew root-finding could be so complicated?
Next part: https://youtu.be/LqbZpur38nw
Special thanks to the following supporters: https://3b1b.co/lessons/newtons-fractal#thanks
An equally valuable form of support is to simply share the videos.
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Interactive for this video:
https://www.3blue1brown.com/lessons/newtons-fractal
On fractal dimension:
https://youtu.be/gB9n2gHsHN4
Mathologer on the cubic formula:
https://youtu.be/N-KXStupwsc
Some articles on Newton's Fractal, and its cousins:
https://www.chiark.greenend.org.uk/~sgtatham/newton/
https://blbadger.github.io/polynomial-roots.html
Some of the videos from this year's Summer of Math Exposition are fairly relevant to the topics covered here. Take a look at these ones,
The Beauty of Bézier Curves
https://youtu.be/aVwxzDHniEw
The insolubility of the quintic:
https://youtu.be/BSHv9Elk1MU
The math behind rasterizing fonts:
https://youtu.be/LaYPoMPRSlk
Viewer-made interactive:
https://codepen.io/mherreshoff/full/RwZPazd
---
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https://www.3blue1brown.com/faq#manim
https://github.com/3b1b/manim
https://github.com/ManimCommunity/manim/
You can find code for specific videos and projects here:
https://github.com/3b1b/videos/
Music by Vincent Rubinetti.
https://www.vincentrubinetti.com/
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
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Timestamps:
0:00 - Intro
0:48 - Roots of polynomials
5:55 - Newton’s method
11:16 - The fractal
17:56 - The boundary property
23:13 - Closing thoughts
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe
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detail

{'title': 'From Newton’s method to Newton’s fractal (which Newton knew nothing about)', 'heatmap': [{'end': 298.297, 'start': 277.017, 'weight': 0.893}, {'end': 706.667, 'start': 670.555, 'weight': 0.843}, {'end': 771.51, 'start': 749.399, 'weight': 0.841}, {'end': 940.349, 'start': 921.641, 'weight': 0.86}, {'end': 1400.027, 'start': 1376.815, 'weight': 1}], 'summary': "Explores an infinite family of fractals, the complexity of computing polynomial roots, newton's method, chaotic fractal patterns, and the mathematical complexity of newton's fractal boundary, offering a comprehensive understanding of their significance in computer graphics and mathematics.", 'chapters': [{'end': 37.099, 'segs': [{'end': 37.099, 'src': 'embed', 'start': 2.591, 'weight': 0, 'content': [{'end': 5.474, 'text': "You've seen the title, so you know this is leading to a certain fractal.", 'start': 2.591, 'duration': 2.883}, {'end': 7.876, 'text': "And actually, it's an infinite family of fractals.", 'start': 5.794, 'duration': 2.082}, {'end': 14.223, 'text': "And yeah, it'll be one of those mind-bogglingly intricate shapes that has infinite detail no matter how far you zoom in.", 'start': 8.697, 'duration': 5.526}, {'end': 18.507, 'text': 'But this is not really a video about generating some pretty picture for us to gawk at.', 'start': 14.823, 'duration': 3.684}, {'end': 25.514, 'text': "Well, okay, maybe that's part of it, but the real story here has a much more pragmatic starting point than the story behind a lot of other fractals.", 'start': 19.048, 'duration': 6.466}, {'end': 32.534, 'text': 'And more than that, the final images that we get to will become a lot more meaningful if we make an effort to understand why,', 'start': 26.468, 'duration': 6.066}, {'end': 37.099, 'text': 'given what they represent, they kind of have to look as complicated as they do,', 'start': 32.534, 'duration': 4.565}], 'summary': 'Exploring an infinite family of intricate fractals with meaningful representations.', 'duration': 34.508, 'max_score': 2.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s2591.jpg'}], 'start': 2.591, 'title': 'Infinite fractal family', 'summary': 'Discusses an infinite family of fractals, emphasizing pragmatic starting point and meaningful complexity of final images.', 'chapters': [{'end': 37.099, 'start': 2.591, 'title': 'Infinite fractal family', 'summary': 'Discusses an infinite family of fractals, highlighting the pragmatic starting point and meaningful complexity of the final images.', 'duration': 34.508, 'highlights': ['The final images become more meaningful if we understand why, given what they represent, they have to look as complicated as they do.', 'The video is not just about generating pretty pictures, but it has a pragmatic starting point.', "It's an infinite family of fractals leading to mind-bogglingly intricate shapes with infinite detail."]}], 'duration': 34.508, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s2591.jpg', 'highlights': ["It's an infinite family of fractals leading to mind-bogglingly intricate shapes with infinite detail.", 'The final images become more meaningful if we understand why, given what they represent, they have to look as complicated as they do.', 'The video is not just about generating pretty pictures, but it has a pragmatic starting point.']}, {'end': 277.017, 'segs': [{'end': 78.079, 'src': 'embed', 'start': 37.099, 'weight': 0, 'content': [{'end': 41.584, 'text': 'and what this complexity reflects about an algorithm that is used all over the place in engineering.', 'start': 37.099, 'duration': 4.485}, {'end': 53.904, 'text': 'The starting point here will be to assume that you have some kind of polynomial and that you want to know when it equals zero.', 'start': 48.26, 'duration': 5.644}, {'end': 59.028, 'text': "So for the one graphed here, you can visually see there's three different places where it crosses the x-axis.", 'start': 54.525, 'duration': 4.503}, {'end': 61.41, 'text': 'You can kind of eyeball what those values might be.', 'start': 59.508, 'duration': 1.902}, {'end': 63.651, 'text': "We'd call those the roots of the polynomial.", 'start': 61.87, 'duration': 1.781}, {'end': 66.333, 'text': 'But how do you actually compute them exactly?', 'start': 64.272, 'duration': 2.061}, {'end': 72.498, 'text': "Now, this is the kind of question where, if you're already bought into math, maybe it's interesting enough in its own right to move forward.", 'start': 67.394, 'duration': 5.104}, {'end': 78.079, 'text': "But if you just pull someone on the street aside and ask them this, I mean they're already falling asleep, because who cares?", 'start': 72.958, 'duration': 5.121}], 'summary': 'Analyzing the complexity of finding roots of a polynomial in engineering and computing exact values.', 'duration': 40.98, 'max_score': 37.099, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s37099.jpg'}, {'end': 123.282, 'src': 'embed', 'start': 94.844, 'weight': 1, 'content': [{'end': 98.305, 'text': 'that somehow involves solving an equation that uses these polynomials.', 'start': 94.844, 'duration': 3.461}, {'end': 100.846, 'text': 'Here, let me give you one fun example.', 'start': 99.485, 'duration': 1.361}, {'end': 106.99, 'text': 'When a computer renders text on the screen, those fonts are typically not defined using pixel values.', 'start': 101.186, 'duration': 5.804}, {'end': 112.474, 'text': "They're defined as a bunch of polynomial curves, or known in the business as Bézier curves.", 'start': 107.471, 'duration': 5.003}, {'end': 119.639, 'text': "And any of you who've messed around with vector graphics, maybe in some design software, would be well familiar with these kinds of curves.", 'start': 113.374, 'duration': 6.265}, {'end': 123.282, 'text': 'But to actually display one of them on the screen,', 'start': 120.419, 'duration': 2.863}], 'summary': 'Rendering fonts on screen involves using bézier curves, familiar to vector graphics users.', 'duration': 28.438, 'max_score': 94.844, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s94844.jpg'}, {'end': 229.84, 'src': 'embed', 'start': 201.509, 'weight': 4, 'content': [{'end': 206.991, 'text': 'Finding this minimum, and hence determining how close the pixel is to the curve and whether it should get filled in,', 'start': 201.509, 'duration': 5.482}, {'end': 208.592, 'text': 'is now just a classic calculus problem.', 'start': 206.991, 'duration': 1.601}, {'end': 215.654, 'text': 'What you do is figure out the slope of this function graph, which is to say its derivative, again some polynomial,', 'start': 209.272, 'duration': 6.382}, {'end': 217.695, 'text': 'and you ask when does that equal zero?', 'start': 215.654, 'duration': 2.041}, {'end': 224.558, 'text': 'So, to actually carry out this seemingly simple task of just displaying a curve,', 'start': 219.296, 'duration': 5.262}, {'end': 229.84, 'text': "wouldn't it be nice if you had a systematic and general way to figure out when a given polynomial equals zero?", 'start': 224.558, 'duration': 5.282}], 'summary': 'Determining pixel proximity to curve involves finding minimum using calculus and derivative of polynomial function.', 'duration': 28.331, 'max_score': 201.509, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s201509.jpg'}, {'end': 277.017, 'src': 'embed', 'start': 243.902, 'weight': 3, 'content': [{'end': 245.983, 'text': 'the task is hardly just an academic one.', 'start': 243.902, 'duration': 2.081}, {'end': 250.345, 'text': 'But again, ask yourself how do you actually compute one of those roots?', 'start': 246.703, 'duration': 3.642}, {'end': 256.848, 'text': "If whatever problem you're working on leads you to a quadratic function, then happy days.", 'start': 252.306, 'duration': 4.542}, {'end': 259.249, 'text': 'You can use the quadratic formula that we all know and love.', 'start': 257.007, 'duration': 2.242}, {'end': 267.167, 'text': 'And as a fun side note, by the way, again relevant to root finding in computer graphics, I once had a Pixar engineer give me the estimate that,', 'start': 260.1, 'duration': 7.067}, {'end': 271.051, 'text': 'considering how many lights were used in some of the scenes for the movie Coco,', 'start': 267.167, 'duration': 3.884}, {'end': 277.017, 'text': 'and given the nature of some of these per pixel calculations when polynomially defined things like spheres are involved,', 'start': 271.051, 'duration': 5.966}], 'summary': 'Root finding in quadratic functions, relevant to computer graphics, estimated by a pixar engineer working on movie coco.', 'duration': 33.115, 'max_score': 243.902, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s243902.jpg'}], 'start': 37.099, 'title': 'Roots of polynomials and their use in computer graphics', 'summary': 'Discusses the complexity of computing the roots of a polynomial, a fundamental task in engineering, highlighting the challenge of exact computation. it also explores the pervasive use of polynomials in computer graphics, particularly in rendering vector graphics, presenting classic calculus problems and practical applications in pixel coloring.', 'chapters': [{'end': 78.079, 'start': 37.099, 'title': 'Roots of a polynomial', 'summary': 'Discusses the complexity of computing the roots of a polynomial, a fundamental task in engineering, and highlights the challenge of exact computation in contrast to visual estimation.', 'duration': 40.98, 'highlights': ['Computing the roots of a polynomial is a fundamental task in engineering and mathematics, reflecting the complexity of an algorithm used extensively in various fields.', 'The challenge lies in the exact computation of the roots, which contrasts with the visual estimation provided by graphing, where three different root values can be visually identified on the x-axis.']}, {'end': 277.017, 'start': 78.82, 'title': 'Use of polynomials in computer graphics', 'summary': 'Discusses the pervasive use of polynomials in computer graphics, particularly in rendering vector graphics, where polynomials are used to define fonts and compute distances to curves for pixel coloring, presenting a classic calculus problem in determining how close the pixel is to the curve and whether it should get filled in.', 'duration': 198.197, 'highlights': ['Polynomials are extensively used in computer graphics, particularly in rendering vector graphics and defining fonts as polynomial curves.', 'Computing distances to curves for pixel coloring presents a classic calculus problem involving determining the minimum of a polynomial function, which is achieved by finding its derivative and solving for when it equals zero.', 'The task of seeking roots of polynomials is not merely academic but is a practical challenge encountered in various disciplines including computer graphics.', 'The quadratic formula is a useful tool for computing the roots of quadratic functions, providing a systematic way to solve for the roots.']}], 'duration': 239.918, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s37099.jpg', 'highlights': ['Computing the roots of a polynomial is a fundamental task in engineering and mathematics, reflecting the complexity of an algorithm used extensively in various fields.', 'Polynomials are extensively used in computer graphics, particularly in rendering vector graphics and defining fonts as polynomial curves.', 'The challenge lies in the exact computation of the roots, which contrasts with the visual estimation provided by graphing, where three different root values can be visually identified on the x-axis.', 'The task of seeking roots of polynomials is not merely academic but is a practical challenge encountered in various disciplines including computer graphics.', 'Computing distances to curves for pixel coloring presents a classic calculus problem involving determining the minimum of a polynomial function, which is achieved by finding its derivative and solving for when it equals zero.', 'The quadratic formula is a useful tool for computing the roots of quadratic functions, providing a systematic way to solve for the roots.']}, {'end': 765.441, 'segs': [{'end': 303.06, 'src': 'heatmap', 'start': 277.017, 'weight': 0, 'content': [{'end': 281.882, 'text': 'the quadratic formula was easily used multiple trillions of times in the production of that film.', 'start': 277.017, 'duration': 4.865}, {'end': 287.508, 'text': 'Now, when your problem leads you to a higher order polynomial, things start to get trickier.', 'start': 283.365, 'duration': 4.143}, {'end': 292.792, 'text': 'For cubic polynomials, there is also a formula, which Mathologer has done a wonderful video on.', 'start': 288.109, 'duration': 4.683}, {'end': 298.297, 'text': "There's even a quartic formula, something that solves degree four polynomials, although honestly,", 'start': 293.353, 'duration': 4.944}, {'end': 303.06, 'text': 'that one is such a god-awful nightmare of a formula that essentially no one actually uses it in practice.', 'start': 298.297, 'duration': 4.763}], 'summary': 'The quadratic formula was used trillions of times, cubic polynomials have a formula, and quartic formula is rarely used.', 'duration': 26.043, 'max_score': 277.017, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s277017.jpg'}, {'end': 357.581, 'src': 'embed', 'start': 317.565, 'weight': 2, 'content': [{'end': 321.869, 'text': 'you can prove that there is no possible way that you can combine those functions together.', 'start': 317.565, 'duration': 4.304}, {'end': 326.473, 'text': 'that allows you to plug in the coefficients of a quintic polynomial and always get out a root.', 'start': 321.869, 'duration': 4.604}, {'end': 333.519, 'text': 'This is known as the unsolvability of the quintic, which is a whole other can of worms, we can hopefully get into it some other time.', 'start': 327.694, 'duration': 5.825}, {'end': 336.562, 'text': "But in practice it kind of doesn't matter,", 'start': 333.999, 'duration': 2.563}, {'end': 342.588, 'text': 'because we have algorithms to approximate solutions to these kinds of equations with whatever level of precision you want.', 'start': 336.562, 'duration': 6.026}, {'end': 347.092, 'text': "A common one, and the main topic for you and me today, is Newton's method.", 'start': 343.248, 'duration': 3.844}, {'end': 350.015, 'text': 'And yes, this is what will lead us to the fractals,', 'start': 347.593, 'duration': 2.422}, {'end': 354.399, 'text': 'but I want you to pay attention to just how innocent and benign the whole procedure seems at first.', 'start': 350.015, 'duration': 4.384}, {'end': 357.581, 'text': 'The algorithm begins with a random guess.', 'start': 355.56, 'duration': 2.021}], 'summary': "Unsolvability of quintic, but algorithms can approximate solutions. focus on newton's method and its innocent appearance.", 'duration': 40.016, 'max_score': 317.565, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s317565.jpg'}, {'end': 599.588, 'src': 'embed', 'start': 571.881, 'weight': 3, 'content': [{'end': 580.867, 'text': "It's only when this process just happens to throw the new guess off far enough to the left by chance that the sequence of new guesses does anything productive and actually approaches that true root.", 'start': 571.881, 'duration': 8.986}, {'end': 587.432, 'text': 'Where things get especially interesting is if we ask about finding roots in the complex plane.', 'start': 582.789, 'duration': 4.643}, {'end': 592.765, 'text': 'Even if a polynomial like the one shown here has only a single real number root,', 'start': 588.484, 'duration': 4.281}, {'end': 599.588, 'text': "you'll always be able to factor this polynomial into five terms like this if you allow these roots to potentially be complex numbers.", 'start': 592.765, 'duration': 6.823}], 'summary': 'Iterative process leads to productive guesses; complex plane allows for factoring into five terms.', 'duration': 27.707, 'max_score': 571.881, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s571881.jpg'}, {'end': 706.667, 'src': 'heatmap', 'start': 670.555, 'weight': 0.843, 'content': [{'end': 674.417, 'text': 'you can see that we land on a value whose corresponding output is essentially zero.', 'start': 670.555, 'duration': 3.862}, {'end': 677.099, 'text': "Now here's the fun part.", 'start': 676.277, 'duration': 0.822}, {'end': 680.886, 'text': "Let's apply this idea to many different possible initial guesses.", 'start': 677.519, 'duration': 3.367}, {'end': 686.456, 'text': "For reference, I'll put up the five true roots of this particular polynomial in the complex plane.", 'start': 681.747, 'duration': 4.709}, {'end': 691.938, 'text': "With each iteration, each one of our little dots takes some step based on Newton's method.", 'start': 687.455, 'duration': 4.483}, {'end': 696.341, 'text': 'Most of the dots will quickly converge to one of the five true roots,', 'start': 692.738, 'duration': 3.603}, {'end': 700.383, 'text': 'but there are some noticeable stragglers which seem to spend a while bouncing around.', 'start': 696.341, 'duration': 4.042}, {'end': 706.667, 'text': 'In particular, notice how the ones that are trapped on the positive real number line, well, they look a little bit lost.', 'start': 701.084, 'duration': 5.583}], 'summary': "Newton's method converges most points to true roots, some stragglers bounce around.", 'duration': 36.112, 'max_score': 670.555, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s670555.jpg'}], 'start': 277.017, 'title': "Quintic polynomials and newton's method", 'summary': "Explores the unsolvability of quintic polynomials using quadratic and cubic formulas, and discusses newton's method for approximating solutions to polynomial equations, its application to finding roots in the complex plane, and how it leads to the generation of fractals.", 'chapters': [{'end': 336.562, 'start': 277.017, 'title': 'Unsolvability of quintic polynomials', 'summary': 'Explores the unsolvability of quintic polynomials, with the quadratic and cubic formulas being used extensively, while the quartic formula is rarely used due to its complexity.', 'duration': 59.545, 'highlights': ['The unsolvability of quintic polynomials is a significant result in math, as there is no formula to solve polynomials of degree five or more, unlike the easily used quadratic formula which has been employed multiple trillions of times in film production.', 'The quartic formula, used to solve degree four polynomials, is considered a god-awful nightmare and essentially not used in practice, unlike the cubic formula which Mathologer has done a wonderful video on.', 'For cubic polynomials, there is a formula, while higher order polynomials lead to trickier problems, showcasing the complexity in solving polynomials of different degrees.']}, {'end': 765.441, 'start': 336.562, 'title': "Newton's method and fractals", 'summary': "Discusses newton's method for approximating solutions to polynomial equations, its application to finding roots in the complex plane, and how it leads to the generation of fractals.", 'duration': 428.879, 'highlights': ["Newton's method for approximating solutions to polynomial equations", 'Application to finding roots in the complex plane', 'Generation of fractals']}], 'duration': 488.424, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s277017.jpg', 'highlights': ['The unsolvability of quintic polynomials is a significant result in math, as there is no formula to solve polynomials of degree five or more, unlike the easily used quadratic formula which has been employed multiple trillions of times in film production.', 'For cubic polynomials, there is a formula, while higher order polynomials lead to trickier problems, showcasing the complexity in solving polynomials of different degrees.', "Newton's method for approximating solutions to polynomial equations", 'Application to finding roots in the complex plane', 'The quartic formula, used to solve degree four polynomials, is considered a god-awful nightmare and essentially not used in practice, unlike the cubic formula which Mathologer has done a wonderful video on.', 'The unsolvability of quintic polynomials using quadratic and cubic formulas']}, {'end': 1040.631, 'segs': [{'end': 822.273, 'src': 'embed', 'start': 803.582, 'weight': 0, 'content': [{'end': 815.849, 'text': 'but picturing all of this in the complex plane really shines a light on just how unpredictable this kind of root finding algorithm can be and how there are whole swaths of initial values where this sort of unpredictability will take place.', 'start': 803.582, 'duration': 12.267}, {'end': 822.273, 'text': "Now, if I grab one of these roots and change it around, meaning that we're using a different polynomial for the process,", 'start': 817.011, 'duration': 5.262}], 'summary': 'Root finding algorithms in complex plane are unpredictable with swaths of initial values leading to unpredictability.', 'duration': 18.691, 'max_score': 803.582, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s803582.jpg'}, {'end': 870.466, 'src': 'embed', 'start': 844.161, 'weight': 3, 'content': [{'end': 848.388, 'text': 'And it seems like no matter where I place these roots, those fractal boundaries are always there.', 'start': 844.161, 'duration': 4.227}, {'end': 852.195, 'text': "It clearly wasn't just some one-off for the polynomial we happen to start with.", 'start': 848.97, 'duration': 3.225}, {'end': 854.84, 'text': 'This seems to be a general fact for any given polynomial.', 'start': 852.355, 'duration': 2.485}, {'end': 862.28, 'text': "Another facet we can tweak here just to better illustrate what's going on is how many steps of Newton's method we're using.", 'start': 856.815, 'duration': 5.465}, {'end': 865.903, 'text': 'For example, if I had the computer just take zero steps,', 'start': 862.98, 'duration': 2.923}, {'end': 870.466, 'text': "meaning it just colors each point of the plane based on whatever route it's already closest to.", 'start': 865.903, 'duration': 4.563}], 'summary': "Fractal boundaries persist for any polynomial, regardless of root placement or step count in newton's method.", 'duration': 26.305, 'max_score': 844.161, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s844161.jpg'}, {'end': 901.183, 'src': 'embed', 'start': 876.151, 'weight': 2, 'content': [{'end': 884.077, 'text': "And if we let each point of the plane take a single step of Newton's method and then color it based on what route that single step result is closest to,", 'start': 876.151, 'duration': 7.926}, {'end': 884.778, 'text': "here's what we would get.", 'start': 884.077, 'duration': 0.701}, {'end': 897.902, 'text': 'Similarly, if we allow for two steps, we get a slightly more intricate pattern, and so on and so on, where the more steps you allow,', 'start': 890.259, 'duration': 7.643}, {'end': 901.183, 'text': 'the more intricate an image you get, bringing us closer to the original fractal.', 'start': 897.902, 'duration': 3.281}], 'summary': "Using newton's method to create intricate fractal patterns on the plane.", 'duration': 25.032, 'max_score': 876.151, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s876151.jpg'}, {'end': 949.512, 'src': 'heatmap', 'start': 921.641, 'weight': 0.86, 'content': [{'end': 926.523, 'text': 'Or maybe you want to dig deeper into what dynamics are exactly possible with these iterated points,', 'start': 921.641, 'duration': 4.882}, {'end': 929.945, 'text': "or see if there's connections with some other pieces of math that have a similar theme.", 'start': 926.523, 'duration': 3.422}, {'end': 935.708, 'text': 'But I think the most pertinent question should be something like what the is going on here?', 'start': 930.925, 'duration': 4.783}, {'end': 940.349, 'text': "I mean all we're doing here is repeatedly solving linear approximations.", 'start': 936.448, 'duration': 3.901}, {'end': 943.63, 'text': "Why would that produce something that's so endlessly complicated?", 'start': 940.869, 'duration': 2.761}, {'end': 949.512, 'text': "It almost feels like the underlying rule here just shouldn't carry enough information to actually produce an image like this.", 'start': 944.23, 'duration': 5.282}], 'summary': 'Exploring the complexity arising from repeatedly solving linear approximations in math.', 'duration': 27.871, 'max_score': 921.641, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s921641.jpg'}, {'end': 1040.631, 'src': 'embed', 'start': 1012.425, 'weight': 4, 'content': [{'end': 1018.267, 'text': 'the relevant picture for all possible starting points forms this fractal pattern with infinite detail.', 'start': 1012.425, 'duration': 5.842}, {'end': 1029.367, 'text': 'However, quadratic polynomials with only two roots are different.', 'start': 1026.107, 'duration': 3.26}, {'end': 1035.29, 'text': "In that case, each seed value does simply tend towards whichever root it's closest to, the way that you might expect.", 'start': 1029.808, 'duration': 5.482}, {'end': 1040.631, 'text': 'There is a little bit of meandering behavior from all the points that are an equal distance from each root.', 'start': 1036.29, 'duration': 4.341}], 'summary': 'Fractal pattern with infinite detail from quadratic polynomials, points tend towards closest root.', 'duration': 28.206, 'max_score': 1012.425, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s1012425.jpg'}], 'start': 766.262, 'title': 'Chaos in fractal patterns and iterated functions', 'summary': 'Discusses the unpredictable nature of root finding algorithms in the complex plane, leading to chaotic fractal patterns and the creation of intricate images through multiple iterations, emphasizing the significance of fractal boundaries and the unexpected complexity arising from repeatedly solving linear approximations.', 'chapters': [{'end': 862.28, 'start': 766.262, 'title': 'Chaos in fractal patterns', 'summary': "Discusses the unpredictable nature of root finding algorithms in the complex plane, where slight adjustments in seed values can drastically change the true roots, leading to chaotic fractal patterns and highlighting the significance of fractal boundaries. it also emphasizes the general impact of these phenomena on any given polynomial and the potential influence of adjusting the number of steps in newton's method.", 'duration': 96.018, 'highlights': ['Regions in the complex plane exhibit chaos when slight adjustments in seed values lead to drastic changes in true roots, emphasizing the unpredictable nature of root finding algorithms (e.g., by one millionth or one trillionth).', 'The visualization of the complex plane illuminates the unpredictable nature of root finding algorithms and the existence of whole swaths of initial values where unpredictability occurs, providing insight into the significance of fractal boundaries.', 'Manipulating roots in the complex plane results in changes to the fractal pattern, with consistent color regions around given roots due to the effectiveness of linear approximation schemes in finding these roots, while chaos primarily occurs at the boundaries between regions.', 'The presence of fractal boundaries appears to be a general phenomenon for any given polynomial, indicating the widespread impact of chaotic behavior in root finding algorithms.', "The chapter introduces the potential impact of adjusting the number of steps in Newton's method as a means to better illustrate the dynamics of root finding algorithms in the complex plane."]}, {'end': 1040.631, 'start': 862.98, 'title': 'Fractal patterns and iterated functions', 'summary': 'Discusses the creation of fractal patterns through iterated functions, particularly focusing on the development of intricate images through multiple iterations, the exploration of different polynomials, and the unexpected complexity arising from repeatedly solving linear approximations.', 'duration': 177.651, 'highlights': ['The creation of intricate fractal patterns through multiple iterations allows for a deeper understanding of the original fractal, with more iterations leading to increasingly complex images.', 'Exploring different polynomials, such as cubic and quadratic, reveals varying behaviors in the iterated points, from chaotic movement to expected convergence towards roots.', "The unexpected complexity arising from repeatedly solving linear approximations prompts questions about the underlying rule's ability to produce such intricate images, challenging initial assumptions about the behavior of seed values and the relationship to unsolvable quintics."]}], 'duration': 274.369, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s766262.jpg', 'highlights': ['The visualization of the complex plane illuminates the unpredictable nature of root finding algorithms and the existence of whole swaths of initial values where unpredictability occurs, providing insight into the significance of fractal boundaries.', 'Regions in the complex plane exhibit chaos when slight adjustments in seed values lead to drastic changes in true roots, emphasizing the unpredictable nature of root finding algorithms (e.g., by one millionth or one trillionth).', 'The creation of intricate fractal patterns through multiple iterations allows for a deeper understanding of the original fractal, with more iterations leading to increasingly complex images.', 'The presence of fractal boundaries appears to be a general phenomenon for any given polynomial, indicating the widespread impact of chaotic behavior in root finding algorithms.', 'Exploring different polynomials, such as cubic and quadratic, reveals varying behaviors in the iterated points, from chaotic movement to expected convergence towards roots.']}, {'end': 1547.152, 'segs': [{'end': 1075.058, 'src': 'embed', 'start': 1041.231, 'weight': 1, 'content': [{'end': 1043.791, 'text': "It's kind of like they're not able to decide which one to go to.", 'start': 1041.231, 'duration': 2.56}, {'end': 1050.574, 'text': "But that's just a single line of points, and when we play the game of coloring, the diagram we end up with is decidedly more boring.", 'start': 1044.232, 'duration': 6.342}, {'end': 1056.779, 'text': 'So something new seems to happen when you jump from 2 to 3, and the question is what exactly?', 'start': 1052.014, 'duration': 4.765}, {'end': 1062.285, 'text': 'And if you had asked me a month ago, I probably would have shrugged and just said you know, math is what it is.', 'start': 1057.7, 'duration': 4.585}, {'end': 1064.587, 'text': 'Sometimes the answers look simple, sometimes not.', 'start': 1062.565, 'duration': 2.022}, {'end': 1068.171, 'text': "It's not always clear what it would mean to ask why in a setting like this.", 'start': 1065.027, 'duration': 3.144}, {'end': 1070.053, 'text': 'But I would have been wrong.', 'start': 1068.891, 'duration': 1.162}, {'end': 1075.058, 'text': 'There actually is a reason that we can give for why this image has to look as complicated as it does.', 'start': 1070.193, 'duration': 4.865}], 'summary': "Transition from 2 to 3 in diagram reveals complexity in math; there's a reason for the image's complexity.", 'duration': 33.827, 'max_score': 1041.231, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s1041231.jpg'}, {'end': 1316.416, 'src': 'embed', 'start': 1283.798, 'weight': 0, 'content': [{'end': 1289.14, 'text': 'So if you believe the property, it explains why the boundary remains rough no matter how far you zoom in.', 'start': 1283.798, 'duration': 5.342}, {'end': 1295.682, 'text': 'And for those of you who are familiar with the concept of fractal dimension, you can measure the dimension of the particular boundary.', 'start': 1290.12, 'duration': 5.562}, {'end': 1299.923, 'text': "I'm showing you right now to be around 1.44..", 'start': 1295.682, 'duration': 4.241}, {'end': 1302.264, 'text': 'Considering what our colors actually represent.', 'start': 1299.923, 'duration': 2.341}, {'end': 1303.664, 'text': "remember, this isn't just a picture.", 'start': 1302.264, 'duration': 1.4}, {'end': 1306.785, 'text': "for picture's sake, think about what the property is really telling us.", 'start': 1303.664, 'duration': 3.121}, {'end': 1316.416, 'text': "It says that if you're near a sensitive point where some of the seed values go to one root but other seed values nearby would go to another root,", 'start': 1308.709, 'duration': 7.707}], 'summary': 'The fractal boundary has a dimension of around 1.44, indicating sensitivity to seed values.', 'duration': 32.618, 'max_score': 1283.798, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s1283798.jpg'}, {'end': 1406.011, 'src': 'heatmap', 'start': 1376.815, 'weight': 1, 'content': [{'end': 1380.515, 'text': "To be clear, it doesn't guarantee that the quadratic case would have a smooth boundary.", 'start': 1376.815, 'duration': 3.7}, {'end': 1384.016, 'text': 'It is perfectly possible to have a fractal boundary between two colors.', 'start': 1380.935, 'duration': 3.081}, {'end': 1392.558, 'text': "It just looks like our Newton's method diagram is not doing anything more complicated than it needs to under the constraint of this strange boundary condition.", 'start': 1384.476, 'duration': 8.082}, {'end': 1400.027, 'text': 'But of course, all of this simply raises the question of why this bizarre boundary property would have to be true in the first place.', 'start': 1393.803, 'duration': 6.224}, {'end': 1406.011, 'text': "Where does it even come from? For that, I'd like to tell you about a field of math which studies this kind of question.", 'start': 1400.428, 'duration': 5.583}], 'summary': "Newton's method diagram shows constraints of strange boundary, raising questions about its origin.", 'duration': 29.196, 'max_score': 1376.815, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s1376815.jpg'}, {'end': 1471.318, 'src': 'embed', 'start': 1444.946, 'weight': 5, 'content': [{'end': 1449.808, 'text': 'But this overextension of nomenclature carries with it what I think is an inspiring point.', 'start': 1444.946, 'duration': 4.862}, {'end': 1454.951, 'text': 'It reflects how even the simple ideas, ones that could be discovered centuries ago,', 'start': 1450.509, 'duration': 4.442}, {'end': 1461.155, 'text': 'often hold within them some new angle or a new domain of relevance that can sit waiting to be discovered hundreds of years later.', 'start': 1454.951, 'duration': 6.204}, {'end': 1465.016, 'text': "It's not just that Newton had no idea about Newton's fractal.", 'start': 1461.915, 'duration': 3.101}, {'end': 1471.318, 'text': "There are probably many other facts about Newton's method or about all sorts of math that may seem like old news,", 'start': 1465.576, 'duration': 5.742}], 'summary': "Simple ideas can hold new relevance, like newton's fractal and other undiscovered facts about math.", 'duration': 26.372, 'max_score': 1444.946, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s1444946.jpg'}, {'end': 1513.924, 'src': 'embed', 'start': 1488.681, 'weight': 4, 'content': [{'end': 1493.663, 'text': "it leads you to a surprising connection with the Mandelbrot set, and we'll talk a bit about that in the next part.", 'start': 1488.681, 'duration': 4.982}, {'end': 1499.244, 'text': "At the time that I'm posting this, that second part, by the way, is available as an early release to patrons.", 'start': 1495.063, 'duration': 4.181}, {'end': 1503.425, 'text': 'I always like to give new content a little bit of time there to gather feedback and catch errors.', 'start': 1499.604, 'duration': 3.821}, {'end': 1505.506, 'text': 'The finalized version should be out shortly.', 'start': 1503.925, 'duration': 1.581}, {'end': 1510.282, 'text': 'And on the topic of patrons, I do just want to say a quick thanks to everyone whose name is on the screen.', 'start': 1506.4, 'duration': 3.882}, {'end': 1513.924, 'text': 'I know that in recent history, new videos have been a little slow coming.', 'start': 1510.782, 'duration': 3.142}], 'summary': 'The next part is available to patrons as an early release; finalized version coming soon.', 'duration': 25.243, 'max_score': 1488.681, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s1488681.jpg'}], 'start': 1041.231, 'title': "Mathematical complexity and newton's fractal boundary", 'summary': "Delves into the complexity of mathematical diagrams and their relationship with simplicity, as well as newton's fractal boundary property, revealing a fractal dimension of approximately 1.44 and its connection with the mandelbrot set.", 'chapters': [{'end': 1075.058, 'start': 1041.231, 'title': 'Complexity in mathematics', 'summary': 'Explores the complexity in mathematical diagrams and the revelation of a reason behind their complicated appearance, challenging the notion of simplicity in mathematics.', 'duration': 33.827, 'highlights': ['The revelation of a reason behind the complicated appearance of mathematical diagrams challenges the notion of simplicity in mathematics, indicating a shift in perspective from a month ago.', 'The question of what exactly happens when transitioning from 2 to 3 in a diagram is raised, prompting a deeper exploration of mathematical concepts and patterns.', 'The speaker illustrates the contrast between simple and complex appearances in mathematical diagrams, emphasizing the intriguing nature of mathematical phenomena.']}, {'end': 1547.152, 'start': 1075.885, 'title': "Newton's fractal boundary property", 'summary': "Explores the peculiar property of newton's fractal, explaining how its boundary consists of sharp corners, leading to a fractal dimension of around 1.44, and how it reflects the accessibility of roots within a small neighborhood, alluding to the connection with the mandelbrot set.", 'duration': 471.267, 'highlights': ["The boundary of Newton's fractal consists entirely of sharp corners, leading to a fractal dimension of around 1.44, which explains why the boundary remains rough no matter how far you zoom in.", "The property implies that every possible root has to be accessible from within a small neighborhood, with any tiny circle tending to just one root or all of the roots, and it also explains why it's okay for things to look normal in the case of quadratic polynomials with just two roots.", "The chapter alludes to the surprising connection between Newton's fractal and the Mandelbrot set, and the availability of the next part as an early release to patrons, as well as the author's plan to shift gears back to making new videos.", 'The author reflects on the overextension of nomenclature in mathematics, highlighting how even simple ideas discovered centuries ago often hold new angles or domains of relevance that can be discovered much later.', "The chapter introduces the concept of holomorphic dynamics as a field of math that studies the kind of questions related to the bizarre boundary property of Newton's fractal, leaving the detailed explanation for a separate video."]}], 'duration': 505.921, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/-RdOwhmqP5s/pics/-RdOwhmqP5s1041231.jpg', 'highlights': ["The boundary of Newton's fractal has a fractal dimension of around 1.44, explaining its rough appearance.", 'The revelation of a reason behind the complicated appearance of mathematical diagrams challenges the notion of simplicity in mathematics.', 'The question of transitioning from 2 to 3 in a diagram prompts a deeper exploration of mathematical concepts and patterns.', 'The contrast between simple and complex appearances in mathematical diagrams emphasizes the intriguing nature of mathematical phenomena.', "The surprising connection between Newton's fractal and the Mandelbrot set is alluded to, with the next part available as an early release to patrons.", 'The overextension of nomenclature in mathematics is reflected upon, highlighting how even simple ideas discovered centuries ago often hold new angles or domains of relevance.']}], 'highlights': ['The unsolvability of quintic polynomials is a significant result in math, as there is no formula to solve polynomials of degree five or more, unlike the easily used quadratic formula which has been employed multiple trillions of times in film production.', "The boundary of Newton's fractal has a fractal dimension of around 1.44, explaining its rough appearance.", 'The visualization of the complex plane illuminates the unpredictable nature of root finding algorithms and the existence of whole swaths of initial values where unpredictability occurs, providing insight into the significance of fractal boundaries.', 'The final images become more meaningful if we understand why, given what they represent, they have to look as complicated as they do.', 'The presence of fractal boundaries appears to be a general phenomenon for any given polynomial, indicating the widespread impact of chaotic behavior in root finding algorithms.']}