title

Visualizing quaternions (4d numbers) with stereographic projection

description

How to think about this 4d number system in our 3d space.
Part 2: https://youtu.be/zjMuIxRvygQ
Interactive version of these visuals: https://eater.net/quaternions
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/quaternion-thanks
Quanta article on quaternions:
https://www.quantamagazine.org/the-strange-numbers-that-birthed-modern-algebra-20180906/
The math of Alice in Wonderland:
https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/
Timestamps:
0:00 - Intro
4:14 - Linus the linelander
11:03 - Felix the flatlander
17:25 - Mapping 4d to 3d
23:18 - The geometry of quaternion multiplication
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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.
Music by Vincent Rubinetti:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
------------------
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detail

{'title': 'Visualizing quaternions (4d numbers) with stereographic projection', 'heatmap': [{'end': 711.506, 'start': 687.27, 'weight': 0.756}, {'end': 1186.827, 'start': 1166.067, 'weight': 1}, {'end': 1495.161, 'start': 1467.421, 'weight': 0.726}], 'summary': 'Explores quaternion mathematics, its resurgence in programming for 3d rotations and quantum mechanics, unit circle projection, 3d rotations visualization, visualizing spherical objects, quaternions in 4d space, and 4d rotations, providing a comprehensive understanding of their history, applications, and visualization methods.', 'chapters': [{'end': 150.51, 'segs': [{'end': 55.139, 'src': 'embed', 'start': 27.231, 'weight': 0, 'content': [{'end': 34.814, 'text': 'Just as complex numbers are a two-dimensional extension of the real numbers, quaternions are a four-dimensional extension of complex numbers.', 'start': 27.231, 'duration': 7.583}, {'end': 37.735, 'text': "But they're not just playful mathematical shenanigans.", 'start': 35.374, 'duration': 2.361}, {'end': 43.858, 'text': 'They have a surprisingly pragmatic utility for describing rotation in three dimensions, and even for quantum mechanics.', 'start': 38.015, 'duration': 5.843}, {'end': 47.733, 'text': 'The story of their discovery is also quite famous in math.', 'start': 45.051, 'duration': 2.682}, {'end': 55.139, 'text': 'The Irish mathematician William Rowan Hamilton spent much of his life seeking a three-dimensional number system, analogous to the complex numbers.', 'start': 48.174, 'duration': 6.965}], 'summary': 'Quaternions are a four-dimensional extension of complex numbers with pragmatic utility in describing three-dimensional rotation and quantum mechanics.', 'duration': 27.908, 'max_score': 27.231, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg27231.jpg'}, {'end': 139.003, 'src': 'embed', 'start': 116.357, 'weight': 2, 'content': [{'end': 125.508, 'text': 'this was balanced with mathematicians on the other side of the fence who believed that the confusing notion of quaternion multiplication was not necessary for describing three dimensions,', 'start': 116.357, 'duration': 9.151}, {'end': 130.073, 'text': 'resulting in some truly hilarious old-timey trash talk, legitimately calling them evil.', 'start': 125.508, 'duration': 4.565}, {'end': 136.681, 'text': "It's even believed that the Mad Hatter scene from Alice in Wonderland, whose author you may know was an Oxford mathematician,", 'start': 130.814, 'duration': 5.867}, {'end': 139.003, 'text': 'was written in reference to quaternions,', 'start': 136.681, 'duration': 2.322}], 'summary': 'Debate over quaternion necessity; mad hatter scene reference to quaternions.', 'duration': 22.646, 'max_score': 116.357, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg116357.jpg'}], 'start': 0.939, 'title': 'Quaternion mathematics', 'summary': 'Delves into the history, significance, and applications of quaternions, a four-dimensional extension of complex numbers, discovered by william rowan hamilton, with pragmatic utility for describing rotation in three dimensions and quantum mechanics.', 'chapters': [{'end': 150.51, 'start': 0.939, 'title': 'Quaternion mathematics and its practical applications', 'summary': 'Discusses the history, significance, and applications of quaternions, a four-dimensional extension of complex numbers, discovered by william rowan hamilton, which have pragmatic utility for describing rotation in three dimensions and even for quantum mechanics.', 'duration': 149.571, 'highlights': ["William Rowan Hamilton's discovery of quaternions and its pragmatic utility for describing rotation in three dimensions and quantum mechanics.", 'The historical significance and the amusing disputes surrounding quaternion mathematics, including the old-timey trash talk and references in literature such as Alice in Wonderland.', 'The practical applications of quaternions in the computing industry.']}], 'duration': 149.571, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg939.jpg', 'highlights': ["William Rowan Hamilton's discovery of quaternions and its pragmatic utility for describing rotation in three dimensions and quantum mechanics.", 'The practical applications of quaternions in the computing industry.', 'The historical significance and the amusing disputes surrounding quaternion mathematics, including the old-timey trash talk and references in literature such as Alice in Wonderland.']}, {'end': 460.835, 'segs': [{'end': 205.383, 'src': 'embed', 'start': 150.51, 'weight': 0, 'content': [{'end': 156.273, 'text': 'a resurgence among programmers who work with graphics and robotics and anything involving orientation in 3D space.', 'start': 150.51, 'duration': 5.763}, {'end': 162.115, 'text': 'And this is because they give an elegant way to describe and to compute 3D rotations,', 'start': 156.973, 'duration': 5.142}, {'end': 168.637, 'text': 'which is computationally more efficient than other methods and which also avoids a lot of the numerical errors that arise in these other methods.', 'start': 162.115, 'duration': 6.522}, {'end': 174.54, 'text': 'The 20th century also brought quaternions some more love from a completely different direction, quantum mechanics.', 'start': 169.218, 'duration': 5.322}, {'end': 175.381, 'text': 'You see,', 'start': 175.16, 'duration': 0.221}, {'end': 185.984, 'text': 'the special actions that quaternions describe in four dimensions are actually quite relevant to the way that two-state systems like spin of an electron or the polarization of a photon are described mathematically.', 'start': 175.381, 'duration': 10.603}, {'end': 192.239, 'text': "What I'll show you here is a way to visualize quaternions in their full four-dimensional glory.", 'start': 187.537, 'duration': 4.702}, {'end': 195.42, 'text': 'It would surprise me if this approach was fully original,', 'start': 192.859, 'duration': 2.561}, {'end': 205.383, 'text': "but I can say that it's certainly not the standard way to teach quaternions and that the specific four-dimensional right-hand rule image that I'd like to build up to is something that I haven't really seen elsewhere.", 'start': 195.42, 'duration': 9.963}], 'summary': 'Quaternions gaining popularity due to efficient 3d rotation computation and relevance to quantum mechanics.', 'duration': 54.873, 'max_score': 150.51, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg150510.jpg'}, {'end': 386.748, 'src': 'embed', 'start': 358.101, 'weight': 4, 'content': [{'end': 362.722, 'text': "And in two dimensions, there is one and only one stretching rotating action on the plane that'll do this.", 'start': 358.101, 'duration': 4.621}, {'end': 367.183, 'text': "This is also how I'll have you thinking about quaternion multiplication later on,", 'start': 363.462, 'duration': 3.721}, {'end': 370.604, 'text': 'where the number on the left acts as a kind of function to the one on the right.', 'start': 367.183, 'duration': 3.421}, {'end': 377.705, 'text': "And we'll understand this function by seeing how it acts by transforming space, although instead of rotating 2D space,", 'start': 371.104, 'duration': 6.601}, {'end': 380.266, 'text': 'it does a sort of double rotation in 4D space.', 'start': 377.705, 'duration': 2.561}, {'end': 386.748, 'text': 'By the way, if you want to review thinking about complex numbers as a kind of action,', 'start': 383.206, 'duration': 3.542}], 'summary': 'Understanding quaternion multiplication in 4d space and its relationship to complex numbers.', 'duration': 28.647, 'max_score': 358.101, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg358101.jpg'}, {'end': 426.692, 'src': 'embed', 'start': 398.891, 'weight': 5, 'content': [{'end': 403.916, 'text': "Maybe it's a little weird for him to think about stretching in multiple dimensions, but it's not fundamentally different.", 'start': 398.891, 'duration': 5.025}, {'end': 407.519, 'text': 'The difficult thing to communicate to Linus is rotation.', 'start': 404.436, 'duration': 3.083}, {'end': 414.206, 'text': 'Specifically, focus on the unit circle of the complex plane, all the numbers, a distance 1 from 0,', 'start': 408.18, 'duration': 6.026}, {'end': 417.569, 'text': 'since multiplication by these numbers corresponds to pure rotation.', 'start': 414.206, 'duration': 3.363}, {'end': 426.692, 'text': 'How would you explain to Linus the look and the feel of multiplying by these numbers? At first, that might seem impossible.', 'start': 418.23, 'duration': 8.462}], 'summary': 'Explaining complex plane rotation to linus using unit circle and multiplying by numbers can be challenging.', 'duration': 27.801, 'max_score': 398.891, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg398891.jpg'}, {'end': 471.017, 'src': 'embed', 'start': 440.169, 'weight': 3, 'content': [{'end': 447.711, 'text': "So in principle, it should be possible to associate the set of all rotations to the one-dimensional continuum that is Linus's world.", 'start': 440.169, 'duration': 7.542}, {'end': 453.513, 'text': "And there are many ways you could do this, but the one I'm going to show you is what's called a stereographic projection.", 'start': 448.512, 'duration': 5.001}, {'end': 460.835, 'text': "It's a special way to map a circle onto a line, or a sphere into a plane, or even a 4D hypersphere into 3D space.", 'start': 454.053, 'duration': 6.782}, {'end': 471.017, 'text': 'For every point on the unit circle, draw a line from negative 1 through that point.', 'start': 466.195, 'duration': 4.822}], 'summary': 'Rotations can be associated to 1d continuum using stereographic projection.', 'duration': 30.848, 'max_score': 440.169, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg440169.jpg'}], 'start': 150.51, 'title': 'Resurgence of quaternions in programming', 'summary': 'Discusses the efficiency of quaternions in describing and computing 3d rotations, their relevance in quantum mechanics, and a unique visualization method in four dimensions. it also explores the concept of stretching and rotating actions in 2d and 4d space, proposing a stereographic projection as a solution for communicating rotation.', 'chapters': [{'end': 356.599, 'start': 150.51, 'title': 'Resurgence of quaternions in programming', 'summary': 'Discusses the resurgence of quaternions in programming and their efficiency in describing and computing 3d rotations, their relevance in quantum mechanics, and a unique visualization method for quaternions in four dimensions.', 'duration': 206.089, 'highlights': ["Quaternions' efficiency in describing and computing 3D rotations is emphasized, making it computationally more efficient and less prone to numerical errors compared to other methods.", 'The relevance of quaternions in quantum mechanics is highlighted, particularly in describing special actions in four dimensions that are relevant to mathematical descriptions of two-state systems.', 'Introduction of a unique approach to visualize quaternions in their full four-dimensional glory, providing a new perspective that may not be standard but offers a natural and satisfying intuition for understanding quaternion multiplication.']}, {'end': 460.835, 'start': 358.101, 'title': 'Quaternion multiplication and 2d rotation', 'summary': 'Discusses the concept of stretching and rotating actions in 2d and 4d space, emphasizing the difficulty in communicating rotation to linus and proposing a stereographic projection as a solution.', 'duration': 102.734, 'highlights': ['The chapter emphasizes the difficulty in communicating rotation to Linus, focusing on the unit circle of the complex plane where multiplication by numbers corresponds to pure rotation.', 'The concept of stretching and rotating actions in 2D and 4D space is discussed, with an emphasis on quaternion multiplication and a comparison to multiplication by real numbers.', 'The chapter proposes a stereographic projection as a solution to the difficulty in communicating rotation, describing it as a special way to map a circle onto a line, or a sphere into a plane, or even a 4D hypersphere into 3D space.']}], 'duration': 310.325, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg150510.jpg', 'highlights': ["Quaternions' efficiency in describing and computing 3D rotations is emphasized, making it computationally more efficient and less prone to numerical errors compared to other methods.", 'The relevance of quaternions in quantum mechanics is highlighted, particularly in describing special actions in four dimensions that are relevant to mathematical descriptions of two-state systems.', 'Introduction of a unique approach to visualize quaternions in their full four-dimensional glory, providing a new perspective that may not be standard but offers a natural and satisfying intuition for understanding quaternion multiplication.', 'The chapter proposes a stereographic projection as a solution to the difficulty in communicating rotation, describing it as a special way to map a circle onto a line, or a sphere into a plane, or even a 4D hypersphere into 3D space.', 'The concept of stretching and rotating actions in 2D and 4D space is discussed, with an emphasis on quaternion multiplication and a comparison to multiplication by real numbers.', 'The chapter emphasizes the difficulty in communicating rotation to Linus, focusing on the unit circle of the complex plane where multiplication by numbers corresponds to pure rotation.']}, {'end': 647.956, 'segs': [{'end': 528.593, 'src': 'embed', 'start': 490.669, 'weight': 2, 'content': [{'end': 499.034, 'text': 'all of the points on that 90-degree arc between 1 and i will get projected somewhere in the interval between where 1 landed and where i landed.', 'start': 490.669, 'duration': 8.365}, {'end': 504.817, 'text': 'As you continue farther around the circle on the arc between i and negative 1,', 'start': 500.074, 'duration': 4.743}, {'end': 508.339, 'text': 'the projected points end up farther and farther away at an increasing rate.', 'start': 504.817, 'duration': 3.522}, {'end': 515.582, 'text': 'Similarly, if you come around the other way towards negative 1, the projected points end up farther and farther on the other end of the line.', 'start': 509.019, 'duration': 6.563}, {'end': 524.029, 'text': 'This line of projected points is what we show to Linus, labeling a few key points, like 1 and i and negative 1 all for reference.', 'start': 516.582, 'duration': 7.447}, {'end': 528.593, 'text': 'Technically, the point at negative 1 has no projection under this map,', 'start': 524.87, 'duration': 3.723}], 'summary': 'Points on 90-degree arc between 1 and i get projected to an interval between 1 and i, with increasing distance as you move towards negative 1.', 'duration': 37.924, 'max_score': 490.669, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg490669.jpg'}, {'end': 568.705, 'src': 'embed', 'start': 544.758, 'weight': 5, 'content': [{'end': 551.9, 'text': "Now it's important to remember and to remind Linus that what he's seeing is only the complex numbers that are a distance 1 from the origin,", 'start': 544.758, 'duration': 7.142}, {'end': 552.56, 'text': 'the unit circle.', 'start': 551.9, 'duration': 0.66}, {'end': 558.822, 'text': "Linus doesn't see most numbers like 0, or 1 plus i or negative, 2 minus i.", 'start': 553.18, 'duration': 5.642}, {'end': 566.564, 'text': "but that's okay, because right now we just want to describe complex number z, where multiplying by z has the effect of a pure rotation.", 'start': 558.822, 'duration': 7.742}, {'end': 568.705, 'text': 'so he only needs to understand the unit circle.', 'start': 566.564, 'duration': 2.141}], 'summary': 'Linus is learning about complex numbers on the unit circle for pure rotation.', 'duration': 23.947, 'max_score': 544.758, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg544758.jpg'}, {'end': 655.298, 'src': 'embed', 'start': 631.033, 'weight': 0, 'content': [{'end': 638.354, 'text': 'For example, multiplying by i four times, which corresponds to rotating by 90 degrees four times in a row, gets us back to where we started.', 'start': 631.033, 'duration': 7.321}, {'end': 640.355, 'text': 'i to the fourth equals one.', 'start': 639.195, 'duration': 1.16}, {'end': 645.735, 'text': 'Here, to get more of a feel for things, let me just show the circle rotated at various different angles.', 'start': 641.294, 'duration': 4.441}, {'end': 647.956, 'text': 'On both the left and the right half of the screen.', 'start': 646.255, 'duration': 1.701}, {'end': 655.298, 'text': "here I'm putting a hand on the point that started at the number 1 to help us and to help Linus keep track of the overall motion.", 'start': 647.956, 'duration': 7.342}], 'summary': 'Multiplying i by itself 4 times rotates 90 degrees each time, reaching the starting point; i to the fourth equals 1.', 'duration': 24.265, 'max_score': 631.033, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg631033.jpg'}], 'start': 466.195, 'title': 'Unit circle projection and complex numbers', 'summary': 'Covers projecting points on the unit circle onto a line, demonstrating the increasing rate of projection as points move around the circle, and explains the relationship between complex numbers and rotations on the unit circle, including the specific rotation caused by multiplying by i and its visualization on the line.', 'chapters': [{'end': 543.486, 'start': 466.195, 'title': 'Unit circle projection', 'summary': 'Explains the concept of projecting points on the unit circle onto a line, where points between 1 and i get projected between 1 and i, and as you continue around the circle, the projected points end up farther and farther away at an increasing rate.', 'duration': 77.291, 'highlights': ['The points on the 90-degree arc between 1 and i will get projected somewhere in the interval between where 1 landed and where i landed.', 'As you continue farther around the circle on the arc between i and negative 1, the projected points end up farther and farther away at an increasing rate.', 'The point at negative 1 has no projection under this map, but it ends up at the point at infinity, a special point added to the line where you would approach it if you walk infinitely far along the line in either direction.']}, {'end': 647.956, 'start': 544.758, 'title': 'Understanding complex numbers and rotations', 'summary': 'Explains the concept of complex numbers and their relationship with rotations on the unit circle, demonstrating how multiplying by i causes specific rotations, such as 90 degrees counterclockwise, and illustrating the resulting morphing action on the line for visualization.', 'duration': 103.198, 'highlights': ['Multiplying by i causes specific rotations, such as 90 degrees counterclockwise, and illustrates the resulting morphing action on the line for visualization.', 'Communicating important ideas to Linus through the peculiar motion of complex numbers, such as demonstrating that multiplying by i four times results in a full rotation back to the starting point.', "Explaining the specific effects of multiplying by i on reference points, such as i times 1 resulting in i and i times i resulting in -1, to help understand the concept of complex numbers' rotation."]}], 'duration': 181.761, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg466195.jpg', 'highlights': ['Multiplying by i causes specific rotations, such as 90 degrees counterclockwise, and illustrates the resulting morphing action on the line for visualization.', "Explaining the specific effects of multiplying by i on reference points, such as i times 1 resulting in i and i times i resulting in -1, to help understand the concept of complex numbers' rotation.", 'The points on the 90-degree arc between 1 and i will get projected somewhere in the interval between where 1 landed and where i landed.', 'As you continue farther around the circle on the arc between i and negative 1, the projected points end up farther and farther away at an increasing rate.', 'The point at negative 1 has no projection under this map, but it ends up at the point at infinity, a special point added to the line where you would approach it if you walk infinitely far along the line in either direction.', 'Communicating important ideas to Linus through the peculiar motion of complex numbers, such as demonstrating that multiplying by i four times results in a full rotation back to the starting point.']}, {'end': 856.248, 'segs': [{'end': 741.951, 'src': 'heatmap', 'start': 687.27, 'weight': 0, 'content': [{'end': 694.892, 'text': 'with a third axis defined by some newly invented constant j sitting one unit away from zero, perpendicular to the complex plane.', 'start': 687.27, 'duration': 7.622}, {'end': 702.338, 'text': "Instead of having this new axis in the z direction, like you might expect, for a better analogy with how we'll visualize quaternions,", 'start': 695.732, 'duration': 6.606}, {'end': 711.506, 'text': "we'll want to orient things so that the i and the j axes sit in the x and the y directions, with the real number line aligned along the z direction.", 'start': 702.338, 'duration': 9.168}, {'end': 721.685, 'text': 'So every point in 3D space is described as some real number plus some real number times i plus some real number times j.', 'start': 713.502, 'duration': 8.183}, {'end': 727.006, 'text': "As it happens, it's not possible to define a notion of multiplication for a 3D number system like this.", 'start': 721.685, 'duration': 5.321}, {'end': 731.388, 'text': 'that would satisfy the usual algebraic properties that make multiplication a useful construct.', 'start': 727.006, 'duration': 4.382}, {'end': 736.829, 'text': "Perhaps I'll outline why this is the case in a follow-on video, but staying focused on our current goal,", 'start': 731.828, 'duration': 5.001}, {'end': 741.951, 'text': 'think about describing 3D rotations in this coordinate system to Felix the Flatlander.', 'start': 736.829, 'duration': 5.122}], 'summary': 'Introducing a 3d coordinate system with a newly invented constant, j, perpendicular to the complex plane, aiming to visualize quaternions in a way to describe 3d rotations.', 'duration': 60.543, 'max_score': 687.27, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg687270.jpg'}, {'end': 856.248, 'src': 'embed', 'start': 767.618, 'weight': 3, 'content': [{'end': 776.78, 'text': 'stereographic projection will associate almost every point on the unit sphere with a unique point on the horizontal plane defined by the i and the j axes.', 'start': 767.618, 'duration': 9.162}, {'end': 784.982, 'text': 'For each point on the sphere, draw a line from negative one at the south pole through that point and see where it intersects the plane.', 'start': 777.5, 'duration': 7.482}, {'end': 795.472, 'text': 'So the point 1 at the North Pole ends up at the center of the plane.', 'start': 792.389, 'duration': 3.083}, {'end': 801.759, 'text': 'All of the points of the Northern Hemisphere get mapped somewhere inside the unit circle of the IJ plane.', 'start': 796.153, 'duration': 5.606}, {'end': 808.125, 'text': 'And that unit circle, which passes through IJ, negative I, and negative J, actually stays fixed in place.', 'start': 802.54, 'duration': 5.585}, {'end': 810.649, 'text': "And that's an important point to make note of.", 'start': 808.806, 'duration': 1.843}, {'end': 817.778, 'text': 'Even though most points and lines and patches that Felix the Flatlander sees are going to be warped projections of the real sphere,', 'start': 811.169, 'duration': 6.609}, {'end': 824.126, 'text': 'this unit circle is the one thing that he has, which is an honest part of our unit sphere, unaltered by projection.', 'start': 817.778, 'duration': 6.348}, {'end': 831.679, 'text': 'All of the points in the southern hemisphere get projected outside that unit circle,', 'start': 827.458, 'duration': 4.221}, {'end': 835.601, 'text': 'each getting farther and farther away as you approach negative 1 at the south pole.', 'start': 831.679, 'duration': 3.922}, {'end': 842.843, 'text': 'And again, negative 1 has no projection under this mapping, but what we say is that it ends up at some point at infinity.', 'start': 836.381, 'duration': 6.462}, {'end': 849.666, 'text': 'That point at infinity is something such that, no matter which direction you walk on the plane, as you go infinitely far out,', 'start': 843.464, 'duration': 6.202}, {'end': 850.806, 'text': "you'll be approaching that point.", 'start': 849.666, 'duration': 1.14}, {'end': 856.248, 'text': "It's analogous to how if you walk any direction away from the north pole, you're approaching the south pole.", 'start': 851.486, 'duration': 4.762}], 'summary': 'Streographic projection maps unit sphere to plane, with north pole at center and southern hemisphere points outside unit circle.', 'duration': 88.63, 'max_score': 767.618, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg767618.jpg'}], 'start': 647.956, 'title': '3d rotations and stereographic projection', 'summary': 'Discusses extending complex numbers to quaternions, emphasizing visualization of 3d rotations, and explains stereographic projection mapping points on a unit sphere to the horizontal plane.', 'chapters': [{'end': 741.951, 'start': 647.956, 'title': 'Understanding 3d rotations with quaternions', 'summary': 'Discusses the extension of complex numbers to quaternions, introducing the concept of a third axis, and the challenges of defining multiplication in a 3d number system, while emphasizing the visualization of 3d rotations in this coordinate system.', 'duration': 93.995, 'highlights': ['Introducing the concept of a third axis in quaternions', 'Challenges of defining multiplication in a 3D number system', 'Visualizing 3D rotations in the coordinate system']}, {'end': 856.248, 'start': 742.771, 'title': 'Stereographic projection of unit sphere', 'summary': 'Explains the stereographic projection of the unit sphere, where points on the sphere are associated with unique points on the horizontal plane, mapping the northern hemisphere inside the unit circle and the southern hemisphere outside, with the north and south poles ending up at the center and infinity, respectively.', 'duration': 113.477, 'highlights': ['The unit sphere is projected to the horizontal plane using stereographic projection, mapping the North Pole to the center and the South Pole to a point at infinity.', 'Points in the Northern Hemisphere are mapped inside the unit circle of the IJ plane, while points in the Southern Hemisphere are projected outside that unit circle.', 'The unit circle on the IJ plane remains unaltered by projection, serving as an honest part of the unit sphere.', 'The point at infinity represents the projection of the South Pole, analogous to how any direction away from the North Pole leads to the South Pole.']}], 'duration': 208.292, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg647956.jpg', 'highlights': ['Visualizing 3D rotations in the coordinate system', 'Introducing the concept of a third axis in quaternions', 'Challenges of defining multiplication in a 3D number system', 'The unit sphere is projected to the horizontal plane using stereographic projection, mapping the North Pole to the center and the South Pole to a point at infinity.', 'The point at infinity represents the projection of the South Pole, analogous to how any direction away from the North Pole leads to the South Pole.', 'Points in the Northern Hemisphere are mapped inside the unit circle of the IJ plane, while points in the Southern Hemisphere are projected outside that unit circle.', 'The unit circle on the IJ plane remains unaltered by projection, serving as an honest part of the unit sphere.']}, {'end': 1057.637, 'segs': [{'end': 958.093, 'src': 'embed', 'start': 859.192, 'weight': 0, 'content': [{'end': 862.255, 'text': 'Now let me just pull up a view of what Felix sees in two dimensions.', 'start': 859.192, 'duration': 3.063}, {'end': 865.597, 'text': 'As I rotate the sphere in various ways,', 'start': 862.935, 'duration': 2.662}, {'end': 872.343, 'text': "the lines of latitude and longitude drawn on that sphere get projected into various circles and lines in Felix's space.", 'start': 865.597, 'duration': 6.746}, {'end': 874.224, 'text': "And the way I've done things up here.", 'start': 873.043, 'duration': 1.181}, {'end': 879.949, 'text': 'the checkerboard pattern on the surface of the sphere is accurately reflected in the projected view that you see with Felix.', 'start': 874.224, 'duration': 5.725}, {'end': 886.194, 'text': 'And the pink dot represents where the point that started at the North Pole ends up after the rotation.', 'start': 880.429, 'duration': 5.765}, {'end': 891.987, 'text': 'and that yellow circle represents where the equator ended up after the projection.', 'start': 888.004, 'duration': 3.983}, {'end': 897.25, 'text': "The more you put yourself in Felix's shoes right now, the easier quaternions will be in a moment.", 'start': 893.048, 'duration': 4.202}, {'end': 902.754, 'text': 'And as with Linus, it helps to focus on a few key reference objects rather than trying to see the whole sphere.', 'start': 897.911, 'duration': 4.843}, {'end': 911.761, 'text': 'This circle passing through 1, i, negative 1, and negative i gets mapped onto a line which Felix sees as the horizontal axis.', 'start': 903.575, 'duration': 8.186}, {'end': 916.523, 'text': "It's important to remind Felix that what he sees is not the same thing as the i-axis.", 'start': 912.521, 'duration': 4.002}, {'end': 926.289, 'text': "Remember, we're only projecting the numbers that have a distance 1 from the origin, so most points on the actual i-axis, like 0 and 2i and 3i, etc.", 'start': 917.024, 'duration': 9.265}, {'end': 927.85, 'text': ', are completely invisible to Felix.', 'start': 926.289, 'duration': 1.561}, {'end': 937.776, 'text': 'Similarly, the circle that passes through 1, j, and gets projected onto what he sees as a vertical line.', 'start': 930.632, 'duration': 7.144}, {'end': 945.844, 'text': 'And in general, any line that Felix sees comes from some circle on the sphere that passes through.', 'start': 938.755, 'duration': 7.089}, {'end': 949.769, 'text': 'In some sense a line is just a circle that passes through the point at infinity.', 'start': 945.844, 'duration': 3.925}, {'end': 958.093, 'text': 'Now think about what Felix sees as we rotate the sphere.', 'start': 955.492, 'duration': 2.601}], 'summary': "Projection of sphere onto 2d for felix's view, mapping of key objects, and visualization of quaternions.", 'duration': 98.901, 'max_score': 859.192, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg859192.jpg'}], 'start': 859.192, 'title': 'Visualizing spherical objects', 'summary': "Explains the visual representation of latitude and longitude lines on a sphere projected into 2d space, demonstrating accuracy through a checkerboard pattern. it also discusses the projection of circles onto lines from felix the flatlander's perspective.", 'chapters': [{'end': 902.754, 'start': 859.192, 'title': 'Visualizing 2d projection of spherical objects', 'summary': 'Explains the visual representation of lines of latitude and longitude on a sphere projected into circles and lines in a two-dimensional space, with the accuracy of the projection demonstrated through the reflection of a checkerboard pattern, using a pink dot to signify the rotation of the north pole and a yellow circle for the equator, to aid in understanding quaternions and emphasizing the importance of focusing on key reference objects.', 'duration': 43.562, 'highlights': ["The lines of latitude and longitude drawn on the sphere get projected into various circles and lines in Felix's space, accurately reflecting the checkerboard pattern on the surface of the sphere in the projected view.", "The pink dot represents the point that started at the North Pole, demonstrating the effect of rotation, while the yellow circle signifies the location of the equator after the projection, aiding in understanding the visual representation of the sphere's projection into a two-dimensional space.", "It is emphasized that putting oneself in Felix's shoes and focusing on key reference objects will facilitate the understanding of quaternions and the visual representation of spherical objects."]}, {'end': 1057.637, 'start': 903.575, 'title': 'Projection of circles onto lines', 'summary': "Discusses the projection of circles onto lines in the context of felix the flatlander's perspective, highlighting the transformation of unit circles into lines and the relationship between rotations and perspective shifts.", 'duration': 154.062, 'highlights': ["The projection of circles onto lines in Felix's perspective is discussed, emphasizing the invisibility of certain points on the actual i-axis to Felix due to projection limitations.", "The relationship between the rotation of the sphere and the perspective shift for Felix is explained, with the example of a 90 degree rotation about the j-axis transforming the unit circle into a vertical line from Felix's viewpoint.", "The transformation of the unit circle into a horizontal line from Felix's perspective due to a rotation about the i-axis is detailed, highlighting the shift in perception caused by the rotation."]}], 'duration': 198.445, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg859192.jpg', 'highlights': ["The lines of latitude and longitude accurately reflect the checkerboard pattern on the sphere's surface in the projected view.", "The pink dot demonstrates the effect of rotation, aiding in understanding the visual representation of the sphere's projection.", "The yellow circle signifies the location of the equator after the projection, aiding in understanding the visual representation of the sphere's projection.", "The projection of circles onto lines in Felix's perspective is discussed, emphasizing the invisibility of certain points on the actual i-axis to Felix due to projection limitations.", "The relationship between the rotation of the sphere and the perspective shift for Felix is explained, with the example of a 90 degree rotation about the j-axis transforming the unit circle into a vertical line from Felix's viewpoint.", "Putting oneself in Felix's shoes and focusing on key reference objects will facilitate the understanding of quaternions and the visual representation of spherical objects."]}, {'end': 1349.285, 'segs': [{'end': 1086.493, 'src': 'embed', 'start': 1057.637, 'weight': 1, 'content': [{'end': 1063.3, 'text': 'and that the not-actually-a-number system thing we had in three dimensions included a second imaginary direction.', 'start': 1057.637, 'duration': 5.663}, {'end': 1071.706, 'text': 'j, the quaternions include the real numbers together with three separate imaginary dimensions represented by the units i, j and k.', 'start': 1063.3, 'duration': 8.406}, {'end': 1080.008, 'text': "Each of these three imaginary dimensions is perpendicular to the real number line, and they're all perpendicular to each other, somehow.", 'start': 1072.606, 'duration': 7.402}, {'end': 1086.493, 'text': 'So, in the same way that complex numbers are represented as a pair of real numbers,', 'start': 1081.829, 'duration': 4.664}], 'summary': 'Quaternions incorporate real numbers with three perpendicular imaginary dimensions.', 'duration': 28.856, 'max_score': 1057.637, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1057637.jpg'}, {'end': 1146.006, 'src': 'embed', 'start': 1118.716, 'weight': 0, 'content': [{'end': 1125.621, 'text': 'This is analogous to defining complex multiplication by saying that i times i is negative 1, and then distributing and simplifying products.', 'start': 1118.716, 'duration': 6.905}, {'end': 1133.883, 'text': 'And indeed this is how you would tell a computer to perform quaternion multiplication and the relative compactness of this operation compared to, say,', 'start': 1126.201, 'duration': 7.682}, {'end': 1139.064, 'text': "matrix multiplication is what's made quaternion so useful for graphics programming and many other things.", 'start': 1133.883, 'duration': 5.181}, {'end': 1146.006, 'text': "There's also a rather elegant form of this multiplication rule, written in terms of the dot product and the cross product, and in some sense,", 'start': 1139.804, 'duration': 6.202}], 'summary': 'Defining quaternion multiplication, its compactness, and elegant form make it useful for graphics programming and other applications.', 'duration': 27.29, 'max_score': 1118.716, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1118716.jpg'}, {'end': 1190.128, 'src': 'heatmap', 'start': 1166.067, 'weight': 1, 'content': [{'end': 1172.769, 'text': 'And just as the magnitude of a complex number, its distance from zero is the square root of the sum of the squares of its component,', 'start': 1166.067, 'duration': 6.702}, {'end': 1175.73, 'text': 'that same operation gives you the magnitude of a quaternion.', 'start': 1172.769, 'duration': 2.961}, {'end': 1186.827, 'text': 'and multiplying one quaternion q1, by another q2, has the effect of scaling q2 by the magnitude of q1,', 'start': 1178.362, 'duration': 8.465}, {'end': 1190.128, 'text': 'followed by a very special type of rotation in four dimensions.', 'start': 1186.827, 'duration': 3.301}], 'summary': 'Calculation of quaternion magnitude and multiplication for scaling and rotation in four dimensions.', 'duration': 24.061, 'max_score': 1166.067, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1166067.jpg'}, {'end': 1286.512, 'src': 'embed', 'start': 1262.066, 'weight': 2, 'content': [{'end': 1269.168, 'text': 'we get a whole sphere passing through I, J and K on that unit hypersphere which stays in place under the projection.', 'start': 1262.066, 'duration': 7.102}, {'end': 1278.411, 'text': 'So what we see as a unit sphere in our 3D space represents the only unaltered part of the hypersphere of quaternions getting projected down onto us.', 'start': 1269.848, 'duration': 8.563}, {'end': 1286.512, 'text': "It's something analogous to the equator of a 3D sphere, and it represents all of the unit quaternions whose real part is zero,", 'start': 1279.111, 'duration': 7.401}], 'summary': 'Projection of the hypersphere onto 3d space yields a unit sphere, representing unaltered part of quaternions.', 'duration': 24.446, 'max_score': 1262.066, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1262066.jpg'}], 'start': 1057.637, 'title': 'Quaternions in 4d space', 'summary': 'Introduces quaternions, a 4d number system, and explains quaternion multiplication, highlighting its application in graphics programming and projection into 3d space.', 'chapters': [{'end': 1103.445, 'start': 1057.637, 'title': 'Quaternions in four dimensions', 'summary': 'Explains the concept of quaternions, a number system that includes a real part and three separate imaginary dimensions, living in four-dimensional space, where each quaternion can be written using four real numbers.', 'duration': 45.808, 'highlights': ['Quaternions include a real part and three separate imaginary dimensions represented by the units i, j, and k, living in four-dimensional space.', 'Each quaternion can be written using four real numbers and is broken up into a real or scalar part, and then a 3D imaginary part.', 'Hamilton used a special word for quaternions that had no real part and just IJK components.']}, {'end': 1349.285, 'start': 1103.445, 'title': 'Understanding quaternion multiplication', 'summary': 'Explains the concept of quaternion multiplication, its relation to complex numbers, and its 4d geometry, emphasizing its application in graphics programming and the projection of the hypersphere into 3d space.', 'duration': 245.84, 'highlights': ["Quaternion multiplication's compactness compared to matrix multiplication is what makes it useful for graphics programming.", 'The special 4D rotations of quaternion multiplication correspond to the hypersphere of quaternions, where unit quaternions with positive real parts are closer to the number one in 3D space.', 'The projection of the hypersphere of quaternions into 3D space results in a unit sphere representing all the unit quaternions whose real part is zero, akin to the equator of a 3D sphere.']}], 'duration': 291.648, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1057637.jpg', 'highlights': ["Quaternion multiplication's compactness makes it useful for graphics programming.", 'Quaternions include a real part and three separate imaginary dimensions.', 'The projection of the hypersphere of quaternions into 3D space results in a unit sphere.']}, {'end': 1899.427, 'segs': [{'end': 1418.797, 'src': 'embed', 'start': 1377.028, 'weight': 0, 'content': [{'end': 1386.112, 'text': 'And that whole sphere gets projected into the plane that we see passing through 1, i, negative i, j, negative j and negative 1, off at infinity,', 'start': 1377.028, 'duration': 9.084}, {'end': 1387.813, 'text': 'what you and I might call the xy-plane.', 'start': 1386.112, 'duration': 1.701}, {'end': 1389.242, 'text': 'In general,', 'start': 1388.621, 'duration': 0.621}, {'end': 1396.748, 'text': 'any plane that you see here really represents the projection of a sphere somewhere up on the hypersphere which passes through the number negative one.', 'start': 1389.242, 'duration': 7.506}, {'end': 1414.675, 'text': 'Now, the action of taking a unit quaternion and multiplying it by any other quaternion from the left can be thought of in terms of two separate 2D rotations happening perpendicular to and in sync with each other in a way that could only ever be possible in four dimensions.', 'start': 1399.468, 'duration': 15.207}, {'end': 1418.797, 'text': "As a first example, let's look at multiplication by i.", 'start': 1415.455, 'duration': 3.342}], 'summary': 'Unit quaternions represent 2d rotations in 4d space.', 'duration': 41.769, 'max_score': 1377.028, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1377028.jpg'}, {'end': 1508.311, 'src': 'heatmap', 'start': 1461.657, 'weight': 3, 'content': [{'end': 1466.941, 'text': 'You can think of the action of i on this perpendicular circle as obeying a certain right-hand rule.', 'start': 1461.657, 'duration': 5.284}, {'end': 1473.845, 'text': "If you'll excuse the intrusion of my ghostly green screen hand into our otherwise pristine platonic mathematical stage,", 'start': 1467.421, 'duration': 6.424}, {'end': 1478.769, 'text': 'you let that thumb of your right hand point from the number 1 to i and you curl your fingers.', 'start': 1473.845, 'duration': 4.924}, {'end': 1483.012, 'text': 'the JK circle will rotate in the direction of that curl.', 'start': 1479.409, 'duration': 3.603}, {'end': 1489.937, 'text': 'How much? Well, by the same amount as the one I circle rotates, which is 90 degrees in this case.', 'start': 1483.672, 'duration': 6.265}, {'end': 1495.161, 'text': 'This is what I meant by two rotations perpendicular to and in sync with each other.', 'start': 1490.617, 'duration': 4.544}, {'end': 1508.311, 'text': 'So J goes to K, K goes to negative J, negative J goes to negative K, and negative K goes to J.', 'start': 1496.001, 'duration': 12.31}], 'summary': 'Rotation of jk circle by 90 degrees in sync with i circle', 'duration': 46.654, 'max_score': 1461.657, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1461657.jpg'}, {'end': 1568.635, 'src': 'embed', 'start': 1538.577, 'weight': 5, 'content': [{'end': 1544.082, 'text': 'so knowing what our transformation does to them gives us the full information about what it does to all of space.', 'start': 1538.577, 'duration': 5.505}, {'end': 1556.092, 'text': 'Geometrically, a four-dimensional creature would be able to look at those two perpendicular rotations that I just described and understand that they lock you into one and only one rigid motion for the hypersphere.', 'start': 1545.089, 'duration': 11.003}, {'end': 1561.113, 'text': 'We might lack the intuitions of such a hypothetical creature, but we can maybe try to get close.', 'start': 1556.712, 'duration': 4.401}, {'end': 1568.635, 'text': "Here's what the action of repeatedly multiplying by i looks like on our stereographic projection of the IJK sphere.", 'start': 1561.933, 'duration': 6.702}], 'summary': 'Exploring the transformative effects of four-dimensional geometry.', 'duration': 30.058, 'max_score': 1538.577, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1538577.jpg'}, {'end': 1686.065, 'src': 'embed', 'start': 1648.09, 'weight': 2, 'content': [{'end': 1658.444, 'text': 'So j times 1 is 1, and j times j is negative 1.', 'start': 1648.09, 'duration': 10.354}, {'end': 1666.115, 'text': 'The circle perpendicular to that one passing through i and k, gets rotated 90 degrees according to this right-hand rule,', 'start': 1658.444, 'duration': 7.671}, {'end': 1668.659, 'text': 'where you point your thumb from 1 to j.', 'start': 1666.115, 'duration': 2.544}, {'end': 1674.547, 'text': 'So j times i is negative k, and j times k is i.', 'start': 1668.659, 'duration': 5.888}, {'end': 1686.065, 'text': 'In general for any other unit quaternion, you see somewhere in space.', 'start': 1682.122, 'duration': 3.943}], 'summary': 'Quaternion multiplication: j^2=-1, j*i=-k, j*k=i.', 'duration': 37.975, 'max_score': 1648.09, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1648090.jpg'}, {'end': 1748.434, 'src': 'embed', 'start': 1720.515, 'weight': 6, 'content': [{'end': 1723.676, 'text': 'One thing worth noticing here is that order of multiplication matters.', 'start': 1720.515, 'duration': 3.161}, {'end': 1726.697, 'text': "It's not, as mathematicians would say, commutative.", 'start': 1724.136, 'duration': 2.561}, {'end': 1734.859, 'text': 'For example, i times j is k, which you might think of in terms of i acting on the quaternion j, rotating it up to k.', 'start': 1727.297, 'duration': 7.562}, {'end': 1740.521, 'text': 'But if you think of j as acting on i, j times i, it rotates i to negative k.', 'start': 1734.859, 'duration': 5.662}, {'end': 1748.434, 'text': 'In fact, commutativity, this ability to swap the order of multiplication, is a way more special property than a lot of people realize.', 'start': 1741.707, 'duration': 6.727}], 'summary': 'Order of multiplication in quaternions is not commutative, i.e., i times j is k, but j times i is -k.', 'duration': 27.919, 'max_score': 1720.515, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1720515.jpg'}, {'end': 1847.683, 'src': 'embed', 'start': 1824.909, 'weight': 8, 'content': [{'end': 1834.535, 'text': 'Understanding this left hand rule for multiplication from the other side will be extremely useful for understanding how unit quaternions describe rotation in three dimensions.', 'start': 1824.909, 'duration': 9.626}, {'end': 1840.74, 'text': "And so far, it's probably not clear how exactly quaternions do describe 3D rotation.", 'start': 1835.598, 'duration': 5.142}, {'end': 1847.683, 'text': "I mean if you consider one of these actions on the unit sphere passing through i, j and k, it doesn't leave that sphere in place.", 'start': 1841.24, 'duration': 6.443}], 'summary': 'Understanding left hand rule for multiplication aids in grasping how unit quaternions describe 3d rotation.', 'duration': 22.774, 'max_score': 1824.909, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1824909.jpg'}], 'start': 1350.868, 'title': 'Quaternions and 4d rotations', 'summary': 'Explores quaternion multiplication, stereographic projection, and the concept of perpendicular circles in four dimensions, demonstrating 90-degree rotations and the right-hand rule. it also explains the projection of 3d planes from a hypersphere in 4d space, while highlighting the non-commutative nature and the significance of left and right multiplication, and teasing the relation to 3d rotation.', 'chapters': [{'end': 1514.891, 'start': 1350.868, 'title': 'Quaternion multiplication and stereographic projection', 'summary': 'Explores quaternion multiplication, stereographic projection, and the concept of perpendicular circles in four dimensions, demonstrating 90-degree rotations and the right-hand rule. it also explains the projection of 3d planes from a hypersphere in 4d space.', 'duration': 164.023, 'highlights': ['The action of taking a unit quaternion and multiplying it by any other quaternion from the left can be thought of in terms of two separate 2D rotations happening perpendicular to and in sync with each other in a way that could only ever be possible in four dimensions.', 'The chapter explains the projection of 3D planes from a hypersphere in 4D space, highlighting the representation of any plane as the projection of a sphere somewhere up on the hypersphere which passes through the number negative one.', 'The chapter demonstrates the 90-degree rotation due to quaternion multiplication, showing that the circle passing through 1 and i rotates such that 1 goes to i, i goes to -1, -1 comes back around to -i, and -i goes to 1.', 'The concept of perpendicular circles in four dimensions is illustrated, with the action of i on the perpendicular circle obeying a certain right-hand rule and resulting in rotations in sync with the 90-degree rotation of the circle passing through 1 and i.', 'The explanation of quaternion multiplication and stereographic projection provides insights into the 4D nature of these operations, highlighting the unique behaviors that are only possible in four dimensions.']}, {'end': 1899.427, 'start': 1515.492, 'title': 'Understanding quaternions and 3d rotation', 'summary': 'Explains how quaternions act on 4-dimensional space, demonstrating rotations and multiplication by i, j, and k, highlighting the non-commutative nature and the significance of left and right multiplication, and teasing the relation to 3d rotation.', 'duration': 383.935, 'highlights': ['Quaternions act on 4-dimensional space, providing full information about what it does to all of space.', 'Demonstration of multiplication by i, j, and k, depicting rotations and their effects on other quaternions.', 'Emphasis on the non-commutative nature of quaternion multiplication and its significance.', 'Importance of left and right multiplication, with specific examples and their implications for understanding 3D rotation.']}], 'duration': 548.559, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/d4EgbgTm0Bg/pics/d4EgbgTm0Bg1350868.jpg', 'highlights': ['The action of taking a unit quaternion and multiplying it by any other quaternion from the left can be thought of in terms of two separate 2D rotations happening perpendicular to and in sync with each other in a way that could only ever be possible in four dimensions.', 'The chapter explains the projection of 3D planes from a hypersphere in 4D space, highlighting the representation of any plane as the projection of a sphere somewhere up on the hypersphere which passes through the number negative one.', 'The chapter demonstrates the 90-degree rotation due to quaternion multiplication, showing that the circle passing through 1 and i rotates such that 1 goes to i, i goes to -1, -1 comes back around to -i, and -i goes to 1.', 'The concept of perpendicular circles in four dimensions is illustrated, with the action of i on the perpendicular circle obeying a certain right-hand rule and resulting in rotations in sync with the 90-degree rotation of the circle passing through 1 and i.', 'The explanation of quaternion multiplication and stereographic projection provides insights into the 4D nature of these operations, highlighting the unique behaviors that are only possible in four dimensions.', 'Quaternions act on 4-dimensional space, providing full information about what it does to all of space.', 'Demonstration of multiplication by i, j, and k, depicting rotations and their effects on other quaternions.', 'Emphasis on the non-commutative nature of quaternion multiplication and its significance.', 'Importance of left and right multiplication, with specific examples and their implications for understanding 3D rotation.']}], 'highlights': ['The action of taking a unit quaternion and multiplying it by any other quaternion from the left can be thought of in terms of two separate 2D rotations happening perpendicular to and in sync with each other in a way that could only ever be possible in four dimensions.', 'The practical applications of quaternions in the computing industry.', "Quaternions' efficiency in describing and computing 3D rotations is emphasized, making it computationally more efficient and less prone to numerical errors compared to other methods.", 'The chapter explains the projection of 3D planes from a hypersphere in 4D space, highlighting the representation of any plane as the projection of a sphere somewhere up on the hypersphere which passes through the number negative one.', 'Quaternions act on 4-dimensional space, providing full information about what it does to all of space.', 'The relevance of quaternions in quantum mechanics is highlighted, particularly in describing special actions in four dimensions that are relevant to mathematical descriptions of two-state systems.', 'The historical significance and the amusing disputes surrounding quaternion mathematics, including the old-timey trash talk and references in literature such as Alice in Wonderland.', 'The concept of stretching and rotating actions in 2D and 4D space is discussed, with an emphasis on quaternion multiplication and a comparison to multiplication by real numbers.', 'The chapter proposes a stereographic projection as a solution to the difficulty in communicating rotation, describing it as a special way to map a circle onto a line, or a sphere into a plane, or even a 4D hypersphere into 3D space.', 'The chapter emphasizes the difficulty in communicating rotation to Linus, focusing on the unit circle of the complex plane where multiplication by numbers corresponds to pure rotation.']}