title

Who cares about topology? (Inscribed rectangle problem)

description

An unsolved conjecture, and a clever topological solution to a similar question.
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This video is based on a proof from H. Vaughan, 1977. To learn more, take a look at this survey:
https://pure.mpg.de/rest/items/item_3120610/component/file_3120611/content
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{'title': 'Who cares about topology? (Inscribed rectangle problem)', 'heatmap': [{'end': 276.251, 'start': 237.338, 'weight': 0.907}, {'end': 716.826, 'start': 698.257, 'weight': 0.774}, {'end': 763.669, 'start': 729.696, 'weight': 0.747}, {'end': 800.793, 'start': 764.709, 'weight': 0.709}, {'end': 912.899, 'start': 853.97, 'weight': 0.704}, {'value': 0.9397533914481632, 'end_time': 912.899, 'start_time': 884.821}], 'summary': 'Explores the unsolved inscribed square problem, presenting a solution for a weaker version and highlighting the relevance of topology in mathematics. additionally, it discusses the loop pairing theorem and its application on 2d surfaces, providing insights into gluing techniques and problem-solving using the mobius strip.', 'chapters': [{'end': 147.05, 'segs': [{'end': 49.054, 'src': 'embed', 'start': 24.827, 'weight': 2, 'content': [{'end': 32.389, 'text': 'I would occasionally find myself in some talk or a seminar where people wanted to get the youth excited about things that mathematicians care about.', 'start': 24.827, 'duration': 7.562}, {'end': 37.471, 'text': 'A very common go-to topic to excite our imaginations was topology.', 'start': 33.27, 'duration': 4.201}, {'end': 44.993, 'text': 'We might be shown something like a Mobius strip, maybe building it out of construction paper by twisting a rectangle and gluing its ends.', 'start': 38.111, 'duration': 6.882}, {'end': 49.054, 'text': "Look, we'd be told, as we were asked to draw a line along the surface.", 'start': 45.613, 'duration': 3.441}], 'summary': 'Mathematicians use topology, like the mobius strip, to excite youth in seminars.', 'duration': 24.227, 'max_score': 24.827, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K824827.jpg'}, {'end': 147.05, 'src': 'embed', 'start': 123.288, 'weight': 0, 'content': [{'end': 132.696, 'text': "I mean just the fact that this is unsolved is interesting that the current tools of math can neither confirm nor deny that there's some loop with no inscribed square in it.", 'start': 123.288, 'duration': 9.408}, {'end': 141.448, 'text': "Now, if we weaken the question a bit and ask about inscribed rectangles instead of inscribed squares, it's still pretty hard,", 'start': 134.045, 'duration': 7.403}, {'end': 147.05, 'text': 'but there is a beautiful, video-worthy solution that might actually be my favorite piece of math.', 'start': 141.448, 'duration': 5.602}], 'summary': 'Unsolved math problem: no inscribed square, beautiful solution for rectangles.', 'duration': 23.762, 'max_score': 123.288, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8123288.jpg'}], 'start': 4.608, 'title': 'The inscribed square problem and topology', 'summary': 'Discusses the unsolved inscribed square problem, presents an elegant solution for a weaker version, and explains the relevance of topology in mathematics, providing insight into why mathematicians care about certain shapes and their properties.', 'chapters': [{'end': 147.05, 'start': 4.608, 'title': 'The inscribed square problem and topology', 'summary': 'Discusses the unsolved inscribed square problem, presents an elegant solution for a weaker version, and explains the relevance of topology in mathematics, providing insight into why mathematicians care about certain shapes and their properties.', 'duration': 142.442, 'highlights': ['The inscribed square problem, asking whether every possible closed loop has at least one inscribed square, remains unsolved, showcasing the limitations of current mathematical tools.', 'A weaker version of the problem involves finding inscribed rectangles, which has a beautiful and video-worthy solution, offering valuable insights into mathematics.', 'The chapter also delves into the relevance of topology in mathematics, shedding light on why mathematicians care about certain shapes and their properties, providing a deeper understanding of the subject.']}], 'duration': 142.442, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K84608.jpg', 'highlights': ['The inscribed square problem remains unsolved, showcasing limitations of current tools.', 'A weaker version involves finding inscribed rectangles, with a beautiful solution.', 'The chapter delves into the relevance of topology in mathematics.']}, {'end': 705.319, 'segs': [{'end': 276.251, 'src': 'heatmap', 'start': 194.535, 'weight': 0, 'content': [{'end': 198.118, 'text': "So what we're going to do is try to prove that for any closed loop,", 'start': 194.535, 'duration': 3.583}, {'end': 204.763, 'text': "it's always possible to find two distinct pairs of points on that loop that share a midpoint and which are the same distance apart.", 'start': 198.118, 'duration': 6.645}, {'end': 207.144, 'text': "Take a moment to make sure that's clear.", 'start': 205.763, 'duration': 1.381}, {'end': 213.429, 'text': "We're finding two distinct pairs of points that share a common midpoint and which are the same distance apart.", 'start': 207.645, 'duration': 5.784}, {'end': 225.609, 'text': "The way we'll go about this is to define a function that takes in pairs of points on the loop and outputs a single point in 3D space,", 'start': 218.362, 'duration': 7.247}, {'end': 228.772, 'text': 'which kind of encodes the midpoint and distance information.', 'start': 225.609, 'duration': 3.163}, {'end': 230.574, 'text': 'It will be sort of like a graph.', 'start': 229.293, 'duration': 1.281}, {'end': 236.517, 'text': 'Consider the closed loop to be sitting on the xy-plane in 3D space.', 'start': 232.694, 'duration': 3.823}, {'end': 246.385, 'text': 'For a given pair of points, label their midpoint m, which will be some point on the xy-plane, and label the distance between them d.', 'start': 237.338, 'duration': 9.047}, {'end': 251.729, 'text': 'Plot the point which is exactly d units above that midpoint m in the z direction.', 'start': 246.385, 'duration': 5.344}, {'end': 260.517, 'text': "As you do this for many possible pairs of points, you'll effectively be drawing through 3D space.", 'start': 255.194, 'duration': 5.323}, {'end': 267.324, 'text': "And if you do it for all possible pairs of points on the loop, you'll draw out some kind of surface above the plane.", 'start': 261.379, 'duration': 5.945}, {'end': 272.288, 'text': 'Now look at the surface, and notice how it seems to hug the loop itself.', 'start': 268.745, 'duration': 3.543}, {'end': 276.251, 'text': "This is actually going to be important later, so let's think about why it happens.", 'start': 273.068, 'duration': 3.183}], 'summary': 'Proving that for any closed loop, we can find two pairs of points sharing a midpoint and the same distance apart through a function that encodes midpoint and distance information in 3d space.', 'duration': 65.982, 'max_score': 194.535, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8194535.jpg'}, {'end': 338.652, 'src': 'embed', 'start': 307.239, 'weight': 2, 'content': [{'end': 310.121, 'text': 'Another important fact is that this function is continuous.', 'start': 307.239, 'duration': 2.882}, {'end': 314.543, 'text': 'And all that really means is that if you slightly adjust a given pair of points,', 'start': 310.741, 'duration': 3.802}, {'end': 318.725, 'text': 'then the corresponding output in 3D space is only slightly adjusted as well.', 'start': 314.543, 'duration': 4.182}, {'end': 321.227, 'text': "There's never a sudden discontinuous jump.", 'start': 319.166, 'duration': 2.061}, {'end': 330.69, 'text': 'Our goal, then, is to show that this function has a collision, that two distinct pairs of points each map to the same spot in 3D space.', 'start': 322.547, 'duration': 8.143}, {'end': 338.652, 'text': 'Because the only way for that to happen is if they share a common midpoint, and if their distance d apart from each other is the same.', 'start': 331.71, 'duration': 6.942}], 'summary': 'Goal: prove function collision with common midpoint and equal distance.', 'duration': 31.413, 'max_score': 307.239, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8307239.jpg'}, {'end': 611.444, 'src': 'embed', 'start': 580.544, 'weight': 3, 'content': [{'end': 586.308, 'text': 'The torus is to pair of points on the loop what the xy-plane is to pairs of points on the real number line.', 'start': 580.544, 'duration': 5.764}, {'end': 598.115, 'text': "The key property of this association is that it's continuous both ways, meaning if you nudge any point on the torus by just a tiny amount,", 'start': 590.086, 'duration': 8.029}, {'end': 602.721, 'text': 'it corresponds to only a very slight nudge to the pair of points on the loop and vice versa.', 'start': 598.115, 'duration': 4.606}, {'end': 611.444, 'text': "So, if the torus is the natural shape for ordered pairs of points on the loop, what's the natural shape for unordered pairs?", 'start': 605.12, 'duration': 6.324}], 'summary': 'The torus is a natural shape for ordered pairs on a loop, with continuous association.', 'duration': 30.9, 'max_score': 580.544, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8580544.jpg'}], 'start': 148.26, 'title': 'Loop pairing theorem', 'summary': "Discusses the loop pairing theorem, proving that for any closed loop, it's always possible to find two distinct pairs of points sharing a midpoint and the same distance apart using a defined function and 2d surfaces, providing a comprehensive understanding of this concept.", 'chapters': [{'end': 330.69, 'start': 148.26, 'title': 'Loop pairing theorem', 'summary': "Discusses the loop pairing theorem, stating that for any closed loop, it's always possible to find two distinct pairs of points on that loop that share a midpoint and which are the same distance apart, achieved by defining a function that encodes the midpoint and distance information of pairs of points on the loop and drawing out a surface above the plane that hugs the loop itself, ultimately proving that this function has a collision.", 'duration': 182.43, 'highlights': ["The Loop Pairing Theorem states that for any closed loop, it's always possible to find two distinct pairs of points on that loop that share a midpoint and which are the same distance apart.", 'By defining a function that takes in pairs of points on the loop and outputs a single point in 3D space encoding the midpoint and distance information, it is proven that this function has a collision, where two distinct pairs of points each map to the same spot in 3D space.', 'The function defined is continuous, meaning that if a given pair of points is slightly adjusted, the corresponding output in 3D space is only slightly adjusted as well, without any sudden discontinuous jump.', 'The process involves plotting the point which is exactly the distance between the pair of points above the midpoint in the z-direction, effectively drawing through 3D space and creating a surface above the plane that hugs the loop itself, demonstrating the possibility of finding two distinct pairs of points on the loop that share a midpoint and are the same distance apart.']}, {'end': 705.319, 'start': 331.71, 'title': 'Understanding pairs of points on a loop', 'summary': 'Discusses the concept of representing pairs of points on a loop using 2d surfaces and shows how the torus represents ordered pairs of points while the folding of a square represents unordered pairs, aiming to demonstrate the shared midpoint and equal distance between distinct pairs.', 'duration': 373.609, 'highlights': ['The torus represents ordered pairs of points on the loop, and every point on the torus corresponds to a unique pair of points on the loop, with the association being continuous both ways.', 'The folding of the square over a diagonal line represents unordered pairs of points on the loop, with the diagonal line representing pairs of points that are actually the same point written twice.', 'The chapter emphasizes the need to understand unordered pairs of points and demonstrates the process of representing them using a folding square, which is crucial to showing the shared midpoint and equal distance between distinct pairs.']}], 'duration': 557.059, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8148260.jpg', 'highlights': ["The Loop Pairing Theorem states that for any closed loop, it's always possible to find two distinct pairs of points on that loop that share a midpoint and which are the same distance apart.", 'The process involves plotting the point which is exactly the distance between the pair of points above the midpoint in the z-direction, effectively drawing through 3D space and creating a surface above the plane that hugs the loop itself, demonstrating the possibility of finding two distinct pairs of points on the loop that share a midpoint and are the same distance apart.', 'The function defined is continuous, meaning that if a given pair of points is slightly adjusted, the corresponding output in 3D space is only slightly adjusted as well, without any sudden discontinuous jump.', 'The torus represents ordered pairs of points on the loop, and every point on the torus corresponds to a unique pair of points on the loop, with the association being continuous both ways.', 'By defining a function that takes in pairs of points on the loop and outputs a single point in 3D space encoding the midpoint and distance information, it is proven that this function has a collision, where two distinct pairs of points each map to the same spot in 3D space.']}, {'end': 974.357, 'segs': [{'end': 763.669, 'src': 'heatmap', 'start': 705.559, 'weight': 0, 'content': [{'end': 707.82, 'text': 'We need to glue that bottom edge to the right edge.', 'start': 705.559, 'duration': 2.261}, {'end': 711.782, 'text': 'And the orientation with which we do this is going to be important.', 'start': 708.9, 'duration': 2.882}, {'end': 716.826, 'text': 'Points towards the left of the bottom edge have to be glued to points towards the bottom of the right edge.', 'start': 712.363, 'duration': 4.463}, {'end': 721.69, 'text': 'And points towards the right of the bottom edge have to be glued to points towards the top of the right edge.', 'start': 717.407, 'duration': 4.283}, {'end': 726.534, 'text': "It's weird to think about, right? Go ahead, pause and ponder this for a moment.", 'start': 722.43, 'duration': 4.104}, {'end': 735.04, 'text': 'The trick, which is kind of clever, is to make a diagonal cut, which we need to remember to glue back in just a moment.', 'start': 729.696, 'duration': 5.344}, {'end': 738.603, 'text': 'After that, we can glue the bottom and the right like so.', 'start': 735.901, 'duration': 2.702}, {'end': 744.055, 'text': 'But notice the orientation of the arrows here.', 'start': 741.932, 'duration': 2.123}, {'end': 749.684, 'text': "To glue back what we just cut, we don't simply connect the edges of this rectangle to get a cylinder.", 'start': 744.696, 'duration': 4.988}, {'end': 751.207, 'text': 'We have to make a twist.', 'start': 750.225, 'duration': 0.982}, {'end': 756.065, 'text': 'Doing this in 3D space, the shape we get is a Mobius strip.', 'start': 752.443, 'duration': 3.622}, {'end': 763.669, 'text': "Isn't that awesome? Evidently, the surface which represents all pairs of unordered points on the loop is the Mobius strip.", 'start': 756.685, 'duration': 6.984}], 'summary': 'Glue bottom and right edges with diagonal cut, creating mobius strip in 3d space.', 'duration': 38.496, 'max_score': 705.559, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8705559.jpg'}, {'end': 798.651, 'src': 'embed', 'start': 764.709, 'weight': 2, 'content': [{'end': 772.954, 'text': 'And notice the edge of this strip shown here in red represents the pairs of points that look like those which are really just a single point,', 'start': 764.709, 'duration': 8.245}, {'end': 773.634, 'text': 'listed twice.', 'start': 772.954, 'duration': 0.68}, {'end': 782.18, 'text': 'The Mobius strip is to unordered pairs of points on the loop what the xy-plane is to pairs of real numbers.', 'start': 776.556, 'duration': 5.624}, {'end': 785.502, 'text': 'That totally blew my mind when I first saw it.', 'start': 782.88, 'duration': 2.622}, {'end': 798.651, 'text': 'Now, with this fact that there is a continuous one-to-one association between unordered pairs of points on the loop and individual points on this Mobius strip,', 'start': 788.985, 'duration': 9.666}], 'summary': 'The mobius strip represents pairs of points on the loop, forming a continuous one-to-one association.', 'duration': 33.942, 'max_score': 764.709, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8764709.jpg'}, {'end': 800.793, 'src': 'heatmap', 'start': 764.709, 'weight': 0.709, 'content': [{'end': 772.954, 'text': 'And notice the edge of this strip shown here in red represents the pairs of points that look like those which are really just a single point,', 'start': 764.709, 'duration': 8.245}, {'end': 773.634, 'text': 'listed twice.', 'start': 772.954, 'duration': 0.68}, {'end': 782.18, 'text': 'The Mobius strip is to unordered pairs of points on the loop what the xy-plane is to pairs of real numbers.', 'start': 776.556, 'duration': 5.624}, {'end': 785.502, 'text': 'That totally blew my mind when I first saw it.', 'start': 782.88, 'duration': 2.622}, {'end': 798.651, 'text': 'Now, with this fact that there is a continuous one-to-one association between unordered pairs of points on the loop and individual points on this Mobius strip,', 'start': 788.985, 'duration': 9.666}, {'end': 800.793, 'text': 'we can solve the inscribed rectangle problem.', 'start': 798.651, 'duration': 2.142}], 'summary': 'Mobius strip relates to unordered pairs, solving inscribed rectangle problem.', 'duration': 36.084, 'max_score': 764.709, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8764709.jpg'}, {'end': 912.899, 'src': 'heatmap', 'start': 853.97, 'weight': 0.704, 'content': [{'end': 859.653, 'text': 'And in the extreme case of pairs of points like the output of the function is exactly on the loop.', 'start': 853.97, 'duration': 5.683}, {'end': 870.39, 'text': 'Since points on this red edge of the Mobius strip correspond to pairs, like when the Mobius strip is mapped onto this surface,', 'start': 861.843, 'duration': 8.547}, {'end': 876.635, 'text': 'it must be done in such a way that the edge of the strip gets mapped right onto that loop in the x-y plane.', 'start': 870.39, 'duration': 6.245}, {'end': 884.821, 'text': 'But if you stand back and think about it for a moment, considering the strange shape of the Mobius strip,', 'start': 879.157, 'duration': 5.664}, {'end': 890.986, 'text': 'there is no way to glue its edge to something two-dimensional without forcing the strip to intersect itself.', 'start': 884.821, 'duration': 6.165}, {'end': 896.866, 'text': 'Since points of the MÃ¶bius strip represent pairs of points on the loop.', 'start': 893.143, 'duration': 3.723}, {'end': 905.153, 'text': 'if the strip intersects itself during this mapping,', 'start': 896.866, 'duration': 8.287}, {'end': 912.899, 'text': 'it means that there are at least two distinct pairs of points that correspond to the same output on this surface,', 'start': 905.153, 'duration': 7.746}], 'summary': 'Mapping mobius strip to surface results in intersection, causing multiple pairs to correspond to same output.', 'duration': 58.929, 'max_score': 853.97, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8853970.jpg'}, {'end': 953.68, 'src': 'embed', 'start': 922.661, 'weight': 3, 'content': [{'end': 929.624, 'text': "Or at least, if you're willing to trust me in saying that you can't glue the edge of a Mobius strip to a plane without forcing it to intersect itself,", 'start': 922.661, 'duration': 6.963}, {'end': 930.545, 'text': "then that's the proof.", 'start': 929.624, 'duration': 0.921}, {'end': 938.149, 'text': 'This fact is intuitively clear looking at the Mobius strip, but in order to make it rigorous,', 'start': 933.345, 'duration': 4.804}, {'end': 941.171, 'text': 'you basically need to start developing the field of topology.', 'start': 938.149, 'duration': 3.022}, {'end': 945.494, 'text': 'In fact, for any of you who have a topology class in your future,', 'start': 942.012, 'duration': 3.482}, {'end': 953.68, 'text': 'going through the exercise of trying to justify this is a good way to gain an appreciation for why topologists choose to make certain definitions.', 'start': 945.494, 'duration': 8.186}], 'summary': 'It is impossible to glue the edge of a mobius strip to a plane without forcing it to intersect itself, leading to the need to develop topology.', 'duration': 31.019, 'max_score': 922.661, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8922661.jpg'}], 'start': 705.559, 'title': 'Gluing and problem solving', 'summary': 'Discusses the importance of orientation in gluing the bottom and right edges, along with a clever trick of making a diagonal cut. it also explores the use of the mobius strip to solve the inscribed rectangle problem, demonstrating a one-to-one association between pairs of points on the loop and points on the mobius strip, leading to the proof of self-intersection and the existence of a rectangle.', 'chapters': [{'end': 744.055, 'start': 705.559, 'title': 'Gluing bottom and right edges', 'summary': 'Discusses the importance of the orientation in gluing the bottom and right edges, emphasizing the specific points to be glued and the clever trick of making a diagonal cut.', 'duration': 38.496, 'highlights': ['The orientation with which we glue the bottom and right edges is crucial, with points towards the left of the bottom edge being glued to points towards the bottom of the right edge, and points towards the right of the bottom edge being glued to points towards the top of the right edge.', 'A clever trick mentioned is to make a diagonal cut and remember to glue it back in a moment, which facilitates the gluing process and ensures proper orientation.', 'The importance of the orientation and specific points to be glued is emphasized, with a reminder to pay attention to the orientation of the arrows when gluing the bottom and right edges.']}, {'end': 974.357, 'start': 744.696, 'title': 'Solving the inscribed rectangle problem', 'summary': 'Discusses the use of the mobius strip to solve the inscribed rectangle problem, demonstrating the one-to-one association between pairs of points on the loop and points on the mobius strip, leading to the proof that the strip must intersect itself when glued to a plane, resulting in the existence of a rectangle.', 'duration': 229.661, 'highlights': ['The Mobius strip represents the pairs of unordered points on the loop, demonstrating a continuous one-to-one association between them.', 'The proof that the Mobius strip must intersect itself when glued to a plane, leading to the existence of a rectangle.', 'The discussion of the torus and the Mobius strip as a natural way to understand pairs of points on a loop.']}], 'duration': 268.798, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/AmgkSdhK4K8/pics/AmgkSdhK4K8705559.jpg', 'highlights': ['The importance of the orientation and specific points to be glued is emphasized, with a reminder to pay attention to the orientation of the arrows when gluing the bottom and right edges.', 'The orientation with which we glue the bottom and right edges is crucial, with points towards the left of the bottom edge being glued to points towards the bottom of the right edge, and points towards the right of the bottom edge being glued to points towards the top of the right edge.', 'The Mobius strip represents the pairs of unordered points on the loop, demonstrating a continuous one-to-one association between them.', 'The proof that the Mobius strip must intersect itself when glued to a plane, leading to the existence of a rectangle.', 'A clever trick mentioned is to make a diagonal cut and remember to glue it back in a moment, which facilitates the gluing process and ensures proper orientation.', 'The discussion of the torus and the Mobius strip as a natural way to understand pairs of points on a loop.']}], 'highlights': ['The inscribed square problem remains unsolved, showcasing limitations of current tools.', "The Loop Pairing Theorem states that for any closed loop, it's always possible to find two distinct pairs of points on that loop that share a midpoint and which are the same distance apart.", 'The importance of the orientation and specific points to be glued is emphasized, with a reminder to pay attention to the orientation of the arrows when gluing the bottom and right edges.']}