title

Thinking outside the 10-dimensional box

description

Visualizing high-dimensional spheres to understand a surprising puzzle.
Help fund future projects: https://www.patreon.com/3blue1brown
This video was sponsored by Brilliant: https://brilliant.org/3b1b
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/high-d-thanks
Home page: https://www.3blue1brown.com/
Podcast! https://www.benbenandblue.com/
Check out Ben Eater's channel: https://www.youtube.com/user/eaterbc
------------------
Animations largely made using manim, a scrappy open source python library. https://github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
------------------
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, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended
Various social media stuffs:
Website: https://www.3blue1brown.com
Twitter: https://twitter.com/3Blue1Brown
Patreon: https://patreon.com/3blue1brown
Facebook: https://www.facebook.com/3blue1brown
Reddit: https://www.reddit.com/r/3Blue1Brown

detail

{'title': 'Thinking outside the 10-dimensional box', 'heatmap': [{'end': 1368.367, 'start': 1348.005, 'weight': 0.802}, {'end': 1513.11, 'start': 1496.84, 'weight': 1}], 'summary': 'Explores the interconnectedness of mathematics and geometric reasoning, limitations of views in higher dimensions, inner circle radius calculation, real estate analogy for x² + y² = 1, and counterintuitive behavior of real estate in higher dimensions.', 'chapters': [{'end': 84.865, 'segs': [{'end': 84.865, 'src': 'embed', 'start': 49.378, 'weight': 0, 'content': [{'end': 56.541, 'text': 'since it offers such a rich library of that special category of cleverness that involves connecting two seemingly disparate ideas.', 'start': 49.378, 'duration': 7.163}, {'end': 62.184, 'text': "And I don't just mean the general back and forth between pairs or triplets of numbers and spatial reasoning.", 'start': 57.361, 'duration': 4.823}, {'end': 66.226, 'text': 'I mean this specific one between sums of squares and circles and spheres.', 'start': 62.184, 'duration': 4.042}, {'end': 72.232, 'text': "It's at the heart of the video I made showing how pi is connected to number theory and primes,", 'start': 67.066, 'duration': 5.166}, {'end': 75.695, 'text': 'and the one showing how to visualize all possible Pythagorean triples.', 'start': 72.232, 'duration': 3.463}, {'end': 84.865, 'text': 'It also underlies the video on the Borsuk-Ulam theorem being used to solve what was basically a counting puzzle by using topological facts about spheres.', 'start': 76.396, 'duration': 8.469}], 'summary': 'The transcript discusses the connection between sums of squares and circles/spheres, and its application in videos about pi, pythagorean triples, and the borsuk-ulam theorem.', 'duration': 35.487, 'max_score': 49.378, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI49378.jpg'}], 'start': 4.289, 'title': 'Math and geometric reasoning', 'summary': 'Delves into the interconnectedness of mathematics and geometric reasoning, showcasing the visualization of shapes through numerical properties and the reciprocal relationship between geometric insights and analytical facts.', 'chapters': [{'end': 84.865, 'start': 4.289, 'title': 'Math and geometric reasoning', 'summary': 'Discusses the beauty of geometric reasoning in mathematics, exemplifying the connection between pairs or triplets of numbers and spatial concepts, with examples such as the visualization of circles and spheres using numerical properties, and the utilization of geometric insights to clarify analytic facts and vice versa.', 'duration': 80.576, 'highlights': ['The visualization of circles and spheres using numerical properties, such as x squared plus y squared equal to one, exemplifies the connection between pairs or triplets of numbers and spatial concepts.', 'Geometric insights clarify analytic facts and vice versa, offering a rich library of cleverness that involves connecting seemingly disparate ideas.', 'The videos showcasing the connection of pi to number theory and primes, as well as the visualization of possible Pythagorean triples, demonstrate the utilization of geometric insights to explore number theory and solve counting puzzles.']}], 'duration': 80.576, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI4289.jpg', 'highlights': ['The videos showcase the connection of pi to number theory and primes, as well as the visualization of possible Pythagorean triples, demonstrating the utilization of geometric insights to explore number theory and solve counting puzzles.', 'The visualization of circles and spheres using numerical properties exemplifies the connection between pairs or triplets of numbers and spatial concepts.', 'Geometric insights clarify analytic facts and vice versa, offering a rich library of cleverness that involves connecting seemingly disparate ideas.']}, {'end': 546.55, 'segs': [{'end': 175.105, 'src': 'embed', 'start': 149.332, 'weight': 0, 'content': [{'end': 154.935, 'text': 'Now, what I want to offer here is a hybrid between the purely geometric and the purely analytic views,', 'start': 149.332, 'duration': 5.603}, {'end': 161.518, 'text': 'a method for making the analytic reasoning a little more visual, in a way that generalizes to arbitrarily high dimensions.', 'start': 154.935, 'duration': 6.583}, {'end': 164.62, 'text': 'And to drive home the value of a tactic like this.', 'start': 162.238, 'duration': 2.382}, {'end': 175.105, 'text': 'I want to share with you a very famous example where analogies with two and three dimensions cannot help because of something extremely counterintuitive that only happens in higher dimensions.', 'start': 164.62, 'duration': 10.485}], 'summary': 'Introducing a hybrid method for visualizing high-dimensional data and illustrating its value with a counterintuitive example.', 'duration': 25.773, 'max_score': 149.332, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI149332.jpg'}, {'end': 316.618, 'src': 'embed', 'start': 270.5, 'weight': 5, 'content': [{'end': 278.161, 'text': "So ask yourself what's a nice way to think about this relation that x squared plus y squared is 1?", 'start': 270.5, 'duration': 7.661}, {'end': 286.727, 'text': 'I like to think of the value of x² as the real estate belonging to x and likewise the value of y² as the real estate belonging to y,', 'start': 278.161, 'duration': 8.566}, {'end': 290.55, 'text': 'and that they have a total of one unit of real estate to share between them.', 'start': 286.727, 'duration': 3.823}, {'end': 298.196, 'text': 'So moving around on the circle corresponds to a constant exchange of real estate between the two variables.', 'start': 291.551, 'duration': 6.645}, {'end': 307.753, 'text': 'Part of the reason I choose this term is that it lets us make a very useful analogy that real estate is cheap near zero and more expensive away from zero.', 'start': 299.488, 'duration': 8.265}, {'end': 316.618, 'text': 'To see this, consider starting off in a position where x equals one and y is zero, meaning x has all of the real estate to itself,', 'start': 308.654, 'duration': 7.964}], 'summary': 'X² and y² on a circle share a total of one unit of real estate, with real estate being cheap near zero and more expensive away from zero.', 'duration': 46.118, 'max_score': 270.5, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI270500.jpg'}, {'end': 369.947, 'src': 'embed', 'start': 343.956, 'weight': 7, 'content': [{'end': 351.782, 'text': 'In other words, x changed a little to give up expensive real estate, so that y could move a lot and gain the same value of cheap real estate.', 'start': 343.956, 'duration': 7.826}, {'end': 357.586, 'text': 'In terms of the usual circle drawing, this corresponds to the steep slope near the right side.', 'start': 352.822, 'duration': 4.764}, {'end': 361.529, 'text': 'A small nudge in x allows for a very big change to y.', 'start': 358.206, 'duration': 3.323}, {'end': 369.947, 'text': "Moving forward, let's add some tick marks to these lines to indicate what 0.05 units of real estate looks like at each point.", 'start': 362.779, 'duration': 7.168}], 'summary': 'A small change in x leads to a large change in y, illustrated by tick marks on the lines.', 'duration': 25.991, 'max_score': 343.956, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI343956.jpg'}, {'end': 438.59, 'src': 'embed', 'start': 410.58, 'weight': 1, 'content': [{'end': 416.925, 'text': 'For a unit sphere in three dimensions, the set of all triplets x, y, z, where the sum of their squares is 1,.', 'start': 410.58, 'duration': 6.345}, {'end': 420.308, 'text': 'all we have to do is add a third slider for z.', 'start': 416.925, 'duration': 3.383}, {'end': 424.311, 'text': 'But these three sliders still only have the one unit of real estate to share between them.', 'start': 420.308, 'duration': 4.003}, {'end': 429.515, 'text': 'To get a feel for this, imagine holding x in place at 0.5, where it occupies 0.25 units of real estate.', 'start': 425.191, 'duration': 4.324}, {'end': 438.59, 'text': 'What this means is that y and z can move around in the same piston-dense motion we saw before,', 'start': 433.168, 'duration': 5.422}], 'summary': 'For a unit sphere in 3d, 3 sliders share 1 unit of real estate. x at 0.5 occupies 0.25 units, allowing y and z to move similarly.', 'duration': 28.01, 'max_score': 410.58, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI410580.jpg'}, {'end': 530.736, 'src': 'embed', 'start': 501.334, 'weight': 4, 'content': [{'end': 504.895, 'text': 'But the fundamental rules of this real estate exchange remain the same.', 'start': 501.334, 'duration': 3.561}, {'end': 509.991, 'text': 'If you fix one slider in place and watch the other three trade off.', 'start': 505.768, 'duration': 4.223}, {'end': 516.155, 'text': 'this is basically what it means to take a slice of the 4D sphere to get a small 3D sphere,', 'start': 509.991, 'duration': 6.164}, {'end': 523.282, 'text': 'in much the same way that fixing one of the sliders for the three-dimensional case gave us a circular slice when the remaining two were free to vary.', 'start': 516.155, 'duration': 7.127}, {'end': 530.736, 'text': 'Now, watching these sliders move about and thinking about the real estate exchange is pretty fun,', 'start': 525.211, 'duration': 5.525}], 'summary': 'Real estate exchange rules remain same, 4d sphere yields small 3d sphere, sliders move for fun', 'duration': 29.402, 'max_score': 501.334, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI501334.jpg'}], 'start': 85.646, 'title': 'Visualizing higher-dimensional spaces and real estate exchange on a circle', 'summary': 'Discusses limitations of geometric and analytic views in higher dimensions, introduces visual analytic reasoning, and focuses on understanding higher-dimensional spheres. it also presents a real estate analogy for x² + y² = 1, highlighting the constant exchange of real estate on a circle and its extension to higher dimensions.', 'chapters': [{'end': 269.619, 'start': 85.646, 'title': 'Visualizing higher-dimensional spaces', 'summary': 'Discusses the limitations of geometric and analytic views in higher dimensions, introduces a method for making analytic reasoning more visual, and focuses on understanding higher-dimensional spheres using a literal approach with sliders.', 'duration': 183.973, 'highlights': ['The chapter discusses the limitations of geometric and analytic views in higher dimensions.', 'Introduces a method for making analytic reasoning more visual.', 'Focuses on understanding higher-dimensional spheres using a literal approach with sliders.']}, {'end': 546.55, 'start': 270.5, 'title': 'Real estate exchange on a circle', 'summary': 'Discusses a real estate analogy for the relation x² + y² = 1, showing how moving on the circle corresponds to a constant exchange of real estate between x and y, with real estate being cheaper near zero and more expensive away from zero, and how this analogy extends to the unit sphere in three dimensions and beyond.', 'duration': 276.05, 'highlights': ['The value of x² as the real estate belonging to x and the value of y² as the real estate belonging to y, with a total of one unit of real estate to share between them.', 'Moving around on the circle corresponds to a constant exchange of real estate between the two variables, with real estate being cheaper near zero and more expensive away from zero.', 'A small nudge in x allows for a very big change to y, corresponding to the steep slope near the right side of the circle.', 'The set of all triplets x, y, z, where the sum of their squares is 1, with the sliders having the one unit of real estate to share between them.', 'In four dimensions and higher, the fundamental rules of the real estate exchange remain the same, with the analogy of taking a slice of a 4D sphere to get a small 3D sphere.']}], 'duration': 460.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI85646.jpg', 'highlights': ['Introduces a method for making analytic reasoning more visual.', 'Focuses on understanding higher-dimensional spheres using a literal approach with sliders.', 'The chapter discusses the limitations of geometric and analytic views in higher dimensions.', 'The set of all triplets x, y, z, where the sum of their squares is 1, with the sliders having the one unit of real estate to share between them.', 'In four dimensions and higher, the fundamental rules of the real estate exchange remain the same, with the analogy of taking a slice of a 4D sphere to get a small 3D sphere.', 'The value of x² as the real estate belonging to x and the value of y² as the real estate belonging to y, with a total of one unit of real estate to share between them.', 'Moving around on the circle corresponds to a constant exchange of real estate between the two variables, with real estate being cheaper near zero and more expensive away from zero.', 'A small nudge in x allows for a very big change to y, corresponding to the steep slope near the right side of the circle.']}, {'end': 731.132, 'segs': [{'end': 609.104, 'src': 'embed', 'start': 576.397, 'weight': 0, 'content': [{'end': 578.359, 'text': 'circles tangent to each one of them.', 'start': 576.397, 'duration': 1.962}, {'end': 585.968, 'text': 'What we want to do for this setup, and for its analogies in higher dimensions, is find the radius of that inner circle.', 'start': 579.366, 'duration': 6.602}, {'end': 588.809, 'text': 'Here in two dimensions.', 'start': 587.889, 'duration': 0.92}, {'end': 595.511, 'text': 'we can use the Pythagorean theorem to see that the distance from the origin to the corner of the box is the square root of 2,,', 'start': 588.809, 'duration': 6.702}, {'end': 598.274, 'text': 'which is around 1.414..', 'start': 595.511, 'duration': 2.763}, {'end': 600.616, 'text': 'Then you can subtract off this portion here.', 'start': 598.274, 'duration': 2.342}, {'end': 609.104, 'text': 'the radius of the corner circle, which by definition is 1, and that means the radius of the inner circle, is square root of 2 minus 1,', 'start': 600.616, 'duration': 8.488}], 'summary': "Using pythagorean theorem to find inner circle's radius in 2d setup", 'duration': 32.707, 'max_score': 576.397, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI576397.jpg'}, {'end': 677.095, 'src': 'embed', 'start': 654.233, 'weight': 1, 'content': [{'end': 664.323, 'text': "I guess I still haven't yet explicitly said that the way distances work in higher dimensions is always to add up the squares of the components in each direction and take the square root.", 'start': 654.233, 'duration': 10.09}, {'end': 670.429, 'text': "If you've never seen why this follows from the Pythagorean theorem just in the two-dimensional case.", 'start': 665.184, 'duration': 5.245}, {'end': 677.095, 'text': "it's actually a really fun puzzle to think about and I've left the relevant image up on the screen for any of you who want to pause and ponder on it.", 'start': 670.429, 'duration': 6.666}], 'summary': 'In higher dimensions, distances are calculated by adding squares of components in each direction and taking the square root.', 'duration': 22.862, 'max_score': 654.233, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI654233.jpg'}, {'end': 713.599, 'src': 'embed', 'start': 689.834, 'weight': 2, 'content': [{'end': 697.295, 'text': 'So the radius of that inner sphere is going to be this quantity minus the radius of a corner sphere, which by definition is 1.', 'start': 689.834, 'duration': 7.461}, {'end': 703.036, 'text': 'And again, 0.73 seems like a reasonable radius for that inner sphere.', 'start': 697.295, 'duration': 5.741}, {'end': 707.077, 'text': 'But what happens to that inner radius as you increase dimensions?', 'start': 703.876, 'duration': 3.201}, {'end': 713.599, 'text': 'Obviously, the reason I bring this up is that something surprising will happen, and some of you might see where this is going.', 'start': 708.075, 'duration': 5.524}], 'summary': 'The inner sphere has a radius of 0.73, a reasonable value, but it changes as dimensions increase, leading to a surprising outcome.', 'duration': 23.765, 'max_score': 689.834, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI689834.jpg'}], 'start': 546.55, 'title': 'Inner circle and distances in higher dimensions', 'summary': "Delves into calculating inner circle radius in higher dimensions, using a 2x2 box in 2 and 3 dimensions to find the radius as approximately 0.414 and extending the analogy to 3 dimensions. it also explains how distances work in higher dimensions by adding squares of components in each direction and taking the square root, illustrating this with the example of a corner in a higher-dimensional space and the surprising behavior of the inner sphere's radius as dimensions increase.", 'chapters': [{'end': 654.233, 'start': 546.55, 'title': 'Calculating inner circle radius in higher dimensions', 'summary': 'Discusses the calculation of the radius of an inner circle in higher dimensions using the example of a 2x2 box in 2 and 3 dimensions, finding the radius of the inner circle as approximately 0.414 in 2 dimensions and extending the analogy to 3 dimensions.', 'duration': 107.683, 'highlights': ['In 2 dimensions, the radius of the inner circle can be found using the Pythagorean theorem, resulting in a radius of approximately 0.414, derived from the distance from the origin to the corner of the box and the radius of the corner circle being 1.', 'Extending the analogy to 3 dimensions, a 2x2x2 cube is used and the radius of the inner sphere is considered, with the calculation being analogous to the 2D case, resulting in a similar approach to finding the radius of the inner circle in 2 dimensions.']}, {'end': 731.132, 'start': 654.233, 'title': 'Distances in higher dimensions', 'summary': "Discusses how distances work in higher dimensions by adding squares of components in each direction and taking the square root, illustrating this with the example of a corner in a higher-dimensional space and the surprising behavior of the inner sphere's radius as dimensions increase.", 'duration': 76.899, 'highlights': ['The distance in higher dimensions is calculated by adding up the squares of the components in each direction and taking the square root, as illustrated by the example of the distance between the origin and the corner in a higher-dimensional space.', 'The radius of the inner sphere in higher dimensions decreases as the dimensions increase, leading to a surprising and counterintuitive behavior.', 'The goal of the discussion is to achieve genuine understanding of the concept, rather than to simply surprise with counterintuitive facts and math.']}], 'duration': 184.582, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI546550.jpg', 'highlights': ['The radius of the inner circle in 2 dimensions is approximately 0.414, derived from the Pythagorean theorem.', 'The distance in higher dimensions is calculated by adding up the squares of the components in each direction and taking the square root.', 'The radius of the inner sphere in higher dimensions decreases as the dimensions increase, leading to surprising behavior.']}, {'end': 1487.603, 'segs': [{'end': 782.232, 'src': 'embed', 'start': 755.127, 'weight': 4, 'content': [{'end': 758.328, 'text': "And it's the same basic idea here as you move around the center.", 'start': 755.127, 'duration': 3.201}, {'end': 762.948, 'text': "it's just that the real estate might be dependent on the distance between each coordinate and some other number.", 'start': 758.328, 'duration': 4.62}, {'end': 772.23, 'text': 'So for this circle, centered at the amount of real estate belonging to x is the square of its distance from 1.', 'start': 763.649, 'duration': 8.581}, {'end': 777.231, 'text': 'Likewise, the real estate belonging to y is the square of its distance from.', 'start': 772.23, 'duration': 5.001}, {'end': 782.232, 'text': 'Other than that, the look and feel with this piston dance trade-off is completely the same.', 'start': 777.231, 'duration': 5.001}], 'summary': 'Real estate allocation based on distance from center, with a circle centered at 1.', 'duration': 27.105, 'max_score': 755.127, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI755127.jpg'}, {'end': 844.591, 'src': 'embed', 'start': 820.109, 'weight': 5, 'content': [{'end': 828.015, 'text': 'Imagine perturbing that point slightly, maybe moving x a little closer to 0, which means y would have to move a little away from 0.', 'start': 820.109, 'duration': 7.906}, {'end': 831.858, 'text': 'The change in X would have to be a little smaller than the change in Y,', 'start': 828.015, 'duration': 3.843}, {'end': 839.263, 'text': 'since the real estate it gains by moving farther away from one is more expensive than the real estate that Y loses by getting closer to one.', 'start': 831.858, 'duration': 7.405}, {'end': 844.591, 'text': 'but from the perspective of the origin point that trade-off is reversed.', 'start': 840.27, 'duration': 4.321}], 'summary': 'Perturbing a point: x moves closer to 0, y moves away. change in x smaller than change in y.', 'duration': 24.482, 'max_score': 820.109, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI820109.jpg'}, {'end': 910.587, 'src': 'embed', 'start': 865.354, 'weight': 2, 'content': [{'end': 869.015, 'text': 'evenly results in an increasing distance from the origin.', 'start': 865.354, 'duration': 3.661}, {'end': 877.429, 'text': 'The reason we care is that this point is tangent to the inner circle, so we can also think about it as being a point of the inner circle.', 'start': 870.367, 'duration': 7.062}, {'end': 880.07, 'text': 'And this will be very useful for higher dimensions.', 'start': 878.13, 'duration': 1.94}, {'end': 884.432, 'text': 'It gives us a reference point to understanding the radius of that inner circle.', 'start': 880.53, 'duration': 3.902}, {'end': 890.974, 'text': 'Specifically, you can ask how much real estate is shared between x and y at this point,', 'start': 885.212, 'duration': 5.762}, {'end': 898.315, 'text': 'when real estate measurements are done with respect to the origin, For example down here in two dimensions,', 'start': 890.974, 'duration': 7.341}, {'end': 902.779, 'text': 'both x and y dip below 0.5 in this configuration.', 'start': 898.315, 'duration': 4.464}, {'end': 910.587, 'text': 'So the total value, x squared plus y squared, is going to be less than 0.5 squared plus 0.5 squared.', 'start': 903.42, 'duration': 7.167}], 'summary': 'Point tangent to inner circle useful for dimensions, real estate measurements, x and y dip below 0.5 in 2 dimensions.', 'duration': 45.233, 'max_score': 865.354, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI865354.jpg'}, {'end': 1084.989, 'src': 'embed', 'start': 1057.296, 'weight': 1, 'content': [{'end': 1065.685, 'text': "with respect to, it's still the case that each of these four coordinates has 0.25 units of real estate,", 'start': 1057.296, 'duration': 8.389}, {'end': 1068.768, 'text': 'making for a total of one shared between the four coordinates.', 'start': 1065.685, 'duration': 3.083}, {'end': 1074.721, 'text': 'In other words, that inner sphere is precisely the same size as the corner spheres.', 'start': 1070.117, 'duration': 4.604}, {'end': 1077.463, 'text': 'This matches with what you see numerically.', 'start': 1075.621, 'duration': 1.842}, {'end': 1084.989, 'text': 'by the way, where you can compute the distance between the origin and the corner 1, 1, 1, 1, is the square root of 4, and then,', 'start': 1077.463, 'duration': 7.526}], 'summary': 'Each of the four coordinates has 0.25 units of real estate, totaling one shared between them. the inner sphere is the same size as the corner spheres.', 'duration': 27.693, 'max_score': 1057.296, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI1057296.jpg'}, {'end': 1334.582, 'src': 'embed', 'start': 1296.553, 'weight': 0, 'content': [{'end': 1298.534, 'text': 'Way up here in ten dimensions.', 'start': 1296.553, 'duration': 1.981}, {'end': 1307.459, 'text': 'that quote-unquote inner sphere is actually large enough to poke outside of that outer bounding box, since it has a diameter bigger than four.', 'start': 1298.534, 'duration': 8.925}, {'end': 1320.136, 'text': 'I know that seems crazy, but you have to realize that the face of the box is always 2 units away from the origin, no matter how high the dimension is,', 'start': 1310.512, 'duration': 9.624}, {'end': 1324.278, 'text': "and fundamentally it's because it only involves moving along a single axis.", 'start': 1320.136, 'duration': 4.142}, {'end': 1334.582, 'text': "But the point which determines the inner sphere's radius, is actually really far away from the center, all the way up here in 10 dimensions.", 'start': 1325.158, 'duration': 9.424}], 'summary': 'In 10 dimensions, the inner sphere exceeds the outer bounding box with a diameter larger than four.', 'duration': 38.029, 'max_score': 1296.553, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI1296553.jpg'}, {'end': 1376.316, 'src': 'heatmap', 'start': 1348.005, 'weight': 0.802, 'content': [{'end': 1351.046, 'text': 'Not only is it poking outside of these boxes,', 'start': 1348.005, 'duration': 3.041}, {'end': 1359.328, 'text': 'but the proportion of the inner sphere lying inside the box decreases exponentially towards zero as the dimension keeps increasing.', 'start': 1351.046, 'duration': 8.282}, {'end': 1362.941, 'text': 'So taking a step back.', 'start': 1361.539, 'duration': 1.402}, {'end': 1368.367, 'text': 'one of the things I like about using this slider method for teaching is that when I shared it with a few friends,', 'start': 1362.941, 'duration': 5.426}, {'end': 1376.316, 'text': 'the way they started to talk about higher dimensions became a little less metaphysical and started to sound more like how you would hear a mathematician talk about the topic.', 'start': 1368.367, 'duration': 7.949}], 'summary': 'Proportion of inner sphere decreases exponentially as dimension increases, making higher dimensions more tangible for mathematicians.', 'duration': 28.311, 'max_score': 1348.005, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI1348005.jpg'}], 'start': 731.132, 'title': 'Analyzing real estate in higher dimensions', 'summary': "Covers analyzing circles centered at specific points, and explains the concept of real estate belonging to x and y. it also explores the concept of real estate in higher dimensions, demonstrating the shared real estate between coordinates in 2, 3, 4, 5, and 10 dimensions, showcasing counterintuitive behavior as dimensions increase and the inner sphere's growth exceeding the outer bounding box.", 'chapters': [{'end': 884.432, 'start': 731.132, 'title': 'Analyzing two and three dimensional cases with sliders', 'summary': 'Discusses analyzing circles centered at specific points, and explains the concept of real estate belonging to x and y, demonstrating the trade-off and perturbation from the origin point.', 'duration': 153.3, 'highlights': ['The real estate belonging to x and y is dependent on their distance from a certain point, demonstrated by the example of a circle centered at a specific point.', 'The concept of real estate and perturbation is demonstrated through the trade-off between x and y coordinates, with an explanation of how a slight perturbation away from a point of shared real estate results in an increasing distance from the origin.', 'The discussion of a circle centered at a specific point provides a reference point to understand the radius of the inner circle, emphasizing its usefulness for higher dimensions.']}, {'end': 1487.603, 'start': 885.212, 'title': 'Real estate in higher dimensions', 'summary': "Explores the concept of real estate in higher dimensions, demonstrating the shared real estate between coordinates in 2, 3, 4, 5, and 10 dimensions, showcasing the counterintuitive behavior as dimensions increase and the inner sphere's growth exceeding the outer bounding box.", 'duration': 602.391, 'highlights': ["The inner sphere in 10 dimensions pokes outside the outer bounding box with a radius of about 2.16, demonstrating the counterintuitive behavior of the inner sphere's growth exceeding the outer bounding box as dimensions increase.", 'In 4 dimensions, the inner sphere is precisely the same size as the corner spheres, indicating a unique relationship between the inner sphere and the corner spheres.', 'In 3 dimensions, the amount of real estate shared between x, y, and z is less than 0.75, as each coordinate has less than 0.25 units of real estate, showcasing the shared real estate between coordinates in 3 dimensions.', 'In 2 dimensions, the total value of x squared plus y squared is less than 0.5 squared plus 0.5 squared, demonstrating the shared real estate between x and y in 2 dimensions.']}], 'duration': 756.471, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/zwAD6dRSVyI/pics/zwAD6dRSVyI731132.jpg', 'highlights': ["The inner sphere in 10 dimensions pokes outside the outer bounding box with a radius of about 2.16, demonstrating the counterintuitive behavior of the inner sphere's growth exceeding the outer bounding box as dimensions increase.", 'In 4 dimensions, the inner sphere is precisely the same size as the corner spheres, indicating a unique relationship between the inner sphere and the corner spheres.', 'In 3 dimensions, the amount of real estate shared between x, y, and z is less than 0.75, as each coordinate has less than 0.25 units of real estate, showcasing the shared real estate between coordinates in 3 dimensions.', 'In 2 dimensions, the total value of x squared plus y squared is less than 0.5 squared plus 0.5 squared, demonstrating the shared real estate between x and y in 2 dimensions.', 'The real estate belonging to x and y is dependent on their distance from a certain point, demonstrated by the example of a circle centered at a specific point.', 'The concept of real estate and perturbation is demonstrated through the trade-off between x and y coordinates, with an explanation of how a slight perturbation away from a point of shared real estate results in an increasing distance from the origin.', 'The discussion of a circle centered at a specific point provides a reference point to understand the radius of the inner circle, emphasizing its usefulness for higher dimensions.']}], 'highlights': ["The inner sphere in 10 dimensions pokes outside the outer bounding box with a radius of about 2.16, demonstrating the counterintuitive behavior of the inner sphere's growth exceeding the outer bounding box as dimensions increase.", 'The videos showcase the connection of pi to number theory and primes, as well as the visualization of possible Pythagorean triples, demonstrating the utilization of geometric insights to explore number theory and solve counting puzzles.', 'The radius of the inner circle in 2 dimensions is approximately 0.414, derived from the Pythagorean theorem.', 'The visualization of circles and spheres using numerical properties exemplifies the connection between pairs or triplets of numbers and spatial concepts.', 'The distance in higher dimensions is calculated by adding up the squares of the components in each direction and taking the square root.', 'In 4 dimensions, the inner sphere is precisely the same size as the corner spheres, indicating a unique relationship between the inner sphere and the corner spheres.', 'The chapter discusses the limitations of geometric and analytic views in higher dimensions.', 'The set of all triplets x, y, z, where the sum of their squares is 1, with the sliders having the one unit of real estate to share between them.', 'The concept of real estate and perturbation is demonstrated through the trade-off between x and y coordinates, with an explanation of how a slight perturbation away from a point of shared real estate results in an increasing distance from the origin.', 'Geometric insights clarify analytic facts and vice versa, offering a rich library of cleverness that involves connecting seemingly disparate ideas.']}