title

How to lie using visual proofs

description

Three false proofs, and what lessons they teach.
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Here's a nice short video on the false pi = 4 proof
https://www.youtube.com/watch?v=6Qnfd5dRyf4
Time stamps:
0:00 - Fake sphere proof
1:39 - Fake pi = 4 proof
5:16 - Fake proof that all triangles are isosceles
9:54 - Sphere "proof" explanation
15:09 - pi = 4 "proof" explanation
16:57 - Triangle "proof" explanation and conclusion
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detail

{'title': 'How to lie using visual proofs', 'heatmap': [{'end': 105.956, 'start': 90.197, 'weight': 0.708}, {'end': 823.466, 'start': 790.164, 'weight': 1}, {'end': 1027.657, 'start': 1017.611, 'weight': 0.713}], 'summary': "'how to lie using visual proofs' covers three fake proofs demonstrating mathematical insights, parametric functions defining shapes, euclid-style proof of isosceles triangles, and limitations in using visual proofs in geometry, emphasizing critical thinking in mathematical reasoning.", 'chapters': [{'end': 314.004, 'segs': [{'end': 119.725, 'src': 'heatmap', 'start': 90.197, 'weight': 1, 'content': [{'end': 97.642, 'text': "I originally saw this example thanks to Henry Reich, and, to be fair, it's not necessarily inconsistent with the 4 pi r-squared formula,", 'start': 90.197, 'duration': 7.445}, {'end': 99.163, 'text': 'just so long as pi is equal to 4..', 'start': 97.642, 'duration': 1.521}, {'end': 105.956, 'text': "For the next proof, I'd like to show you a simple argument for the fact that pi is equal to 4.", 'start': 99.163, 'duration': 6.793}, {'end': 110.619, 'text': 'We start off with a circle, say with radius 1, and we ask how can we figure out its circumference?', 'start': 105.956, 'duration': 4.663}, {'end': 115.382, 'text': 'After all, pi is by definition the ratio of this circumference to the diameter of the circle.', 'start': 110.979, 'duration': 4.403}, {'end': 119.725, 'text': 'We start off by drawing the square, whose side lengths are all tangent to that circle.', 'start': 115.983, 'duration': 3.742}], 'summary': "A simple argument claims pi equals 4 by showing a circle's circumference ratio to its diameter.", 'duration': 39.654, 'max_score': 90.197, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY90197.jpg'}, {'end': 219.008, 'src': 'embed', 'start': 189.64, 'weight': 3, 'content': [{'end': 193.724, 'text': 'the claim of the example is not that any one of these approximations equals the curve.', 'start': 189.64, 'duration': 4.084}, {'end': 197.147, 'text': "It's that the limit of all of the approximations equals our circle.", 'start': 194.104, 'duration': 3.043}, {'end': 200.411, 'text': 'And to appreciate the lesson that this example teaches us.', 'start': 197.588, 'duration': 2.823}, {'end': 206.017, 'text': "it's worth taking a moment to be a little more mathematically precise about what I mean by the limit of a sequence of curves.", 'start': 200.411, 'duration': 5.606}, {'end': 211.684, 'text': "Let's say we describe the very first shape, this square, as a parametric function,", 'start': 207.321, 'duration': 4.363}, {'end': 219.008, 'text': 'something that has an input t and it outputs a point in 2D space so that as t ranges from 0 to 1, it traces that square.', 'start': 211.684, 'duration': 7.324}], 'summary': 'The limit of all approximations equals our circle, teaching a lesson in mathematical precision.', 'duration': 29.368, 'max_score': 189.64, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY189640.jpg'}, {'end': 299.415, 'src': 'embed', 'start': 270.746, 'weight': 0, 'content': [{'end': 274.569, 'text': 'at any input t is whatever this limiting value for all the curves is.', 'start': 270.746, 'duration': 3.823}, {'end': 276.551, 'text': "So, here's the point.", 'start': 275.29, 'duration': 1.261}, {'end': 280.574, 'text': 'That limiting function, c infinity, is the circle.', 'start': 277.151, 'duration': 3.423}, {'end': 284.679, 'text': "It's not an approximation of the circle, it's not some jagged version of the circle.", 'start': 280.834, 'duration': 3.845}, {'end': 288.623, 'text': 'it is the genuine smooth circular curve whose perimeter we want to know.', 'start': 284.679, 'duration': 3.944}, {'end': 295.511, 'text': "And what's also true is that the limit of the lengths of all of our curves really is 8,", 'start': 289.664, 'duration': 5.847}, {'end': 299.415, 'text': 'because each individual curve really does have a perimeter of 8..', 'start': 295.511, 'duration': 3.904}], 'summary': 'The limiting function, c infinity, represents the genuine smooth circular curve with a perimeter of 8.', 'duration': 28.669, 'max_score': 270.746, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY270746.jpg'}], 'start': 0.149, 'title': 'Mathematical insights and parametric functions', 'summary': "Delves into three fake proofs highlighting mathematical insights, and explores parametric functions defining shapes and the 'c infinity' limiting function, challenging approximations with a genuine smooth circular curve of perimeter 8.", 'chapters': [{'end': 206.017, 'start': 0.149, 'title': 'Fake proofs and mathematical insights', 'summary': 'Discusses three fake proofs, including the surface area of a sphere and the value of pi, to illustrate mathematical insights and misconceptions.', 'duration': 205.868, 'highlights': ['The false proof for the surface area of a sphere cleverly translates the problem into a more understandable shape and area calculation, resulting in an elegant but incorrect result of pi squared times r squared, showcasing the misconception of the true surface area being 4 pi r-squared.', "A simplistic argument attempting to prove that pi is equal to 4 is presented by producing a sequence of curves that closely approximate the circle's perimeter, emphasizing the misconception of the limit of these approximations equaling the circle, teaching a mathematical lesson about the concept of limits and approximations.", 'The example of the sequence of curves approximating the circle highlights the misconception that the limit of all the approximations equals the circle, rather than any single approximation being equal to the curve, emphasizing the mathematical precision required to understand the concept of a limit of a sequence of curves.']}, {'end': 314.004, 'start': 207.321, 'title': 'Parametric functions and limiting curves', 'summary': "Explores the concept of parametric functions representing shapes and the idea of a limiting function, 'c infinity', which defines the genuine smooth circular curve with a perimeter of 8, challenging the use of approximation through limits.", 'duration': 106.683, 'highlights': ["The limiting function, 'c infinity', represents the genuine smooth circular curve with a perimeter of 8, not an approximation or jagged version of the circle.", 'The limit of the lengths of all curves is 8, as each individual curve has a perimeter of 8.', 'The concept challenges the typical calculus approach of approximating a complex shape through the limit of simpler components, raising the question of why such a method does not hold in this case.', 'The chapter introduces the concept of parametrizing shapes as functions, such as c0 and c1, with an input t ranging from 0 to 1, tracing the respective shapes in 2D space.', 'The discussion revolves around the consideration of a particular value of t, such as 0.2, and the sequence of points obtained by evaluating the sequence of functions at this specific point.']}], 'duration': 313.855, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY149.jpg', 'highlights': ["The limiting function, 'c infinity', represents the genuine smooth circular curve with a perimeter of 8.", 'The false proof for the surface area of a sphere results in an elegant but incorrect result of pi squared times r squared.', "A simplistic argument attempting to prove that pi is equal to 4 is presented by producing a sequence of curves that closely approximate the circle's perimeter.", 'The example of the sequence of curves approximating the circle highlights the misconception that the limit of all the approximations equals the circle.', 'The concept challenges the typical calculus approach of approximating a complex shape through the limit of simpler components.']}, {'end': 582.525, 'segs': [{'end': 356.281, 'src': 'embed', 'start': 333.506, 'weight': 1, 'content': [{'end': 342.313, 'text': "I would like to do something that doesn't lean as hard on visual intuition and instead give a Euclid-style proof for the claim that all triangles are isosceles.", 'start': 333.506, 'duration': 8.807}, {'end': 346.935, 'text': "The way this will work is we'll take any particular triangle and make no assumptions about it.", 'start': 342.873, 'duration': 4.062}, {'end': 350.097, 'text': "I'll label its vertices A, B, and C.", 'start': 347.116, 'duration': 2.981}, {'end': 356.281, 'text': 'And what I would like to prove for you is that the side length AB is necessarily equal to the side length AC.', 'start': 350.097, 'duration': 6.184}], 'summary': 'Proof that all triangles are isosceles using euclid-style reasoning.', 'duration': 22.775, 'max_score': 333.506, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY333506.jpg'}, {'end': 448.767, 'src': 'embed', 'start': 426.281, 'weight': 0, 'content': [{'end': 435.883, 'text': 'My first claim is that this triangle here, which is AFP, is the same, or at least congruent, to this triangle over here, AEP.', 'start': 426.281, 'duration': 9.602}, {'end': 439.524, 'text': 'Essentially, this follows from symmetry across that angle bisector.', 'start': 436.364, 'duration': 3.16}, {'end': 445.846, 'text': 'More specifically, we can say they share a side length, and then they both have an angle alpha and both have an angle 90 degrees.', 'start': 440.245, 'duration': 5.601}, {'end': 448.767, 'text': 'So it follows by the side-angle-angle congruence relation.', 'start': 446.046, 'duration': 2.721}], 'summary': 'Triangles afp and aep are congruent due to shared side length and angles.', 'duration': 22.486, 'max_score': 426.281, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY426281.jpg'}, {'end': 527.134, 'src': 'embed', 'start': 501.936, 'weight': 2, 'content': [{'end': 507.838, 'text': "And the reasoning here is they both have that triple ticked side, a double ticked side, and they're both 90 degree triangles.", 'start': 501.936, 'duration': 5.902}, {'end': 511.36, 'text': 'So this follows by the side-side angle congruence relation.', 'start': 508.379, 'duration': 2.981}, {'end': 514.129, 'text': 'And all of those are valid congruence relations.', 'start': 512.168, 'duration': 1.961}, {'end': 516.409, 'text': "I'm not pulling the wool over your eyes with one of those.", 'start': 514.229, 'duration': 2.18}, {'end': 521.571, 'text': 'And all of this will basically be enough to show us why AB has to be the same as BC.', 'start': 517.07, 'duration': 4.501}, {'end': 527.134, 'text': 'That first pair of triangles implies that the length AF is the same as the length AE.', 'start': 522.412, 'duration': 4.722}], 'summary': 'Two triangles with congruent sides and angles prove ab=bc and af=ae.', 'duration': 25.198, 'max_score': 501.936, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY501936.jpg'}, {'end': 589.028, 'src': 'embed', 'start': 562.329, 'weight': 5, 'content': [{'end': 566.971, 'text': 'And because we made no assumptions about the triangle, this implies that any triangle is isosceles.', 'start': 562.329, 'duration': 4.642}, {'end': 574.002, 'text': 'Actually, for that matter, since we made no assumptions about the specific two sides we chose, it implies that any triangle is equilateral.', 'start': 567.699, 'duration': 6.303}, {'end': 579.024, 'text': 'So this leaves us, somewhat disturbingly, with three different possibilities.', 'start': 575.602, 'duration': 3.422}, {'end': 582.525, 'text': "All triangles really are equilateral, that's just the truth of the universe.", 'start': 579.464, 'duration': 3.061}, {'end': 586.787, 'text': 'Or, you can use Euclid-style reasoning to derive false results.', 'start': 583.105, 'duration': 3.682}, {'end': 589.028, 'text': "Or, there's something wrong in the proof.", 'start': 587.327, 'duration': 1.701}], 'summary': 'The proof implies any triangle could be isosceles or equilateral, leading to three possibilities.', 'duration': 26.699, 'max_score': 562.329, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY562329.jpg'}], 'start': 317.034, 'title': 'Triangle congruence', 'summary': 'Explores euclid-style proof of isosceles triangles and the process of using congruence relations to prove triangles are equilateral, providing logical steps and visual aids for understanding.', 'chapters': [{'end': 469.993, 'start': 317.034, 'title': 'Euclid-style proof of isosceles triangle', 'summary': 'Explains a euclid-style proof that all triangles are isosceles, challenging the obvious false result, and presents step-by-step logic and visual aids to support the proof.', 'duration': 152.959, 'highlights': ['The chapter presents a Euclid-style proof that challenges the obvious false result that all triangles are isosceles, and encourages the audience to identify the flaw in the proof, adding an element of challenge and engagement.', 'A step-by-step logic with visual aids, like drawing perpendicular bisector and angle bisector, is used to support the proof of the claim that all triangles are isosceles, enhancing the understanding of the process.', 'The proof includes the explanation of the congruence of triangles AFP and AEP, supported by the side-angle-angle congruence relation, providing a clear and logical progression in the argument.']}, {'end': 582.525, 'start': 469.993, 'title': 'Triangle congruence and isosceles triangles', 'summary': 'Demonstrates the process of using side-angle-side and side-side angle congruence relations to prove the congruence of sides in triangles, resulting in the conclusion that all triangles are equilateral.', 'duration': 112.532, 'highlights': ['Using side-angle-side and side-side angle congruence relations to prove the congruence of sides in triangles', 'Concluding that all triangles are equilateral', 'Demonstrating the isosceles nature of any triangle']}], 'duration': 265.491, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY317034.jpg', 'highlights': ['The proof includes the explanation of the congruence of triangles AFP and AEP, supported by the side-angle-angle congruence relation, providing a clear and logical progression in the argument.', 'A step-by-step logic with visual aids, like drawing perpendicular bisector and angle bisector, is used to support the proof of the claim that all triangles are isosceles, enhancing the understanding of the process.', 'Using side-angle-side and side-side angle congruence relations to prove the congruence of sides in triangles', 'Demonstrating the isosceles nature of any triangle', 'The chapter presents a Euclid-style proof that challenges the obvious false result that all triangles are isosceles, and encourages the audience to identify the flaw in the proof, adding an element of challenge and engagement.', 'Concluding that all triangles are equilateral']}, {'end': 1090.696, 'segs': [{'end': 682.575, 'src': 'embed', 'start': 657.762, 'weight': 0, 'content': [{'end': 667.267, 'text': "The main problem with the sphere argument is that when we flatten out all of those orange wedges if we were to do it accurately in a way that preserves their area they don't look like triangles.", 'start': 657.762, 'duration': 9.505}, {'end': 668.608, 'text': 'They should bulge outward.', 'start': 667.367, 'duration': 1.241}, {'end': 670.129, 'text': 'And if you want to see this,', 'start': 669.028, 'duration': 1.101}, {'end': 677.573, 'text': "let's think really critically about just one particular one of those wedges on the sphere and ask yourself how does the width across that wedge,", 'start': 670.129, 'duration': 7.444}, {'end': 682.575, 'text': 'this little portion of a line of latitude, vary as you go up and down the wedge?', 'start': 677.573, 'duration': 5.002}], 'summary': 'Flattening spheres into triangles results in bulging outward, challenging the traditional triangular shape.', 'duration': 24.813, 'max_score': 657.762, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY657762.jpg'}, {'end': 798.408, 'src': 'embed', 'start': 771.828, 'weight': 6, 'content': [{'end': 777.113, 'text': 'It reminds me of one of those rearrangement puzzles where you have a number of pieces and just by moving them around,', 'start': 771.828, 'duration': 5.285}, {'end': 779.296, 'text': 'you can seemingly create area out of nowhere.', 'start': 777.113, 'duration': 2.183}, {'end': 785.482, 'text': "For example, right now I've arranged all these pieces to form a triangle, except it's missing two units of area in the middle.", 'start': 779.616, 'duration': 5.866}, {'end': 789.144, 'text': 'Now I want you to focus on the vertices of that triangle, these white dots.', 'start': 785.882, 'duration': 3.262}, {'end': 790.164, 'text': "Those don't move.", 'start': 789.364, 'duration': 0.8}, {'end': 795.427, 'text': "I'm not pulling any trickery with that, but I can rearrange all of the pieces back to how they originally were,", 'start': 790.164, 'duration': 5.263}, {'end': 798.408, 'text': 'so that those two units of area in the middle seem to disappear.', 'start': 795.427, 'duration': 2.981}], 'summary': 'A rearrangement puzzle creates a triangle missing 2 units of area, but can be rearranged to make the area disappear.', 'duration': 26.58, 'max_score': 771.828, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY771828.jpg'}, {'end': 823.466, 'src': 'heatmap', 'start': 790.164, 'weight': 1, 'content': [{'end': 795.427, 'text': "I'm not pulling any trickery with that, but I can rearrange all of the pieces back to how they originally were,", 'start': 790.164, 'duration': 5.263}, {'end': 798.408, 'text': 'so that those two units of area in the middle seem to disappear.', 'start': 795.427, 'duration': 2.981}, {'end': 806.372, 'text': 'All the constituent parts remain the same, the triangle that they form remains the same, and yet two units of area seem to appear out of nowhere.', 'start': 798.688, 'duration': 7.684}, {'end': 811.334, 'text': "If you've never seen this one before, by the way, I highly encourage you to pause and try to think it through.", 'start': 807.252, 'duration': 4.082}, {'end': 812.815, 'text': "It's a very fun little puzzle.", 'start': 811.514, 'duration': 1.301}, {'end': 823.466, 'text': "The answer starts to reveal itself if we carefully draw the edges of this triangle and zoom in close enough to see that our pieces don't actually fit inside the triangle.", 'start': 813.915, 'duration': 9.551}], 'summary': 'Rearranging pieces creates illusion of area change in triangle puzzle.', 'duration': 33.302, 'max_score': 790.164, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY790164.jpg'}, {'end': 866.182, 'src': 'embed', 'start': 841.72, 'weight': 2, 'content': [{'end': 849.462, 'text': 'The slope of the edge of this blue triangle works out to be 5, divided by 2, whereas the slope of the edge of this red triangle works out to be 7,', 'start': 841.72, 'duration': 7.742}, {'end': 850.742, 'text': 'divided by 3..', 'start': 849.462, 'duration': 1.28}, {'end': 856.303, 'text': 'Those numbers are close enough to look similar as slope, but they allow for this denting inward and the bulging outward.', 'start': 850.742, 'duration': 5.561}, {'end': 862.764, 'text': "You have to be wary of lines that are made to look straight when you haven't had explicit confirmation that they actually are straight.", 'start': 856.803, 'duration': 5.961}, {'end': 866.182, 'text': 'One quick added comment on the sphere.', 'start': 864.62, 'duration': 1.562}], 'summary': 'The blue triangle has a slope of 5/2 and the red triangle has a slope of 7/3, cautioning against assuming straight lines without confirmation.', 'duration': 24.462, 'max_score': 841.72, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY841720.jpg'}, {'end': 924.212, 'src': 'embed', 'start': 894.255, 'weight': 1, 'content': [{'end': 898.378, 'text': "It's only when you zoom in close to a curved surface that it appears locally flat.", 'start': 894.255, 'duration': 4.123}, {'end': 906.303, 'text': 'The issue with our orange wedge argument is that our pieces never got exposed to that local flatness because they only got thin in one direction.', 'start': 899.119, 'duration': 7.184}, {'end': 908.744, 'text': 'They maintain the curvature in that other direction.', 'start': 906.503, 'duration': 2.241}, {'end': 916.808, 'text': "Now, on the topic of the subtlety of limiting arguments, let's turn back to our limit of jagged curves that approaches the smooth circular curve.", 'start': 909.484, 'duration': 7.324}, {'end': 924.212, 'text': 'As I said, the limiting curve really is a circle, and the limiting value for the length of your approximations really is 8.', 'start': 917.248, 'duration': 6.964}], 'summary': 'Zooming in on a curved surface reveals local flatness; limiting jagged curves approach a smooth circular curve with a limiting value of 8.', 'duration': 29.957, 'max_score': 894.255, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY894255.jpg'}, {'end': 1027.657, 'src': 'heatmap', 'start': 996.66, 'weight': 4, 'content': [{'end': 1002.705, 'text': 'Essentially, the deviation between the curve and our approximating rectangles sits strictly inside that red region.', 'start': 996.66, 'duration': 6.045}, {'end': 1010.79, 'text': 'And then what you would want to argue is that in this limiting process, the cumulative area of all of those red rectangles has to approach zero.', 'start': 1003.125, 'duration': 7.665}, {'end': 1021.454, 'text': 'Now as to the final example, our proof that all triangles are isosceles.', 'start': 1017.611, 'duration': 3.843}, {'end': 1027.657, 'text': "let me show you what it looks like if I'm a little bit more careful about actually constructing the angle bisector rather than just eyeballing it.", 'start': 1021.454, 'duration': 6.203}], 'summary': 'The cumulative area of red rectangles approaches zero in the limiting process.', 'duration': 24.794, 'max_score': 996.66, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY996660.jpg'}, {'end': 1090.696, 'src': 'embed', 'start': 1081.513, 'weight': 5, 'content': [{'end': 1086.034, 'text': 'visual arguments and snazzy diagrams will never obviate the need for critical thinking.', 'start': 1081.513, 'duration': 4.521}, {'end': 1090.696, 'text': 'In math, you cannot escape the need to look out for hidden assumptions and edge cases.', 'start': 1086.474, 'duration': 4.222}], 'summary': 'Critical thinking is essential in math to account for hidden assumptions and edge cases.', 'duration': 9.183, 'max_score': 1081.513, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY1081513.jpg'}], 'start': 583.105, 'title': 'Limitations in geometry and limiting arguments', 'summary': 'Discusses limitations in using visual proofs in geometry, such as discrepancies in area calculations for circles and spheres. it also explores the subtleties of limiting arguments in mathematics, emphasizing the concept of gaussian curvature and the importance of critical thinking in mathematical reasoning.', 'chapters': [{'end': 856.303, 'start': 583.105, 'title': 'Visual proofs in geometry', 'summary': 'Discusses the limitations in using visual proofs in geometry, highlighting the discrepancies between the validity of proofs for area calculations in circles and spheres, and their underlying mathematical explanations. it also delves into a rearrangement puzzle to illustrate how subtle differences in visual arrangements can affect the calculated area.', 'duration': 273.198, 'highlights': ["The main problem with the sphere argument is that when we flatten out all of those orange wedges if we were to do it accurately in a way that preserves their area they don't look like triangles.", 'It reminds me of one of those rearrangement puzzles where you have a number of pieces and just by moving them around, you can seemingly create area out of nowhere.', 'The slope of the edge of this blue triangle works out to be 5, divided by 2, whereas the slope of the edge of this red triangle works out to be 7, divided by 3.']}, {'end': 1090.696, 'start': 856.803, 'title': 'Limiting arguments in mathematics', 'summary': 'Explores the subtleties of limiting arguments in mathematics, highlighting the concept of gaussian curvature, the need for care in applying limiting arguments, and the importance of critical thinking in mathematical reasoning.', 'duration': 233.893, 'highlights': ['The issue with our orange wedge argument is that our pieces never got exposed to local flatness because they only got thin in one direction, maintaining the curvature in the other direction.', 'The limiting curve of jagged curves approaching the smooth circular curve is a circle, and the limiting value for the length of the approximations is 8.', 'The deviation between the curve and approximating rectangles sits strictly inside the red region, and in the limiting process, the cumulative area of all of those red rectangles has to approach zero.', 'The proof that all triangles are isosceles breaks down due to the hidden assumption that the relevant point sits in between them, emphasizing the need for critical thinking in mathematical reasoning.']}], 'duration': 507.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/VYQVlVoWoPY/pics/VYQVlVoWoPY583105.jpg', 'highlights': ['The main problem with the sphere argument is the inaccurate preservation of area in flattened wedges.', 'The issue with the orange wedge argument is the lack of exposure to local flatness due to maintaining curvature.', "The slope of the edge of the blue triangle is 5 divided by 2, while the red triangle's slope is 7 divided by 3.", 'The limiting curve of jagged curves approaching the smooth circular curve is a circle, with a limiting value for the length of the approximations being 8.', 'The deviation between the curve and approximating rectangles sits strictly inside the red region, with the cumulative area of all red rectangles approaching zero in the limiting process.', 'The proof that all triangles are isosceles breaks down due to the hidden assumption, emphasizing the need for critical thinking in mathematical reasoning.', 'The sphere argument resembles rearrangement puzzles where seemingly creating area out of nowhere is possible.']}], 'highlights': ['The proof that all triangles are isosceles breaks down due to the hidden assumption, emphasizing the need for critical thinking in mathematical reasoning.', 'The concept challenges the typical calculus approach of approximating a complex shape through the limit of simpler components.', 'The proof includes the explanation of the congruence of triangles AFP and AEP, supported by the side-angle-angle congruence relation, providing a clear and logical progression in the argument.', 'The limiting curve of jagged curves approaching the smooth circular curve is a circle, with a limiting value for the length of the approximations being 8.', 'A step-by-step logic with visual aids, like drawing perpendicular bisector and angle bisector, is used to support the proof of the claim that all triangles are isosceles, enhancing the understanding of the process.', 'The false proof for the surface area of a sphere results in an elegant but incorrect result of pi squared times r squared.', 'The main problem with the sphere argument is the inaccurate preservation of area in flattened wedges.', 'The issue with the orange wedge argument is the lack of exposure to local flatness due to maintaining curvature.', 'The example of the sequence of curves approximating the circle highlights the misconception that the limit of all the approximations equals the circle.', "The limiting function, 'c infinity', represents the genuine smooth circular curve with a perimeter of 8."]}