title

A tale of two problem solvers (Average cube shadows)

description

What's the average area of a cube's shadow?
Numberphile video on Bertrand's paradox: https://youtu.be/mZBwsm6B280
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share the videos.
There's a small error at 19:30, I say "Divide the total by 1/2", but of course meant to say "Multiply..."
Curious why a sphere's surface area is exactly four times its shadow?
https://youtu.be/GNcFjFmqEc8
If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts:
Part 1: https://youtu.be/rAHcZGjKVvg
Part 2: https://youtu.be/cEhLNS5AHss
I first heard this puzzle in a problem-solving seminar at Stanford, but the general result about all convex solids was originally proved by Cauchy.
MÃ©moire sur la rectification des courbes et la quadrature des surfaces courbes par M. Augustin Cauchy
https://ia600208.us.archive.org/27/items/bub_gb_EomNI7m8__UC/bub_gb_EomNI7m8__UC.pdf
The artwork in this video was done by Kurt Bruns
-------------------
Timestamps
0:00 - The players
5:22 - How to start
9:12 - Alice's initial thoughts
13:37 - Piecing together the cube
22:11 - Bob's conclusion
29:58 - Alice's conclusion
34:09 - Which is better?
38:59 - Homework
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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/
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https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
------------------
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detail

{'title': 'A tale of two problem solvers (Average cube shadows)', 'heatmap': [{'end': 1806.203, 'start': 1774.213, 'weight': 1}], 'summary': "Compares alice and bob's problem-solving styles, discusses finding average shadow area for a cube, explores shadow proportionality in 3d space, examines shadow area and convexity, and delves into computation and approximation of shadow area, emphasizing the significance of understanding problem shapes, cube orientations, and the influence of convexity on shadow areas.", 'chapters': [{'end': 66.319, 'segs': [{'end': 51.288, 'src': 'embed', 'start': 15.756, 'weight': 0, 'content': [{'end': 22.64, 'text': "In fact, let's anthropomorphize those two different styles by imagining two students, Alice and Bob, that embody each one of the approaches.", 'start': 15.756, 'duration': 6.884}, {'end': 27.02, 'text': 'So Bob will be the kind of student who really loves calculation.', 'start': 23.419, 'duration': 3.601}, {'end': 33.142, 'text': "As soon as there's a moment when he can dig into the details and get a very concrete view of the concrete situation in front of him.", 'start': 27.36, 'duration': 5.782}, {'end': 34.262, 'text': "that's where he's the most pleased.", 'start': 33.142, 'duration': 1.12}, {'end': 42.645, 'text': "Alice, on the other hand, is more inclined to procrastinate the computations, not because she doesn't know how to do them or doesn't want to per se,", 'start': 35.063, 'duration': 7.582}, {'end': 49.087, 'text': "but she prefers to get a nice high-level general overview of the kind of problem she's dealing with, the general shape that it has,", 'start': 42.645, 'duration': 6.442}, {'end': 51.288, 'text': 'before she digs into the computations themselves.', 'start': 49.087, 'duration': 2.201}], 'summary': 'Two student styles: bob loves calculations, alice prefers high-level overviews.', 'duration': 35.532, 'max_score': 15.756, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi015756.jpg'}], 'start': 0.129, 'title': 'Problem-solving styles: alice vs. bob', 'summary': 'Discusses the problem-solving styles of alice and bob, emphasizing the importance of understanding problem shapes and potential for swift computations. it highlights the contrast between detailed calculations and high-level overviews.', 'chapters': [{'end': 66.319, 'start': 0.129, 'title': 'Problem-solving styles: alice vs. bob', 'summary': 'Discusses two distinct problem-solving styles, represented by two students, alice and bob, with bob favoring detailed calculations and alice preferring a high-level general overview, highlighting the importance of understanding the general shape of problems and the potential for more swift and elegant computations.', 'duration': 66.19, 'highlights': ['Bob favors detailed calculations and concrete views of situations, while Alice prefers a high-level general overview and understanding the broadest possible way to generalize a problem, potentially leading to more swift and elegant computations.', "Alice's preference for a high-level understanding and general view of problems can lead to more swift and elegant computations, once she carries them out.", 'The chapter discusses the importance of understanding the general shape of problems and the potential for more swift and elegant computations.']}], 'duration': 66.19, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0129.jpg', 'highlights': ['Alice prefers high-level overviews for swift computations', 'Bob favors detailed calculations and concrete views', 'Understanding problem shapes leads to swift computations']}, {'end': 329.744, 'segs': [{'end': 103.025, 'src': 'embed', 'start': 73.325, 'weight': 0, 'content': [{'end': 79.13, 'text': 'Now the puzzle that both of them are going to be faced with is to find the average area for the shadow of a cube.', 'start': 73.325, 'duration': 5.805}, {'end': 85.436, 'text': 'So if I have a cube kind of sitting here hovering in space, there are a few things that influence the area of its shadow.', 'start': 79.831, 'duration': 5.605}, {'end': 89.239, 'text': 'One obvious one would be the size of the cube, smaller cube, smaller shadow.', 'start': 86.076, 'duration': 3.163}, {'end': 96.085, 'text': "But also if it's sitting at different orientations, those orientations correspond to different particular shadows with different areas.", 'start': 89.819, 'duration': 6.266}, {'end': 103.025, 'text': 'And when I say find the average here, what I mean is average over all possible orientations for a particular size of the cube.', 'start': 96.724, 'duration': 6.301}], 'summary': 'Find average shadow area for cube, considering size and orientations.', 'duration': 29.7, 'max_score': 73.325, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi073325.jpg'}, {'end': 166.229, 'src': 'embed', 'start': 142.705, 'weight': 1, 'content': [{'end': 149.271, 'text': 'So, just to get our bearings, the easiest situation to think about would be if the cube is straight up, with two of its faces parallel to the ground.', 'start': 142.705, 'duration': 6.566}, {'end': 152.774, 'text': 'In that case, this flat projection shadow is simply a square.', 'start': 149.851, 'duration': 2.923}, {'end': 158.018, 'text': 'And if we say the side lengths of the cube are S, then the area of that shadow is S squared.', 'start': 152.794, 'duration': 5.224}, {'end': 162.802, 'text': 'And, by the way, any time that I have a label up on these animations, like the one down here,', 'start': 158.739, 'duration': 4.063}, {'end': 166.229, 'text': "I'll be assuming that the relevant cube has a side length of 1..", 'start': 162.802, 'duration': 3.427}], 'summary': 'When the cube is straight up, its shadow is a square with an area of s squared.', 'duration': 23.524, 'max_score': 142.705, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0142705.jpg'}, {'end': 212.56, 'src': 'embed', 'start': 173.576, 'weight': 2, 'content': [{'end': 180.003, 'text': 'In that case, the shadow actually looks like a regular hexagon and if you use some of the methods that we will develop in a few minutes,', 'start': 173.576, 'duration': 6.427}, {'end': 185.769, 'text': 'you can compute that the area of that shadow is exactly the square root of 3 times the area of one of the square faces.', 'start': 180.003, 'duration': 5.766}, {'end': 191.192, 'text': 'But of course, more often, the actual shadow will be not so regular as a square or a hexagon.', 'start': 186.67, 'duration': 4.522}, {'end': 196.215, 'text': "It's some harder-to-think-about shape based on some harder-to-think-about orientation for this cube.", 'start': 191.673, 'duration': 4.542}, {'end': 202.138, 'text': 'Earlier, I casually threw out this phrase of averaging over all possible orientations.', 'start': 197.276, 'duration': 4.862}, {'end': 205.2, 'text': 'but you could rightly ask what exactly is that supposed to mean?', 'start': 202.138, 'duration': 3.062}, {'end': 212.56, 'text': 'I think a lot of us have an intuitive feel for what we want it to mean, at least in the sense of what experiment would you do to verify it?', 'start': 206.158, 'duration': 6.402}], 'summary': 'The shadow of a cube can be a regular hexagon, with its area being the square root of 3 times the area of one of its square faces.', 'duration': 38.984, 'max_score': 173.576, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0173576.jpg'}, {'end': 268.707, 'src': 'embed', 'start': 236.587, 'weight': 4, 'content': [{'end': 237.71, 'text': 'approaching infinitely many.', 'start': 236.587, 'duration': 1.123}, {'end': 242.564, 'text': 'Even still the sticklers among you could complain.', 'start': 240.703, 'duration': 1.861}, {'end': 247.788, 'text': "that doesn't really answer the question, because it leaves open the issue of how we're defining a random toss.", 'start': 242.564, 'duration': 5.224}, {'end': 255.374, 'text': 'The proper way to answer this, if we want it to be more formal, would be to first describe the space of all possible orientations,', 'start': 248.409, 'duration': 6.965}, {'end': 257.515, 'text': 'which mathematicians have actually given a fancy name.', 'start': 255.374, 'duration': 2.141}, {'end': 261.618, 'text': 'They call it SO typically defined in terms of a certain family of 3x3 matrices.', 'start': 257.595, 'duration': 4.023}, {'end': 268.707, 'text': 'And the question we want to answer is what probability distribution are we putting to this entire space?', 'start': 263.103, 'duration': 5.604}], 'summary': 'Mathematicians define the space of possible orientations as so, typically in terms of a family of 3x3 matrices, to determine the probability distribution.', 'duration': 32.12, 'max_score': 236.587, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0236587.jpg'}, {'end': 338.473, 'src': 'embed', 'start': 307.773, 'weight': 5, 'content': [{'end': 312.998, 'text': 'And, as with any lesson on problem solving, the goal here is not to get to the answer as quickly as we can,', 'start': 307.773, 'duration': 5.225}, {'end': 315.68, 'text': 'but hopefully for you to feel like you found the answer yourself.', 'start': 312.998, 'duration': 2.682}, {'end': 320.705, 'text': "So if ever there's a point when you feel like you might have an idea, give yourself the freedom to pause and try to think it through.", 'start': 315.961, 'duration': 4.744}, {'end': 329.744, 'text': 'As a first step, and this is really independent of any particular problem-solving styles.', 'start': 325.7, 'duration': 4.044}, {'end': 332.066, 'text': 'just anytime you find a hard question,', 'start': 329.744, 'duration': 2.322}, {'end': 338.473, 'text': "a good thing that you can do is ask what's the simplest possible non-trivial variant of the problem that you can try to solve?", 'start': 332.066, 'duration': 6.407}], 'summary': 'Goal: encourage problem solving by aiming for self-discovered answers and considering simplified problem variants.', 'duration': 30.7, 'max_score': 307.773, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0307773.jpg'}], 'start': 73.325, 'title': 'Cube shadow area', 'summary': 'Covers finding the average shadow area for a cube, considering size, orientation, and light source position, with examples of square and hexagon shadows. it also discusses understanding cube orientations through random tosses to determine experimental mean and emphasizes problem-solving mindset.', 'chapters': [{'end': 191.192, 'start': 73.325, 'title': 'Finding average shadow area of a cube', 'summary': 'Explores the challenge of finding the average area for the shadow of a cube, considering the influence of size, orientation, and light source position, with specific examples of square and hexagon shadows.', 'duration': 117.867, 'highlights': ['The shadow of a cube is influenced by its size, orientation, and light source position, with the average area computed over all possible orientations for a particular size.', 'The specific example of a cube with two faces parallel to the ground results in a flat projection shadow that is simply a square, with an area of S squared.', 'When the long diagonal of the cube is parallel to the direction of the light, the shadow takes the form of a regular hexagon, with an area of square root of 3 times the area of one of the square faces.']}, {'end': 329.744, 'start': 191.673, 'title': 'Understanding cube orientations', 'summary': 'Discusses the concept of averaging over all possible orientations of a cube by repeating random tosses and determining the experimental mean, while also addressing the issue of defining a random toss and the space of all possible orientations. it emphasizes the importance of problem-solving mindset and encourages pausing to think through ideas.', 'duration': 138.071, 'highlights': ['The chapter discusses the concept of averaging over all possible orientations of a cube by repeating random tosses and determining the experimental mean.', 'Addressing the issue of defining a random toss and the space of all possible orientations.', 'Emphasizing the importance of problem-solving mindset and encouraging pausing to think through ideas.']}], 'duration': 256.419, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi073325.jpg', 'highlights': ['The shadow of a cube is influenced by its size, orientation, and light source position', 'The specific example of a cube with two faces parallel to the ground results in a flat projection shadow that is simply a square, with an area of S squared', 'When the long diagonal of the cube is parallel to the direction of the light, the shadow takes the form of a regular hexagon, with an area of square root of 3 times the area of one of the square faces', 'The chapter discusses the concept of averaging over all possible orientations of a cube by repeating random tosses and determining the experimental mean', 'Addressing the issue of defining a random toss and the space of all possible orientations', 'Emphasizing the importance of problem-solving mindset and encouraging pausing to think through ideas']}, {'end': 825.291, 'segs': [{'end': 505.611, 'src': 'embed', 'start': 471.933, 'weight': 0, 'content': [{'end': 474.555, 'text': 'theta, which remembers the adjacent over the hypotenuse.', 'start': 471.933, 'duration': 2.622}, {'end': 478.179, 'text': "It's literally the ratio between the size of this shadow and the size of the slice.", 'start': 474.635, 'duration': 3.544}, {'end': 484.265, 'text': 'So, the factor by which the slice gets squished down in this direction is exactly cosine of theta.', 'start': 478.96, 'duration': 5.305}, {'end': 490.142, 'text': 'And if we broaden our view to the entire square, all the slices in that direction get scaled by the same factor.', 'start': 485.199, 'duration': 4.943}, {'end': 494.164, 'text': 'But in the other direction, in the one perpendicular to that slice,', 'start': 490.682, 'duration': 3.482}, {'end': 498.127, 'text': 'there is no stretching or squishing because the face is not at all tilted in that direction.', 'start': 494.164, 'duration': 3.963}, {'end': 505.611, 'text': 'So overall, the two-dimensional shadow of our two-dimensional face should also be scaled down by this factor of a cosine of theta.', 'start': 498.687, 'duration': 6.924}], 'summary': 'The two-dimensional shadow of a face gets scaled down by the cosine of the angle theta.', 'duration': 33.678, 'max_score': 471.933, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0471933.jpg'}, {'end': 554.051, 'src': 'embed', 'start': 526.596, 'weight': 3, 'content': [{'end': 531.057, 'text': "But of course, we don't want to consider the shadow to have negative area, at least not in a problem like this.", 'start': 526.596, 'duration': 4.461}, {'end': 533.258, 'text': "So there's two different ways you could solve this.", 'start': 531.777, 'duration': 1.481}, {'end': 538.319, 'text': 'You could say we only ever want to consider the normal vector that is pointing up, that has a positive z component.', 'start': 533.298, 'duration': 5.021}, {'end': 544.581, 'text': 'Or more simply, we could say, just take the absolute value of that cosine, and that gives us a valid formula.', 'start': 539.039, 'duration': 5.542}, {'end': 550.829, 'text': "So Bob's happy because he has a precise formula describing the area of the shadow.", 'start': 546.926, 'duration': 3.903}, {'end': 554.051, 'text': 'But Alice starts to think about it a little bit differently.', 'start': 551.61, 'duration': 2.441}], 'summary': 'Two ways to solve shadow area: positive z component or absolute value of cosine. bob has precise formula, alice has different perspective.', 'duration': 27.455, 'max_score': 526.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0526596.jpg'}, {'end': 592.799, 'src': 'embed', 'start': 565.06, 'weight': 4, 'content': [{'end': 568.443, 'text': 'And what stands out to her is that both of these are linear transformations.', 'start': 565.06, 'duration': 3.383}, {'end': 570.324, 'text': 'That means that, in principle,', 'start': 569.083, 'duration': 1.241}, {'end': 576.249, 'text': 'you could describe each one of them with a matrix and that the overall transformation would look like the product of those two matrices.', 'start': 570.324, 'duration': 5.925}, {'end': 582.913, 'text': 'What Alice knows from one of her favorite subjects, linear algebra, is that if you take some shape and you consider its area,', 'start': 576.969, 'duration': 5.944}, {'end': 590.237, 'text': 'then you apply some linear transformation, then the area of that output looks like some constant times the original area of the shape.', 'start': 582.913, 'duration': 7.324}, {'end': 592.799, 'text': 'More specifically, we have a name for that constant.', 'start': 590.857, 'duration': 1.942}], 'summary': 'Linear transformations can be described by matrices and result in a constant times the original area of the shape.', 'duration': 27.739, 'max_score': 565.06, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0565060.jpg'}, {'end': 642.999, 'src': 'embed', 'start': 618.944, 'weight': 1, 'content': [{'end': 627.009, 'text': 'Instead, the thing that she writes down is how this proportionality constant between our original shape and its shadow does not depend on the original shape.', 'start': 618.944, 'duration': 8.065}, {'end': 632.593, 'text': "We could be talking about the shadow of this cat outline, or anything else, and the size of it doesn't really matter.", 'start': 627.389, 'duration': 5.204}, {'end': 637.396, 'text': "The only thing affecting that proportionality constant is what transformation we're applying.", 'start': 633.133, 'duration': 4.263}, {'end': 642.999, 'text': 'which in this context means we could write it down as some factor that depends on the rotation being applied to the shape.', 'start': 637.816, 'duration': 5.183}], 'summary': 'Proportionality constant between original shape and shadow is independent of shape size, only depends on applied rotation.', 'duration': 24.055, 'max_score': 618.944, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0618944.jpg'}, {'end': 698.703, 'src': 'embed', 'start': 668.171, 'weight': 2, 'content': [{'end': 671.575, 'text': "Now it's easy to look at this and say, okay, well Alice isn't really doing anything then.", 'start': 668.171, 'duration': 3.404}, {'end': 675.378, 'text': 'Of course the area of the shadow is proportional to the area of the original shape.', 'start': 671.775, 'duration': 3.603}, {'end': 677.28, 'text': "They're both two-dimensional quantities.", 'start': 675.839, 'duration': 1.441}, {'end': 679.483, 'text': 'They should both scale like two-dimensional things.', 'start': 677.34, 'duration': 2.143}, {'end': 685.657, 'text': 'But keep in mind, this would not at all be true if we were dealing with the harder case that has a closer light source.', 'start': 680.154, 'duration': 5.503}, {'end': 687.938, 'text': 'In that case, the projection is not linear.', 'start': 685.997, 'duration': 1.941}, {'end': 693.16, 'text': 'So, for example, if I rotate this cat so that its tail ends up quite close to the light source,', 'start': 688.538, 'duration': 4.622}, {'end': 698.703, 'text': 'then if I stretch the original shape uniformly in the x-direction, say by a factor of 1.5,', 'start': 693.16, 'duration': 5.543}], 'summary': 'The shadow area is proportional to the original shape; it scales like two-dimensional things.', 'duration': 30.532, 'max_score': 668.171, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0668171.jpg'}], 'start': 329.744, 'title': 'Shadow proportionality in 3d space', 'summary': 'Discusses finding area of shadow on a cube using trigonometry, linear transformations and area proportionality, and the universal proportionality constant that is independent of the size and shape of the 2d object, providing a comprehensive understanding of shadow proportionality in 3d space.', 'chapters': [{'end': 544.581, 'start': 329.744, 'title': 'Finding area of shadow on a cube', 'summary': 'Discusses finding the area of the shadow cast by a tilted face of a cube using trigonometry, showing that the area is scaled down by the cosine of the angle theta, and addressing the issue of negative area when theta is greater than 90 degrees.', 'duration': 214.837, 'highlights': ["The area of the shadow is scaled down by the cosine of the angle theta, as demonstrated through a two-dimensional variant of the problem, providing a formula for the shadow's area (s squared * |cos(theta)|) based on the angle theta.", "The chapter explains the issue of negative area when theta is greater than 90 degrees and presents two solutions: considering only the normal vector pointing up or taking the absolute value of the cosine, ensuring a valid formula for the shadow's area.", 'The discussion emphasizes the use of trigonometry to find the area of the shadow, showing that the area is zero when theta is 90 degrees and equal to the area of the face when theta is 0 degrees, providing a detailed and proof-oriented approach to deriving the formula.']}, {'end': 666.712, 'start': 546.926, 'title': 'Linear transformations and area proportionality', 'summary': 'Discusses how linear transformations in 3d space can be described using matrices, and the concept of determinant as the proportionality constant between the original shape and its shadow, which does not depend on the original shape.', 'duration': 119.786, 'highlights': ['Alice realizes that both the rotation and flat projection are linear transformations, describable with matrices, and their product represents the overall transformation.', 'In linear algebra, the area of a shape under a linear transformation is proportional to the original area, where the proportionality constant is the determinant of the transformation.', 'The proportionality constant between the original shape and its shadow does not depend on the original shape, but on the transformation being applied, specifically related to the rotation in this context.']}, {'end': 825.291, 'start': 668.171, 'title': 'Shadow proportionality and universality', 'summary': 'Discusses the proportionality of shadow areas to original shapes, the non-linear projection with a closer light source, and the universal proportionality constant that is independent of the size and shape of the 2d object, leading to a more elegant deduction of the constant.', 'duration': 157.12, 'highlights': ['The area of the shadow is proportional to the area of the original shape, and they both scale like two-dimensional things.', 'The projection is not linear with a closer light source, leading to disproportionate effects on the ultimate shadow.', 'The universal proportionality constant is independent of the size and shape of the 2D object, providing a more elegant deduction of the constant.']}], 'duration': 495.547, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0329744.jpg', 'highlights': ["The area of the shadow is scaled down by the cosine of the angle theta, providing a formula for the shadow's area (s squared * |cos(theta)|).", 'The proportionality constant between the original shape and its shadow does not depend on the original shape, but on the transformation being applied.', 'The area of the shadow is proportional to the area of the original shape, and they both scale like two-dimensional things.', 'The chapter explains the issue of negative area when theta is greater than 90 degrees and presents two solutions: considering only the normal vector pointing up or taking the absolute value of the cosine.', 'Alice realizes that both the rotation and flat projection are linear transformations, describable with matrices, and their product represents the overall transformation.']}, {'end': 1306.055, 'segs': [{'end': 893.348, 'src': 'embed', 'start': 866.342, 'weight': 0, 'content': [{'end': 869.725, 'text': 'how on earth we would average that across all of the different orientations.', 'start': 866.342, 'duration': 3.383}, {'end': 875.332, 'text': 'But Alice has about three clever insights through this whole problem, and this is the first one of them.', 'start': 870.606, 'duration': 4.726}, {'end': 880.779, 'text': 'She says actually, if we think about the whole cube, not just a pair of faces,', 'start': 875.933, 'duration': 4.846}, {'end': 887.969, 'text': 'we can conclude that the area of the shadow for a given orientation is exactly one half the sum of the areas of all of the faces.', 'start': 880.779, 'duration': 7.19}, {'end': 893.348, 'text': 'Intuitively, you can maybe guess that half of them are bathed in the light and half of them are not.', 'start': 889.406, 'duration': 3.942}], 'summary': 'Alice suggests that the shadow area is half the sum of all faces, with three clever insights.', 'duration': 27.006, 'max_score': 866.342, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0866342.jpg'}, {'end': 940.725, 'src': 'embed', 'start': 908.216, 'weight': 3, 'content': [{'end': 913.319, 'text': 'So every point in that shadow corresponds to exactly two faces above it.', 'start': 908.216, 'duration': 5.103}, {'end': 916.079, 'text': "Well, okay, that's not exactly true.", 'start': 914.318, 'duration': 1.761}, {'end': 922.305, 'text': "If that beam of light happened to go through the edge of one of the squares, there's a little bit of ambiguity on how many faces it's passing.", 'start': 916.22, 'duration': 6.085}, {'end': 928.91, 'text': "But those account for zero area inside the shadow, so we're safe to ignore them if the thing we're trying to do is compute the area.", 'start': 922.765, 'duration': 6.145}, {'end': 940.725, 'text': 'If Alice is pressed and she needs to justify why exactly this is true which is important for understanding how the problem might generalize she can appeal to the idea of convexity.', 'start': 931.3, 'duration': 9.425}], 'summary': 'Shadow points correspond to two faces, with some ambiguity near edges. alice justifies using convexity for generalization.', 'duration': 32.509, 'max_score': 908.216, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0908216.jpg'}, {'end': 978.183, 'src': 'embed', 'start': 947.269, 'weight': 2, 'content': [{'end': 951.531, 'text': 'they never dent inward, but mathematicians have a pretty clever way of formalizing it.', 'start': 947.269, 'duration': 4.262}, {'end': 952.972, 'text': "that's helpful for actual proofs.", 'start': 951.531, 'duration': 1.441}, {'end': 961.495, 'text': 'They say that a set is convex if the line that connects any two points inside that set is entirely contained within the set itself.', 'start': 953.712, 'duration': 7.783}, {'end': 969.619, 'text': 'So a square is convex because no matter where you put two points inside that square, the line connecting them is entirely contained inside the square.', 'start': 962.136, 'duration': 7.483}, {'end': 972.66, 'text': 'But something like the symbol pi is not convex.', 'start': 970.239, 'duration': 2.421}, {'end': 978.183, 'text': 'I can easily find two different points so that the line connecting them has to peak outside of the set itself.', 'start': 972.84, 'duration': 5.343}], 'summary': 'Mathematicians formalize convexity, aiding in proofs. convex set contains line between points. a square is convex, but pi symbol is not.', 'duration': 30.914, 'max_score': 947.269, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0947269.jpg'}, {'end': 1266.197, 'src': 'embed', 'start': 1233.595, 'weight': 1, 'content': [{'end': 1237.577, 'text': 'Like, of course the average shadow area should be proportional to the surface area.', 'start': 1233.595, 'duration': 3.982}, {'end': 1241.959, 'text': "They're both two-dimensional quantities, so they should scale in lockstep with each other.", 'start': 1237.957, 'duration': 4.002}, {'end': 1244.38, 'text': "I mean, it's not obvious.", 'start': 1241.979, 'duration': 2.401}, {'end': 1247.141, 'text': "After all, for a closer light source, it simply wouldn't be true.", 'start': 1244.64, 'duration': 2.501}, {'end': 1254.627, 'text': 'And also, this business where we added up the grid column by column versus row by row is a little more nuanced than it might look at first.', 'start': 1248.081, 'duration': 6.546}, {'end': 1258.15, 'text': "There's a subtle, hidden assumption underlying all of this,", 'start': 1255.187, 'duration': 2.963}, {'end': 1266.197, 'text': 'which carries a special significance when we choose to revisit the question of what probability distribution is being taken across the space of all orientations.', 'start': 1258.15, 'duration': 8.047}], 'summary': 'Surface area and shadow area scale proportionally, but with nuanced assumptions and implications for probability distributions.', 'duration': 32.602, 'max_score': 1233.595, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01233595.jpg'}, {'end': 1294.079, 'src': 'embed', 'start': 1268.732, 'weight': 4, 'content': [{'end': 1275.397, 'text': "the reason that it's not obvious is that the significance of this result right here is not merely that these two values are proportional.", 'start': 1268.732, 'duration': 6.665}, {'end': 1281.442, 'text': "It's that an analogous fact will hold true for any convex solids and, crucially,", 'start': 1276.038, 'duration': 5.404}, {'end': 1287.727, 'text': "the actual content of what Alice has built up so far is that it'll be the same proportionality, constant across all of them.", 'start': 1281.442, 'duration': 6.285}, {'end': 1294.079, 'text': 'Now if you really mull over that, some of you may be able to predict the way that Alice is able to finish things off from here.', 'start': 1289.232, 'duration': 4.847}], 'summary': 'The significance lies in the universal proportionality of values for convex solids.', 'duration': 25.347, 'max_score': 1268.732, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01268732.jpg'}], 'start': 825.291, 'title': 'Shadow area and convexity', 'summary': 'Explores the relationship between the shadow area of a cube and its faces, demonstrating that the shadow area is half the sum of the areas of all faces. it also delves into the concept of convexity and its influence on shadow areas, highlighting a proportional relationship between average shadow area and surface area for convex solids.', 'chapters': [{'end': 928.91, 'start': 825.291, 'title': 'Shadow area of cube', 'summary': 'Discusses the relationship between the area of the shadow of a cube and the area of its individual faces, concluding that the area of the shadow for a given orientation is exactly one half the sum of the areas of all faces, supported by the insight that every point in the shadow corresponds to exactly two faces above it due to the passage of light through the cube.', 'duration': 103.619, 'highlights': ["Alice's insight that the area of the shadow for a given orientation is exactly one half the sum of the areas of all of the faces, supported by the fact that every point in the shadow corresponds to exactly two faces above it due to the passage of light through the cube.", 'The discussion about the overlap between the shadows of different faces, with the observation that the area of overlap seems tricky to think about, and the challenge of averaging it across all different orientations.', 'The consideration of a particular ray of light passing through the cube at exactly two points, supporting the conclusion that the area of the shadow for a given orientation is one half the sum of the areas of all faces.']}, {'end': 1306.055, 'start': 931.3, 'title': 'The convexity and shadows', 'summary': 'Discusses the concept of convexity and its application in understanding the shadow areas of convex solids, revealing a proportional relationship between the average shadow area and the surface area, with implications for all convex solids.', 'duration': 374.755, 'highlights': ['A set is convex if the line connecting any two points inside that set is entirely contained within the set itself, illustrated by the example of a square being convex and the symbol pi not being convex.', 'The average of the sum of the face shadows is the same as the sum of the average of the face shadows, leading to the revelation that the average shadow area is proportional to the surface area of the convex solid.', "The significance lies in the fact that an analogous fact will hold true for any convex solids, with the same proportionality constant across all of them, hinting at the broader implications of Alice's findings."]}], 'duration': 480.764, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi0825291.jpg', 'highlights': ["The area of the shadow for a given orientation is exactly one half the sum of the areas of all of the faces. (Alice's insight)", 'The average shadow area is proportional to the surface area of the convex solid. (Revelation)', 'A set is convex if the line connecting any two points inside that set is entirely contained within the set itself. (Definition)', 'The consideration of a particular ray of light passing through the cube at exactly two points supports the conclusion about the shadow area. (Supporting evidence)', 'An analogous fact will hold true for any convex solids, with the same proportionality constant across all of them. (Broader implications)']}, {'end': 1756.798, 'segs': [{'end': 1344.095, 'src': 'embed', 'start': 1319.808, 'weight': 0, 'content': [{'end': 1329.797, 'text': "which is how to take the formula that he found for the area of a square's shadow and taking the natural next step of trying to find the average of that square's shadow averaged over all possible orientations.", 'start': 1319.808, 'duration': 9.989}, {'end': 1339.191, 'text': "So the way Bob starts, if he's thinking about all the different possible orientations for this square,", 'start': 1334.508, 'duration': 4.683}, {'end': 1344.095, 'text': 'is to ask what are all the different normal vectors that that square can have in all these orientations?', 'start': 1339.191, 'duration': 4.904}], 'summary': "Bob aims to find the average area of a square's shadow across all orientations.", 'duration': 24.287, 'max_score': 1319.808, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01319808.jpg'}, {'end': 1401.01, 'src': 'embed', 'start': 1373.385, 'weight': 1, 'content': [{'end': 1377.369, 'text': 'because in the uncountable infinity of points on the sphere, that would be zero and unhelpful.', 'start': 1373.385, 'duration': 3.984}, {'end': 1378.63, 'text': 'So instead,', 'start': 1378.07, 'duration': 0.56}, {'end': 1389.321, 'text': 'the more precise way to phrase this uniformity would be to say the probability that our normal vector lands in any given patch of area on the sphere should be proportional to that area itself.', 'start': 1378.63, 'duration': 10.691}, {'end': 1395.085, 'text': 'More specifically, it should equal the area of that little patch divided by the total surface area of the sphere.', 'start': 1389.901, 'duration': 5.184}, {'end': 1401.01, 'text': "If that's true, no matter what patch of area we're considering, that's what we mean by a uniform distribution on the sphere.", 'start': 1395.666, 'duration': 5.344}], 'summary': 'Uniform distribution on sphere: probability proportional to patch area', 'duration': 27.625, 'max_score': 1373.385, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01373385.jpg'}, {'end': 1448.402, 'src': 'embed', 'start': 1422.324, 'weight': 3, 'content': [{'end': 1426.466, 'text': "It's only dependent on the cosine of the angle between that normal vector and the vertical.", 'start': 1422.324, 'duration': 4.142}, {'end': 1427.847, 'text': 'Which is kind of neat.', 'start': 1427.126, 'duration': 0.721}, {'end': 1430.047, 'text': 'All those shadows are genuinely different shapes.', 'start': 1427.947, 'duration': 2.1}, {'end': 1430.828, 'text': "They're not the same.", 'start': 1430.088, 'duration': 0.74}, {'end': 1433.469, 'text': 'But the area of each of them will be the same.', 'start': 1431.248, 'duration': 2.221}, {'end': 1440.014, 'text': 'What this means is that when Bob wants this average shadow area over all possible orientations,', 'start': 1435.049, 'duration': 4.965}, {'end': 1446.62, 'text': 'all he really needs to know is the average value of this absolute value of cosine of theta for all different possible normal vectors,', 'start': 1440.014, 'duration': 6.606}, {'end': 1448.402, 'text': 'all different possible points on the sphere.', 'start': 1446.62, 'duration': 1.782}], 'summary': "Shadows' areas are different shapes, but have same area. bob can find average shadow area by averaging absolute value of cosine of theta for all possible normal vectors and points on the sphere.", 'duration': 26.078, 'max_score': 1422.324, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01422324.jpg'}, {'end': 1605.842, 'src': 'embed', 'start': 1582.574, 'weight': 4, 'content': [{'end': 1589.496, 'text': 'Based on our assumption that the distribution along the sphere should be uniform, that probability comes down to knowing the area of this band.', 'start': 1582.574, 'duration': 6.922}, {'end': 1596.679, 'text': 'More specifically, the chances that a randomly chosen vector lands in that band should be that area divided by the total surface area of the sphere.', 'start': 1589.977, 'duration': 6.702}, {'end': 1598.339, 'text': 'To figure out that area.', 'start': 1597.359, 'duration': 0.98}, {'end': 1605.842, 'text': "let's first think of the radius of that band, which, if the radius of our sphere is 1, is definitely going to be smaller than 1..", 'start': 1598.339, 'duration': 7.503}], 'summary': 'Probability of vector landing in band is area of band / total surface area of sphere.', 'duration': 23.268, 'max_score': 1582.574, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01582574.jpg'}, {'end': 1741.384, 'src': 'embed', 'start': 1714.522, 'weight': 5, 'content': [{'end': 1719.345, 'text': 'And to make it a little more analogous to calculus with integrals, let me just swap the main terms inside the sum here.', 'start': 1714.522, 'duration': 4.823}, {'end': 1725.85, 'text': "What we now have, this sum that's going to approximate the answer to our question, is almost what an integral is.", 'start': 1719.906, 'duration': 5.944}, {'end': 1734, 'text': "Instead of writing the sigma for sum, we write the integral symbol, this kind of elongated Leibnizian s, showing us that we're going from zero to pi.", 'start': 1726.676, 'duration': 7.324}, {'end': 1741.384, 'text': 'And instead of describing the step size as delta theta, a concrete finite amount, we instead describe it as d theta,', 'start': 1734.68, 'duration': 6.704}], 'summary': 'Analogous to calculus with integrals, swapping terms inside the sum approximates the answer as an integral.', 'duration': 26.862, 'max_score': 1714.522, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01714522.jpg'}], 'start': 1306.756, 'title': 'Computation and approximation of shadow area', 'summary': "Explores bob's computation to find the average area of a square's shadow over all possible orientations based on the uniform distribution of normal vectors on a sphere. it also discusses the process of approximating the shadow area of a square using integrals and probabilities, demonstrating the mathematical steps involved and the relationship between the angles, probabilities, and surface areas of a sphere.", 'chapters': [{'end': 1495.992, 'start': 1306.756, 'title': "Bob's computation on shadow area", 'summary': "Delves into bob's computation to find the average area of a square's shadow over all possible orientations, based on the uniform distribution of normal vectors on a sphere.", 'duration': 189.236, 'highlights': ["Bob aims to find the average area of a square's shadow over all possible orientations by considering the uniform distribution of normal vectors on a sphere.", 'The probability that the normal vector lands in any given patch of area on the sphere should be proportional to that area itself.', 'The area of each shadow is dependent on the cosine of the angle between the normal vector and the vertical, resulting in different shapes with the same area.']}, {'end': 1756.798, 'start': 1496.592, 'title': 'Approximating shadow area', 'summary': 'Discusses the process of approximating the shadow area of a square using integrals and probabilities, demonstrating the mathematical steps involved and the relationship between the angles, probabilities, and surface areas of a sphere.', 'duration': 260.206, 'highlights': ['The process of approximating the shadow area of a square using integrals and probabilities is discussed.', 'The relationship between the angles, probabilities, and surface areas of a sphere is explained.', 'The use of integrals and probabilities in the approximation process is highlighted.']}], 'duration': 450.042, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01306756.jpg', 'highlights': ["Bob aims to find the average area of a square's shadow over all possible orientations by considering the uniform distribution of normal vectors on a sphere.", 'The probability that the normal vector lands in any given patch of area on the sphere should be proportional to that area itself.', 'The process of approximating the shadow area of a square using integrals and probabilities is discussed.', 'The area of each shadow is dependent on the cosine of the angle between the normal vector and the vertical, resulting in different shapes with the same area.', 'The relationship between the angles, probabilities, and surface areas of a sphere is explained.', 'The use of integrals and probabilities in the approximation process is highlighted.']}, {'end': 2381.617, 'segs': [{'end': 1795.964, 'src': 'embed', 'start': 1774.213, 'weight': 1, 'content': [{'end': 1783.478, 'text': "what Bob finds after doing this is the surprisingly clean fact that the average area for a square's shadow is precisely one half the area of that square.", 'start': 1774.213, 'duration': 9.265}, {'end': 1787.44, 'text': "This is the mystery constant, which Alice doesn't yet know.", 'start': 1784.598, 'duration': 2.842}, {'end': 1792.702, 'text': "If Bob were to look over her shoulder and see the work that she's done, he could finish out the problem right now.", 'start': 1788.22, 'duration': 4.482}, {'end': 1795.964, 'text': 'He plugs in the constant that he just found and he knows the final answer.', 'start': 1792.962, 'duration': 3.002}], 'summary': "The average area for a square's shadow is precisely half the area of the square, a mystery constant in a problem solved by bob.", 'duration': 21.751, 'max_score': 1774.213, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01774213.jpg'}, {'end': 1806.203, 'src': 'heatmap', 'start': 1774.213, 'weight': 1, 'content': [{'end': 1783.478, 'text': "what Bob finds after doing this is the surprisingly clean fact that the average area for a square's shadow is precisely one half the area of that square.", 'start': 1774.213, 'duration': 9.265}, {'end': 1787.44, 'text': "This is the mystery constant, which Alice doesn't yet know.", 'start': 1784.598, 'duration': 2.842}, {'end': 1792.702, 'text': "If Bob were to look over her shoulder and see the work that she's done, he could finish out the problem right now.", 'start': 1788.22, 'duration': 4.482}, {'end': 1795.964, 'text': 'He plugs in the constant that he just found and he knows the final answer.', 'start': 1792.962, 'duration': 3.002}, {'end': 1806.203, 'text': 'And now, finally, with all of this as backdrop, what is it that Alice does to carry out the final solution?', 'start': 1800.58, 'duration': 5.623}], 'summary': 'Bob finds the mystery constant, and if he shares it with alice, they can solve the problem together.', 'duration': 31.99, 'max_score': 1774.213, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01774213.jpg'}, {'end': 1844.436, 'src': 'embed', 'start': 1817.188, 'weight': 0, 'content': [{'end': 1821.67, 'text': 'But this is a case where the generalization itself draws her to a quantitative result.', 'start': 1817.188, 'duration': 4.482}, {'end': 1827.391, 'text': "Remember the substance of what she's found so far is that if you look at any convex solid,", 'start': 1822.31, 'duration': 5.081}, {'end': 1831.633, 'text': 'then the average area for its shadow is going to be proportional to its surface area.', 'start': 1827.391, 'duration': 4.242}, {'end': 1836.454, 'text': "And critically, it'll be the same proportionality constant across all of these solids.", 'start': 1832.073, 'duration': 4.381}, {'end': 1844.436, 'text': 'So all Alice needs to do is find just a single convex solid out there where she already knows the average area of its shadow.', 'start': 1837.054, 'duration': 7.382}], 'summary': 'The average shadow area of any convex solid is proportional to its surface area, with a constant ratio.', 'duration': 27.248, 'max_score': 1817.188, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01817188.jpg'}, {'end': 1940.419, 'src': 'embed', 'start': 1912.461, 'weight': 2, 'content': [{'end': 1919.405, 'text': "Now, that's all very pretty, but some of you might complain that this isn't really a valid argument, because spheres don't have flat faces.", 'start': 1912.461, 'duration': 6.944}, {'end': 1923.648, 'text': "When I said Alice's argument generalizes to any convex solid.", 'start': 1920.066, 'duration': 3.582}, {'end': 1928.952, 'text': 'if we actually look at the argument itself, it definitely depends on the use of a finite number of flat faces.', 'start': 1923.648, 'duration': 5.304}, {'end': 1931.853, 'text': 'For example, if we were mapping it to a dodecahedron,', 'start': 1929.512, 'duration': 2.341}, {'end': 1940.419, 'text': 'you would start by saying that the area of a particular shadow of that dodecahedron looks like exactly one half times the sum of the areas of the shadows of all its faces.', 'start': 1931.853, 'duration': 8.566}], 'summary': "Alice's argument applies to convex solids with finite flat faces, demonstrated with a dodecahedron example.", 'duration': 27.958, 'max_score': 1912.461, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi01912461.jpg'}, {'end': 2097.045, 'src': 'embed', 'start': 2068.572, 'weight': 5, 'content': [{'end': 2074.014, 'text': "I think there's an important way that popularizations of math differ from the feeling of actually doing math.", 'start': 2068.572, 'duration': 5.442}, {'end': 2080.898, 'text': "There's this bias towards showing the slick proofs, the arguments with some clever keen insight that lets you avoid doing calculations.", 'start': 2074.315, 'duration': 6.583}, {'end': 2087.201, 'text': "I could just be projecting, since I'm very guilty of this, but what I can tell you, sitting on the other side of the screen here,", 'start': 2081.578, 'duration': 5.623}, {'end': 2091.803, 'text': "is that it feels a lot more attractive to make a video about Alice's approach than Bob's.", 'start': 2087.201, 'duration': 4.602}, {'end': 2097.045, 'text': "For one thing, in Alice's approach, the line of reasoning is fun, it has these nice aha moments.", 'start': 2092.422, 'duration': 4.623}], 'summary': 'Popularizations of math emphasize slick proofs and clever insights, but actual math involves calculations and fun reasoning.', 'duration': 28.473, 'max_score': 2068.572, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi02068572.jpg'}, {'end': 2194.243, 'src': 'embed', 'start': 2164.873, 'weight': 4, 'content': [{'end': 2170.377, 'text': 'Right here at the top of page 11, you can see what is essentially the same integral that you and I set up in the middle.', 'start': 2164.873, 'duration': 5.504}, {'end': 2177.142, 'text': 'On the other hand, the whole framing of the paper is to find a general fact, not something specific like the case of a cube.', 'start': 2171.317, 'duration': 5.825}, {'end': 2183.371, 'text': 'So if we were asking the question which of these two mindsets correlates with the act of discovering new math?', 'start': 2177.745, 'duration': 5.626}, {'end': 2186.335, 'text': 'the right answer would almost certainly have to be a blend of both.', 'start': 2183.371, 'duration': 2.964}, {'end': 2194.243, 'text': "But I would suggest that many people don't sign enough weight to the part of that blend where you're eager to dive into calculations.", 'start': 2187.276, 'duration': 6.967}], 'summary': 'Balancing general concepts with specific cases leads to new mathematical discoveries.', 'duration': 29.37, 'max_score': 2164.873, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi02164873.jpg'}, {'end': 2269.28, 'src': 'embed', 'start': 2236.896, 'weight': 3, 'content': [{'end': 2241.037, 'text': 'In their own words, recognizing this completely reshaped their outlook and their results.', 'start': 2236.896, 'duration': 4.141}, {'end': 2246.722, 'text': 'And if you look at the famous mathematicians through history, you know Newton, Euler, Gauss, all of them.', 'start': 2241.977, 'duration': 4.745}, {'end': 2250.627, 'text': 'they all have this seemingly infinite patience for doing tedious calculations.', 'start': 2246.722, 'duration': 3.905}, {'end': 2262.661, 'text': 'The irony of being biased to show insights that let us avoid calculations is that the way people often train up the intuitions to find those insights in the first place is by doing piles and piles of calculations.', 'start': 2251.328, 'duration': 11.333}, {'end': 2269.28, 'text': 'All that said, something would definitely be missing without the Alice mindset here.', 'start': 2265.017, 'duration': 4.263}], 'summary': 'Famous mathematicians demonstrated infinite patience for tedious calculations, reshaping outlook and results.', 'duration': 32.384, 'max_score': 2236.896, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/ltLUadnCyi0/pics/ltLUadnCyi02236896.jpg'}], 'start': 1758.999, 'title': 'Shadow area and math popularization', 'summary': "Discusses the surprising fact that the average area for a square's shadow is precisely one half the area of the square, and how alice generalizes this finding to show that for any convex solid, the average shadow area is proportional to its surface area. it also explores the bias in popularizations of math towards slick proofs, the importance of diving into calculations, and the impact of alice and bob mindsets on problem solving, with emphasis on the significance of doing computations.", 'chapters': [{'end': 2040.666, 'start': 1758.999, 'title': 'Shadow area proportional to surface area', 'summary': "Discusses how bob and alice use calculus and geometric reasoning to find the surprising fact that the average area for a square's shadow is precisely one half the area of the square, and how alice generalizes this finding to show that for any convex solid, the average shadow area is proportional to its surface area.", 'duration': 281.667, 'highlights': ["Alice finds that the average area for a square's shadow is precisely one half the area of the square.", 'Alice generalizes the finding to show that for any convex solid, the average shadow area is proportional to its surface area.', 'The use of a finite number of flat faces in the argument is critical for the validity of the generalization.']}, {'end': 2381.617, 'start': 2041.046, 'title': 'Math popularization and problem solving', 'summary': "Discusses the bias in popularizations of math towards slick proofs, the importance of diving into calculations, and the impact of alice and bob mindsets on problem solving, with emphasis on the significance of doing computations, as highlighted by a mathematician's podcast and historical mathematicians.", 'duration': 340.571, 'highlights': ['The bias in popularizations of math towards slick proofs can lead to a disingenuous focus on problem solving.', "The importance of diving into calculations is underscored by a mathematician's podcast and the historical mathematicians Newton, Euler, and Gauss.", 'The impact of Alice and Bob mindsets on problem solving is discussed, emphasizing the blend of both mindsets for discovering new math.']}], 'duration': 622.618, 'thumbnail': '', 'highlights': ['Alice generalizes the finding to show that for any convex solid, the average shadow area is proportional to its surface area.', "The average area for a square's shadow is precisely one half the area of the square.", 'The use of a finite number of flat faces in the argument is critical for the validity of the generalization.', "The importance of diving into calculations is underscored by a mathematician's podcast and the historical mathematicians Newton, Euler, and Gauss.", 'The impact of Alice and Bob mindsets on problem solving is discussed, emphasizing the blend of both mindsets for discovering new math.', 'The bias in popularizations of math towards slick proofs can lead to a disingenuous focus on problem solving.']}], 'highlights': ['The average shadow area is proportional to the surface area of the convex solid. (Revelation)', 'Alice generalizes the finding to show that for any convex solid, the average shadow area is proportional to its surface area.', "The area of the shadow for a given orientation is exactly one half the sum of the areas of all of the faces. (Alice's insight)", 'The shadow of a cube is influenced by its size, orientation, and light source position', 'The specific example of a cube with two faces parallel to the ground results in a flat projection shadow that is simply a square, with an area of S squared', "The area of the shadow is scaled down by the cosine of the angle theta, providing a formula for the shadow's area (s squared * |cos(theta)|)", "Bob aims to find the average area of a square's shadow over all possible orientations by considering the uniform distribution of normal vectors on a sphere", 'The process of approximating the shadow area of a square using integrals and probabilities is discussed', 'Understanding problem shapes leads to swift computations', 'The shadow area is proportional to the area of the original shape, and they both scale like two-dimensional things']}