title

The Brachistochrone, with Steven Strogatz

description

Steven Strogatz and I talk about a famous historical math problem, a clever solution, and a modern twist.
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{'title': 'The Brachistochrone, with Steven Strogatz', 'heatmap': [{'end': 505.377, 'start': 487.313, 'weight': 1}], 'summary': "Explores the history, modern insights, and applications of the brachistochrone problem, including solutions by johann bernoulli, fermat's principle, and the application of math in science, focusing on the geometry of a cycloid and its relationship to the principle of constant velocity over sine theta.", 'chapters': [{'end': 60.874, 'segs': [{'end': 60.874, 'src': 'embed', 'start': 21.675, 'weight': 0, 'content': [{'end': 25.858, 'text': "To put it shortly, he's one of the great mass communicators of math in our time.", 'start': 21.675, 'duration': 4.183}, {'end': 30.69, 'text': 'In our conversation we talked about a lot of things,', 'start': 27.967, 'duration': 2.723}, {'end': 35.916, 'text': 'but it was all centering around this one very famous problem in the history of math the Brachistochrone.', 'start': 30.69, 'duration': 5.226}, {'end': 40.842, 'text': "And for the first two-thirds or so of the video, I'm just going to play some of that conversation.", 'start': 36.657, 'duration': 4.185}, {'end': 47.269, 'text': 'We lay out the problem, talk about some of its history, and go through the solution by Johann Bernoulli from the 17th century.', 'start': 41.462, 'duration': 5.807}, {'end': 51.211, 'text': "After that, I'm going to show this proof that Steve showed me.", 'start': 48.55, 'duration': 2.661}, {'end': 57.953, 'text': "It's by a modern mathematician, Mark Levy, and it gives a certain geometric insight to Johann Bernoulli's original solution.", 'start': 51.571, 'duration': 6.382}, {'end': 60.874, 'text': 'Then at the very end, I have a little challenge for you.', 'start': 58.673, 'duration': 2.201}], 'summary': "He's a great communicator of math, discussing the brachistochrone problem and its historical solution by johann bernoulli, along with a modern geometric insight by mark levy.", 'duration': 39.199, 'max_score': 21.675, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU21675.jpg'}], 'start': 4.465, 'title': 'The brachistochrone problem', 'summary': 'Discusses the history and modern insights of the brachistochrone problem, including solutions by johann bernoulli and modern geometric insight by mark levy, culminating in a challenge.', 'chapters': [{'end': 60.874, 'start': 4.465, 'title': 'The brachistochrone problem and its modern insights', 'summary': 'Delves into a conversation with mathematician steven strogatz about the brachistochrone problem, discussing its history, the solution by johann bernoulli, and a modern geometric insight by mark levy, culminating in a challenge.', 'duration': 56.409, 'highlights': ['Steven Strogatz, a mathematician at Cornell, discusses the famous Brachistochrone problem and its solution by Johann Bernoulli from the 17th century.', "Mark Levy provides a modern geometric insight into Johann Bernoulli's original solution for the Brachistochrone problem.", 'The video presents a challenge related to the Brachistochrone problem at the end.']}], 'duration': 56.409, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU4465.jpg', 'highlights': ["Mark Levy provides a modern geometric insight into Johann Bernoulli's original solution for the Brachistochrone problem.", 'Steven Strogatz, a mathematician at Cornell, discusses the famous Brachistochrone problem and its solution by Johann Bernoulli from the 17th century.', 'The video presents a challenge related to the Brachistochrone problem at the end.']}, {'end': 609.756, 'segs': [{'end': 180.709, 'src': 'embed', 'start': 147.903, 'weight': 2, 'content': [{'end': 149.343, 'text': 'I mean, you could picture it sliding.', 'start': 147.903, 'duration': 1.44}, {'end': 151.104, 'text': "That doesn't really matter how we phrase it.", 'start': 149.403, 'duration': 1.701}, {'end': 158.627, 'text': 'So Galileo had thought about this himself much earlier than Johann Bernoulli in 1638.', 'start': 151.764, 'duration': 6.863}, {'end': 162.772, 'text': 'And Galileo thought that an arc of a circle would be the best thing.', 'start': 158.627, 'duration': 4.145}, {'end': 164.934, 'text': 'So he had the idea that a bit of curvature might help.', 'start': 162.812, 'duration': 2.122}, {'end': 168.318, 'text': 'And it turns out that the arc of the circle is not the right answer.', 'start': 165.595, 'duration': 2.723}, {'end': 170.821, 'text': "It's good, but there are better solutions.", 'start': 168.679, 'duration': 2.142}, {'end': 176.207, 'text': 'And the history of real solutions starts with Johann Bernoulli posing this as a challenge.', 'start': 171.662, 'duration': 4.545}, {'end': 180.709, 'text': "So that's then in June of 1696.", 'start': 177.248, 'duration': 3.461}], 'summary': "Johann bernoulli proposed a challenge in june 1696, seeking better solutions than galileo's arc of a circle.", 'duration': 32.806, 'max_score': 147.903, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU147903.jpg'}, {'end': 270.69, 'src': 'embed', 'start': 241.158, 'weight': 0, 'content': [{'end': 243.94, 'text': 'especially by somebody that he considered beneath him.', 'start': 241.158, 'duration': 2.782}, {'end': 246.201, 'text': 'I mean, he considered pretty much everybody beneath him.', 'start': 243.98, 'duration': 2.221}, {'end': 256.087, 'text': 'But yeah, Newton stayed up all night, solved it, and then sent it in anonymously to the Philosophical Transactions, the journal at the time.', 'start': 246.721, 'duration': 9.366}, {'end': 258.666, 'text': 'And it was published anonymously.', 'start': 256.927, 'duration': 1.739}, {'end': 262.508, 'text': 'And so Newton complained in a letter to a friend of his.', 'start': 258.747, 'duration': 3.761}, {'end': 267.729, 'text': 'He said, I do not love to be dunned and teased by foreigners about mathematical things.', 'start': 262.548, 'duration': 5.181}, {'end': 270.69, 'text': "So he didn't enjoy this challenge, but he did solve it.", 'start': 268.429, 'duration': 2.261}], 'summary': 'Newton anonymously solved mathematical challenge and published it, despite not enjoying the challenge.', 'duration': 29.532, 'max_score': 241.158, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU241158.jpg'}, {'end': 421.009, 'src': 'embed', 'start': 384.043, 'weight': 1, 'content': [{'end': 387.264, 'text': 'And I think before we dive into that case, we should look at something simpler.', 'start': 384.043, 'duration': 3.221}, {'end': 390.805, 'text': "So at this point in the conversation, we talked for a while about Snell's law.", 'start': 387.284, 'duration': 3.521}, {'end': 397.688, 'text': 'This is a result in physics that describes how light bends when it goes from one material into another, where its speed changes.', 'start': 391.326, 'duration': 6.362}, {'end': 403.144, 'text': "I made a separate video out of this, talking about how you can prove it using Fermat's principle,", 'start': 398.703, 'duration': 4.441}, {'end': 407.085, 'text': 'together with a very neat argument using imaginary constant tension springs.', 'start': 403.144, 'duration': 3.941}, {'end': 410.786, 'text': "But for now, all you need to know is the statement of Snell's law itself.", 'start': 407.625, 'duration': 3.161}, {'end': 421.009, 'text': 'When a beam of light passes from one medium into another and you consider the angle that it makes with a line perpendicular to the boundary between those two materials,', 'start': 411.666, 'duration': 9.343}], 'summary': "Discussed snell's law in physics, relating to light bending between materials.", 'duration': 36.966, 'max_score': 384.043, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU384043.jpg'}, {'end': 518.192, 'src': 'heatmap', 'start': 487.313, 'weight': 1, 'content': [{'end': 491.855, 'text': 'The velocity in the first one is v1 and the next one is v2 and the next one is v3.', 'start': 487.313, 'duration': 4.542}, {'end': 496.837, 'text': 'And these are all going to be proportional to the square root of y1 or y2 or y3.', 'start': 492.235, 'duration': 4.602}, {'end': 505.377, 'text': 'And, in principle, you should be thinking about a limiting process where you have infinitely many, infinitely thin layers,', 'start': 498.689, 'duration': 6.688}, {'end': 508.2, 'text': 'and this is kind of a continuous change for the speed of light.', 'start': 505.377, 'duration': 2.823}, {'end': 518.192, 'text': "And so then his question is if light is always instantaneously obeying Snell's law as it goes from one medium to the next,", 'start': 509.362, 'duration': 8.83}], 'summary': 'Velocity v1, v2, and v3 proportional to square root of y1, y2, y3, discussing continuous change for the speed of light.', 'duration': 30.879, 'max_score': 487.313, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU487313.jpg'}, {'end': 588.818, 'src': 'embed', 'start': 565.559, 'weight': 3, 'content': [{'end': 572.745, 'text': 'he just recognized it as the differential equation for a cycloid the shape traced by the point on the rim of a rolling wheel.', 'start': 565.559, 'duration': 7.186}, {'end': 582.953, 'text': "But it's not obvious, certainly not obvious to me, why this sine of theta over square root y property has anything to do with rolling wheels.", 'start': 573.525, 'duration': 9.428}, {'end': 588.818, 'text': "It's not at all obvious, but this is, again, the genius of Mark Levy to the rescue.", 'start': 584.194, 'duration': 4.624}], 'summary': 'Recognized differential equation for a cycloid, related to rolling wheels.', 'duration': 23.259, 'max_score': 565.559, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU565559.jpg'}], 'start': 65.031, 'title': "Brachistochrone problem and fermat's principle", 'summary': "Covers the history and solving of the brachistochrone problem, including the challenge posed by johann bernoulli, involvement of galileo and rivalry with isaac newton, and the application of fermat's principle of least time to understand the behavior of light, including its relation to snell's law and the discovery of the differential equation for a cycloid.", 'chapters': [{'end': 305.598, 'start': 65.031, 'title': 'Brachistochrone problem: history and solutions', 'summary': "Discusses the history and solving of the brachistochrone problem, including the challenge posed by johann bernoulli, the involvement of galileo and the famous rivalry with isaac newton, culminating in newton's anonymous publication of the solution.", 'duration': 240.567, 'highlights': ['Johann Bernoulli posed the Brachistochrone problem as a challenge to the mathematical world in June of 1696, with the intention of proving himself smarter than his brother and rival, Jacob Bernoulli.', "Isaac Newton anonymously solved the Brachistochrone problem in one night, despite Johann Bernoulli taking two weeks to solve it, leading to a legendary quote by Johann on recognizing the solution as Newton's.", 'The involvement of Galileo in the Brachistochrone problem, who initially thought that an arc of a circle would be the best path, but it was later discovered that there are better solutions.', 'The qualitative and quantitative aspects of the Brachistochrone problem, where the challenge lies in finding the balance between a short path and the object gaining speed, making it an intriguing and non-obvious problem to solve.', 'The description of the Brachistochrone problem, involving finding the path for a particle moving down a chute pulled by gravity, connecting two points in the shortest amount of time, emphasizing the difficulty in making a specific curve quantitative and finding the optimal path.']}, {'end': 609.756, 'start': 305.598, 'title': "Fermat's principle of least time", 'summary': "Discusses the application of fermat's principle of least time to understand the behavior of light, including its relation to snell's law and the discovery of the differential equation for a cycloid.", 'duration': 304.158, 'highlights': ["The chapter discusses the application of Fermat's principle of least time to understand the behavior of light, including its relation to Snell's law. The concept of Fermat's principle of least time is applied to explain the behavior of light, specifically its interaction with different mediums and the bending of light as described by Snell's law.", 'The discovery of the differential equation for a cycloid is discussed, relating to the sine of theta over square root y property. The differential equation for a cycloid, which is the shape traced by the point on the rim of a rolling wheel, is linked to the sine of theta over square root y property, revealing a connection between light behavior and the motion of rolling wheels.']}], 'duration': 544.725, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU65031.jpg', 'highlights': ["Isaac Newton anonymously solved the Brachistochrone problem in one night, despite Johann Bernoulli taking two weeks to solve it, leading to a legendary quote by Johann on recognizing the solution as Newton's.", "The chapter discusses the application of Fermat's principle of least time to understand the behavior of light, including its relation to Snell's law.", 'The involvement of Galileo in the Brachistochrone problem, who initially thought that an arc of a circle would be the best path, but it was later discovered that there are better solutions.', 'The discovery of the differential equation for a cycloid is discussed, relating to the sine of theta over square root y property.']}, {'end': 952.058, 'segs': [{'end': 654.276, 'src': 'embed', 'start': 610.257, 'weight': 0, 'content': [{'end': 614.099, 'text': "That is, rather than math in the service of science, it's science in the service of math.", 'start': 610.257, 'duration': 3.842}, {'end': 622.705, 'text': 'And as an example of the kinds of clever things that he does, he recently published a little note, very short,', 'start': 614.819, 'duration': 7.886}, {'end': 628.909, 'text': 'showing that if you look at the geometry of a cycloid, just drawing the correct lines in the right places,', 'start': 622.705, 'duration': 6.204}, {'end': 638.316, 'text': 'that this principle of velocity over sine theta being constant, is built in to the motion of the cycloid itself.', 'start': 628.909, 'duration': 9.407}, {'end': 646.971, 'text': 'So in that conversation, we never actually talked about the details of the proof itself.', 'start': 642.648, 'duration': 4.323}, {'end': 649.653, 'text': "It's kind of a hard thing to do without visuals.", 'start': 647.491, 'duration': 2.162}, {'end': 654.276, 'text': 'But I think a lot of you out there enjoy seeing the math and not just talking about the math.', 'start': 650.273, 'duration': 4.003}], 'summary': 'Science serves math, as seen in the geometry of a cycloid.', 'duration': 44.019, 'max_score': 610.257, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU610257.jpg'}, {'end': 702.847, 'src': 'embed', 'start': 677.798, 'weight': 4, 'content': [{'end': 682.941, 'text': "It's as if, for that moment, P is on the end of a pendulum whose base is at C.", 'start': 677.798, 'duration': 5.143}, {'end': 693.905, 'text': 'Since the tangent line of any circle is always perpendicular to the radius, the tangent line of the cycloid path of P is perpendicular to the line PC.', 'start': 684.423, 'duration': 9.482}, {'end': 702.847, 'text': 'This gives us a right angle inside of the circle, and any right triangle inscribed in a circle must have the diameter as its hypotenuse.', 'start': 694.946, 'duration': 7.901}], 'summary': 'P moves on cycloid path with right angles inside circle.', 'duration': 25.049, 'max_score': 677.798, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU677798.jpg'}, {'end': 755.355, 'src': 'embed', 'start': 732.098, 'weight': 2, 'content': [{'end': 740.152, 'text': 'this length times sine of theta again gives the distance between P and the ceiling the distance that we were calling y earlier.', 'start': 732.098, 'duration': 8.054}, {'end': 750.011, 'text': 'Rearranging this, we see that sine of theta divided by the square root of y is equal to 1 divided by the square root of the diameter.', 'start': 740.173, 'duration': 9.838}, {'end': 755.355, 'text': 'Since the diameter of a circle, of course, stays constant throughout the rotation.', 'start': 751.032, 'duration': 4.323}], 'summary': 'Sine of theta divided by square root of y = 1 divided by square root of diameter', 'duration': 23.257, 'max_score': 732.098, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU732098.jpg'}, {'end': 852.814, 'src': 'embed', 'start': 824.386, 'weight': 1, 'content': [{'end': 829.268, 'text': 'It turns out the wheel will rotate at a constant rate, which is surprising.', 'start': 824.386, 'duration': 4.882}, {'end': 835.971, 'text': 'This means that gravity pulls you along a cycloid in precisely the same way that a constantly rotating wheel would.', 'start': 829.808, 'duration': 6.163}, {'end': 840.553, 'text': 'The warm-up part of this challenge is just confirm this for yourself.', 'start': 837.451, 'duration': 3.102}, {'end': 843.294, 'text': "It's kind of fun to see how it falls out of the equations.", 'start': 840.933, 'duration': 2.361}, {'end': 845.207, 'text': 'But this got me thinking.', 'start': 844.326, 'duration': 0.881}, {'end': 852.814, 'text': 'If we look back at our original Brachistochrone problem, asking about the path of fastest descent between two given points,', 'start': 845.988, 'duration': 6.826}], 'summary': 'Gravity pulls along a cycloid at a constant rate, surprising result.', 'duration': 28.428, 'max_score': 824.386, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU824386.jpg'}], 'start': 610.257, 'title': 'Mathematics serving science', 'summary': 'Delves into the application of math in science, focusing on the geometry of a cycloid and its relationship to the principle of constant velocity over sine theta, and exploring the properties of the cycloid path as well as rethinking the brachistochrone problem in terms of the velocity vector angle.', 'chapters': [{'end': 654.276, 'start': 610.257, 'title': 'Math in the service of science', 'summary': 'Discusses the concept of science in the service of math, exemplified by a recent note on the geometry of a cycloid and its relation to the principle of velocity over sine theta being constant.', 'duration': 44.019, 'highlights': ['The concept of science in the service of math is exemplified by a recent note on the geometry of a cycloid and its relation to the principle of velocity over sine theta being constant.', 'The author recently published a note showing that the principle of velocity over sine theta being constant is built into the motion of the cycloid itself.', 'The conversation did not delve into the details of the proof due to the visual nature of the topic, but highlights the enjoyment in seeing math rather than just talking about it.']}, {'end': 952.058, 'start': 654.896, 'title': 'The brachistochrone and cycloid', 'summary': 'Discusses the concept of cycloid, exploring the geometry of a wheel rolling on the ceiling, proving the properties of the cycloid path, and rethinking the brachistochrone problem in terms of the angle of the velocity vector.', 'duration': 297.162, 'highlights': ['The geometry of a wheel rolling on the ceiling, and the properties of the cycloid path, including the right angle inside the circle and the tangent line always intersecting the bottom of the circle.', "The relationship between the angle theta, the distance to the ceiling y, and the diameter, leading to the constant property of sine of theta divided by square root of y on a cycloid, which aligns with Snell's Law.", 'The surprising revelation that a wheel rotating at a constant rate mimics the descent of an object due to gravity along a cycloid, and the challenge to reframe the Brachistochrone problem in terms of the angle of the velocity vector.']}], 'duration': 341.801, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/Cld0p3a43fU/pics/Cld0p3a43fU610257.jpg', 'highlights': ['The author recently published a note showing that the principle of velocity over sine theta being constant is built into the motion of the cycloid itself.', 'The surprising revelation that a wheel rotating at a constant rate mimics the descent of an object due to gravity along a cycloid, and the challenge to reframe the Brachistochrone problem in terms of the angle of the velocity vector.', "The relationship between the angle theta, the distance to the ceiling y, and the diameter, leading to the constant property of sine of theta divided by square root of y on a cycloid, which aligns with Snell's Law.", 'The concept of science in the service of math is exemplified by a recent note on the geometry of a cycloid and its relation to the principle of velocity over sine theta being constant.', 'The geometry of a wheel rolling on the ceiling, and the properties of the cycloid path, including the right angle inside the circle and the tangent line always intersecting the bottom of the circle.']}], 'highlights': ['The author recently published a note showing that the principle of velocity over sine theta being constant is built into the motion of the cycloid itself.', 'The concept of science in the service of math is exemplified by a recent note on the geometry of a cycloid and its relation to the principle of velocity over sine theta being constant.', "The relationship between the angle theta, the distance to the ceiling y, and the diameter, leading to the constant property of sine of theta divided by square root of y on a cycloid, which aligns with Snell's Law.", 'The surprising revelation that a wheel rotating at a constant rate mimics the descent of an object due to gravity along a cycloid, and the challenge to reframe the Brachistochrone problem in terms of the angle of the velocity vector.', 'The video presents a challenge related to the Brachistochrone problem at the end.', "Isaac Newton anonymously solved the Brachistochrone problem in one night, despite Johann Bernoulli taking two weeks to solve it, leading to a legendary quote by Johann on recognizing the solution as Newton's.", "The chapter discusses the application of Fermat's principle of least time to understand the behavior of light, including its relation to Snell's law.", 'The involvement of Galileo in the Brachistochrone problem, who initially thought that an arc of a circle would be the best path, but it was later discovered that there are better solutions.', "Mark Levy provides a modern geometric insight into Johann Bernoulli's original solution for the Brachistochrone problem.", 'Steven Strogatz, a mathematician at Cornell, discusses the famous Brachistochrone problem and its solution by Johann Bernoulli from the 17th century.']}