title

Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces

description

Solving a discrete math puzzle using topology
I was originally inspired to cover this thanks to a Quora post by Alon Amit
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An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/borsuk-thanks
Home page: https://www.3blue1brown.com
Want more fair division math fun? Check out this Mathologer video
https://youtu.be/7s-YM-kcKME
(Seriously, Mathologer is great)
These videos are supported by the community.
https://www.patreon.com/3blue1brown
The original 1986 by Alon and West with this proof
https://m.tau.ac.il/~nogaa/PDFS/Publications/The%20Borsuk-Ulam%20Theorem%20and%20bisection%20of%20necklaces.pdf
VSauce on fixed points
https://youtu.be/csInNn6pfT4
EE Paper using ideas related to this puzzle
https://dl.acm.org/citation.cfm?id=802179
I first came across this paper thanks to Alon Amit's answer on this Quora post
https://www.quora.com/As-of-2016-what-do-mathematicians-on-Quora-think-of-the-3Blue1Brown-maths-videos
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
Time stamps:
0:00 - Introduction
0:36 - The stolen necklace problem
3:08 - The Borsuk Ulam theorem
9:15 - The continuous necklace problem
13:19 - The connection
17:30 - Higher dimensions
------------------
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detail

{'title': 'Using topology for discrete problems | The Borsuk-Ulam theorem and stolen necklaces', 'heatmap': [{'end': 477.292, 'start': 383.167, 'weight': 0.74}, {'end': 548.868, 'start': 498.217, 'weight': 0.717}], 'summary': "Explore the connection between the stolen necklace problem and the borsuk-ulam theorem, presenting a fair division puzzle with the possibility of achieving fair division with n cuts or fewer. understand the borsuk-ulam theorem's implications in continuous mappings from a 3d sphere onto a 2d plane, illustrated by real-world examples like earth's temperature and pressure pairs. delve into sphere geometry, algebraic concepts, and the correlation of the stolen necklace problem to the borsuk-ulam theorem, generalizing it to higher dimensional spheres.", 'chapters': [{'end': 194.384, 'segs': [{'end': 38.103, 'src': 'embed', 'start': 3.527, 'weight': 2, 'content': [{'end': 8.449, 'text': 'You know that feeling you get when things that seem completely unrelated turn out to have a key connection?', 'start': 3.527, 'duration': 4.922}, {'end': 14.412, 'text': "In math especially, there's a certain tingly sensation I get whenever one of those connections starts to fall into place.", 'start': 9.07, 'duration': 5.342}, {'end': 17.234, 'text': 'This is what I have in store for you today.', 'start': 15.213, 'duration': 2.021}, {'end': 18.915, 'text': 'It takes some time to set up.', 'start': 17.894, 'duration': 1.021}, {'end': 24.317, 'text': 'I have to introduce a fair division puzzle from discrete math called the stolen necklace problem,', 'start': 19.175, 'duration': 5.142}, {'end': 29.2, 'text': "as well as a topological fact about spheres that we'll use to solve it, called the Borsuk-Ulam theorem.", 'start': 24.317, 'duration': 4.883}, {'end': 35.462, 'text': 'But trust me, seeing these two seemingly disconnected pieces of math come together is well worth the setup.', 'start': 29.7, 'duration': 5.762}, {'end': 38.103, 'text': "Let's start with the puzzle that we're going to solve.", 'start': 36.322, 'duration': 1.781}], 'summary': 'Discover the connection between a fair division puzzle and the borsuk-ulam theorem in math.', 'duration': 34.576, 'max_score': 3.527, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c3527.jpg'}, {'end': 134.602, 'src': 'embed', 'start': 101.804, 'weight': 0, 'content': [{'end': 104.485, 'text': 'two diamonds and three rubies.', 'start': 101.804, 'duration': 2.681}, {'end': 106.025, 'text': 'The claim.', 'start': 105.245, 'duration': 0.78}, {'end': 111.307, 'text': 'the thing that I want to prove in this video is that if there are N different jewel types,', 'start': 106.025, 'duration': 5.282}, {'end': 115.909, 'text': "it's always possible to do this fair division with only N cuts or fewer.", 'start': 111.307, 'duration': 4.602}, {'end': 121.171, 'text': 'So, with four jewel types in this example, no matter what random ordering of the jewels,', 'start': 116.609, 'duration': 4.562}, {'end': 128.152, 'text': 'it should be possible to cut it in four places and divvy up the five necklace pieces so that each thief has the same number of each jewel type.', 'start': 121.171, 'duration': 6.981}, {'end': 134.602, 'text': 'With five jewel types, you should be able to do it with five cuts, no matter the arrangement, and so on.', 'start': 129.898, 'duration': 4.704}], 'summary': 'Proving fair division with n cuts or fewer for n jewel types.', 'duration': 32.798, 'max_score': 101.804, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c101804.jpg'}, {'end': 187.479, 'src': 'embed', 'start': 157.838, 'weight': 1, 'content': [{'end': 162.441, 'text': 'these are the kind of optimization issues that actually come up quite frequently in practical applications.', 'start': 157.838, 'duration': 4.603}, {'end': 169.346, 'text': "For the computer system folks among you, I'm sure you can imagine how this is analogous to kinds of efficient memory allocation problems.", 'start': 163.062, 'duration': 6.284}, {'end': 176.132, 'text': "Also, for the curious among you, I've left a link in the description to an electrical engineering paper that applies this specific problem.", 'start': 170.047, 'duration': 6.085}, {'end': 180.034, 'text': 'Independent from the usefulness, though, it certainly does make for a good puzzle.', 'start': 176.892, 'duration': 3.142}, {'end': 187.479, 'text': "Can you always find a fair division using only as many cuts as there are types of jewels? So, that's the puzzle.", 'start': 180.635, 'duration': 6.844}], 'summary': 'Optimization issues in practical applications, analogous to efficient memory allocation problems. puzzle: finding fair division using as many cuts as types of jewels.', 'duration': 29.641, 'max_score': 157.838, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c157838.jpg'}], 'start': 3.527, 'title': 'Mathematical connections and fair division puzzle', 'summary': 'Delves into the connection between stolen necklace problem and borsuk-ulam theorem, highlighting the interplay between mathematical concepts. it also presents a fair division puzzle demonstrating its practical applications and the possibility to achieve fair division with n cuts or fewer, where n represents the number of different jewel types.', 'chapters': [{'end': 38.103, 'start': 3.527, 'title': 'Mathematical connections', 'summary': 'Explores the intriguing connection between the stolen necklace problem from discrete math and the borsuk-ulam theorem, showcasing the fascinating interplay between seemingly unrelated mathematical concepts.', 'duration': 34.576, 'highlights': ['Introducing the stolen necklace problem from discrete math and the Borsuk-Ulam theorem, showcasing their seemingly disconnected nature but their eventual interplay.', 'The captivating interrelation between seemingly unrelated mathematical concepts, highlighting the appeal of discovering connections within math.']}, {'end': 194.384, 'start': 38.443, 'title': 'Fair division puzzle', 'summary': "Presents a puzzle where a necklace containing a variety of jewels needs to be divided equally between two individuals with as few cuts as possible, demonstrating that for n different jewel types, it's always possible to achieve fair division with n cuts or fewer, and the puzzle's practical applications in optimization issues.", 'duration': 155.941, 'highlights': ["It's always possible to do fair division with only N cuts or fewer for N different jewel types.", 'Practical applications in optimization issues.', "The puzzle's practical applications."]}], 'duration': 190.857, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c3527.jpg', 'highlights': ["It's always possible to do fair division with only N cuts or fewer for N different jewel types.", 'Practical applications in optimization issues.', 'The captivating interrelation between seemingly unrelated mathematical concepts, highlighting the appeal of discovering connections within math.', 'Introducing the stolen necklace problem from discrete math and the Borsuk-Ulam theorem, showcasing their seemingly disconnected nature but their eventual interplay.']}, {'end': 553.773, 'segs': [{'end': 241.844, 'src': 'embed', 'start': 214.235, 'weight': 0, 'content': [{'end': 218.739, 'text': 'As you do this, many different pairs of points will land on top of each other once they hit the plane.', 'start': 214.235, 'duration': 4.504}, {'end': 220.661, 'text': "And, you know, that's not really a big deal.", 'start': 219.099, 'duration': 1.562}, {'end': 225.846, 'text': "The special fact that we're going to use, known as the Borsuk-Ulam theorem,", 'start': 221.201, 'duration': 4.645}, {'end': 232.211, 'text': 'is that you will always be able to find a pair of points that started off on the exact opposite sides of the sphere,', 'start': 225.846, 'duration': 6.365}, {'end': 233.953, 'text': 'which land on each other during the mapping.', 'start': 232.211, 'duration': 1.742}, {'end': 241.844, 'text': 'Points on the exact opposite like this are called antipodes, or antipodal points.', 'start': 237.321, 'duration': 4.523}], 'summary': 'Borsuk-ulam theorem: pairs of antipodal points always intersect during mapping.', 'duration': 27.609, 'max_score': 214.235, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c214235.jpg'}, {'end': 323.432, 'src': 'embed', 'start': 303.177, 'weight': 1, 'content': [{'end': 314.247, 'text': 'This is because associating each point on the surface of the Earth with a pair of numbers temperature and pressure is the same thing as mapping the surface of the Earth onto a 2D coordinate plane,', 'start': 303.177, 'duration': 11.07}, {'end': 317.53, 'text': 'where the first coordinate represents temperature and the second represents pressure.', 'start': 314.247, 'duration': 3.283}, {'end': 323.432, 'text': 'The implicit assumption here is that temperature and pressure each vary continuously as you walk around the Earth.', 'start': 318.411, 'duration': 5.021}], 'summary': "Mapping earth's surface into 2d plane using temp and pressure pairs.", 'duration': 20.255, 'max_score': 303.177, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c303177.jpg'}, {'end': 477.292, 'src': 'heatmap', 'start': 383.167, 'weight': 0.74, 'content': [{'end': 386.649, 'text': 'where negative p is the antipodal point on the other side of the sphere.', 'start': 383.167, 'duration': 3.482}, {'end': 397.822, 'text': 'The key idea here, which might seem small at first, is to rearrange this and say f of p minus, f of negative p equals 0, 0,', 'start': 389.315, 'duration': 8.507}, {'end': 400.565, 'text': 'and focus on a new function g of p.', 'start': 397.822, 'duration': 2.743}, {'end': 404.728, 'text': "that's defined to be this left-hand side here f of p minus, f of negative p.", 'start': 400.565, 'duration': 4.163}, {'end': 411.331, 'text': 'This way what we need to show is that g maps some point of the sphere onto the origin in 2D space.', 'start': 405.609, 'duration': 5.722}, {'end': 415.773, 'text': 'So, rather than finding a pair of colliding points which could land anywhere,', 'start': 412.051, 'duration': 3.722}, {'end': 419.694, 'text': 'this helps limit our focus to just one point of the output space the origin.', 'start': 415.773, 'duration': 3.921}, {'end': 426.016, 'text': 'This function g has a pretty special property which is going to help us out.', 'start': 422.635, 'duration': 3.381}, {'end': 433.799, 'text': 'That g is equal to negative g Basically negating the input involves swapping these terms.', 'start': 426.596, 'duration': 7.203}, {'end': 444.344, 'text': 'In other words, going to the antipodal point of the sphere results in reflecting the output of g through the origin of the output space.', 'start': 436.115, 'duration': 8.229}, {'end': 448.929, 'text': 'Or maybe you think of it as rotating that output 180 degrees around the origin.', 'start': 444.905, 'duration': 4.024}, {'end': 455.717, 'text': 'Notice what this means if you were to continuously walk around the equator and look at the outputs of g.', 'start': 450.071, 'duration': 5.646}, {'end': 467.122, 'text': 'What happens when you go halfway around? Well, the output needs to have wandered to the reflection of the starting point through the origin.', 'start': 455.717, 'duration': 11.405}, {'end': 477.292, 'text': 'Then, as you continue walking around the other half, the second half of your output path must be the reflection of the first half, or equivalently,', 'start': 467.943, 'duration': 9.349}], 'summary': 'Rearranging the function to focus on a new function g of p, showing g maps a point of the sphere onto the origin in 2d space.', 'duration': 94.125, 'max_score': 383.167, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c383167.jpg'}, {'end': 444.344, 'src': 'embed', 'start': 412.051, 'weight': 4, 'content': [{'end': 415.773, 'text': 'So, rather than finding a pair of colliding points which could land anywhere,', 'start': 412.051, 'duration': 3.722}, {'end': 419.694, 'text': 'this helps limit our focus to just one point of the output space the origin.', 'start': 415.773, 'duration': 3.921}, {'end': 426.016, 'text': 'This function g has a pretty special property which is going to help us out.', 'start': 422.635, 'duration': 3.381}, {'end': 433.799, 'text': 'That g is equal to negative g Basically negating the input involves swapping these terms.', 'start': 426.596, 'duration': 7.203}, {'end': 444.344, 'text': 'In other words, going to the antipodal point of the sphere results in reflecting the output of g through the origin of the output space.', 'start': 436.115, 'duration': 8.229}], 'summary': 'Function g has the special property of being equal to its negative, limiting focus to the origin.', 'duration': 32.293, 'max_score': 412.051, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c412051.jpg'}, {'end': 548.868, 'src': 'heatmap', 'start': 484.099, 'weight': 3, 'content': [{'end': 490.723, 'text': "Now, there's a slim possibility that one of these points happens to pass through the origin, in which case you've lucked out and were done early.", 'start': 484.099, 'duration': 6.624}, {'end': 496.486, 'text': 'But otherwise what we have here is a path that winds around the origin at least once.', 'start': 491.343, 'duration': 5.143}, {'end': 505.182, 'text': 'Now look at that path on the equator and imagine continuously deforming it up to the north pole, cinching that loop tight.', 'start': 498.217, 'duration': 6.965}, {'end': 514.227, 'text': 'As you do this, the resulting path in the output space is also continuously deforming to a point, since the function g is continuous.', 'start': 506.042, 'duration': 8.185}, {'end': 521.412, 'text': 'Now, because it wound around the origin, at some point during this process, it must cross the origin.', 'start': 515.325, 'duration': 6.087}, {'end': 533.993, 'text': 'And this means there is some point p on the sphere where g has the coordinates, which means f of p minus f of negative p equals 0,0,', 'start': 522.133, 'duration': 11.86}, {'end': 538.698, 'text': "meaning f of p is the same as f of negative p, the antipodal collision that we're looking for.", 'start': 533.993, 'duration': 4.705}, {'end': 543.743, 'text': "Isn't that clever? And it's a pretty common style of argument in the context of topology.", 'start': 539.559, 'duration': 4.184}, {'end': 548.868, 'text': "It doesn't matter what particular continuous function from the sphere to the plane you define,", 'start': 544.323, 'duration': 4.545}], 'summary': 'Topology argument shows antipodal collision is possible.', 'duration': 49.894, 'max_score': 484.099, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c484099.jpg'}], 'start': 194.784, 'title': 'Borsuk-ulam theorem and topology', 'summary': "Explains the borsuk-ulam theorem, stating that in any continuous mapping from a 3d sphere onto a 2d plane, there will always be a pair of antipodal points that land on top of each other, illustrated by examples like earth's temperature and pressure pairs.", 'chapters': [{'end': 365.895, 'start': 194.784, 'title': 'Borsuk-ulam theorem and topology', 'summary': "Explains the borsuk-ulam theorem, stating that in any continuous mapping from a 3d sphere onto a 2d plane, there will always be a pair of antipodal points that land on top of each other, illustrated by examples like earth's temperature and pressure pairs.", 'duration': 171.111, 'highlights': ['The Borsuk-Ulam Theorem guarantees that in any continuous mapping from a 3D sphere onto a 2D plane, there will always be a pair of antipodal points that land on top of each other.', "Illustration of the theorem using Earth's temperature and barometric pressure pairs, demonstrating that there must exist a pair of antipodal points on the opposite side of the Earth with precisely the same temperature and pressure.", "Explanation of how the association of each point on the Earth's surface with a pair of temperature and pressure numbers is equivalent to a continuous mapping from the sphere onto a 2D coordinate plane."]}, {'end': 553.773, 'start': 365.895, 'title': 'Topology and antipodal points', 'summary': 'Discusses the concept of antipodal points on a sphere, demonstrating that for any continuous function from the sphere to a plane, there will always be a pair of antipodal points that map to the same output, using the function g to show that the path winds around the origin and eventually crosses it.', 'duration': 187.878, 'highlights': ['The key idea is to rearrange the function to g(p) = f(p) - f(-p) and show that g maps some point of the sphere onto the origin in 2D space.', 'The function g has the property g = -g, meaning that going to the antipodal point of the sphere results in reflecting the output of g through the origin of the output space.', 'The path of the function g on the equator winds around the origin at least once, and when continuously deformed to the north pole, it must cross the origin, resulting in a point p on the sphere where f(p) = f(-p).']}], 'duration': 358.989, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c194784.jpg', 'highlights': ['The Borsuk-Ulam Theorem guarantees a pair of antipodal points in a 3D sphere map onto a 2D plane', "Illustration using Earth's temperature and pressure pairs demonstrates the theorem", "Explanation of associating each point on Earth's surface with temperature and pressure", 'The function g maps some point of the sphere onto the origin in 2D space', 'The function g has the property g = -g, reflecting the output through the origin', 'The path of the function g on the equator winds around the origin at least once', 'When continuously deformed to the north pole, it must cross the origin']}, {'end': 1140.147, 'segs': [{'end': 665.904, 'src': 'embed', 'start': 641.825, 'weight': 1, 'content': [{'end': 649.83, 'text': 'Part of the reason these two things feel so very unrelated is that the necklace problem is discrete, while the Borsuk-Ulam theorem is continuous.', 'start': 641.825, 'duration': 8.005}, {'end': 654.713, 'text': 'So our first step is to translate the stolen necklace problem into a continuous version,', 'start': 650.39, 'duration': 4.323}, {'end': 657.835, 'text': 'seeking the connection between necklace divisions and points on the sphere.', 'start': 654.713, 'duration': 3.122}, {'end': 665.904, 'text': "For right now let's limit ourselves to the case where there's only two jewel types, say sapphires and emeralds,", 'start': 660.12, 'duration': 5.784}], 'summary': 'Translating discrete necklace problem into continuous version, focusing on two jewel types', 'duration': 24.079, 'max_score': 641.825, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c641825.jpg'}, {'end': 703.891, 'src': 'embed', 'start': 677.151, 'weight': 2, 'content': [{'end': 686.417, 'text': 'this means the goal is to cut the necklace in two different spots and divvy up those three segments so that each thief ends up with half of the sapphires and half of the emeralds.', 'start': 677.151, 'duration': 9.266}, {'end': 691.14, 'text': 'Notice the top and the bottom each have four sapphires and five emeralds.', 'start': 687.137, 'duration': 4.003}, {'end': 694.763, 'text': 'For our continuousification.', 'start': 692.882, 'duration': 1.881}, {'end': 703.891, 'text': 'think of the necklace as a line with length, one with the jewels sitting evenly spaced on it, and divide up that line into 18 evenly sized segments,', 'start': 694.763, 'duration': 9.128}], 'summary': 'Goal: divide necklace into 18 segments for equal distribution of sapphires and emeralds.', 'duration': 26.74, 'max_score': 677.151, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c677151.jpg'}, {'end': 1049.046, 'src': 'embed', 'start': 1022.737, 'weight': 3, 'content': [{'end': 1027.281, 'text': "And the animation you're looking at right now is that literal map for the necklace I was showing.", 'start': 1022.737, 'duration': 4.544}, {'end': 1040.776, 'text': 'So the Borsuk-Ulam theorem guarantees that some antipodal pair of points on the sphere land on each other in the plane,', 'start': 1033.907, 'duration': 6.869}, {'end': 1043.88, 'text': 'which means there must be some necklace division using two cuts.', 'start': 1040.776, 'duration': 3.104}, {'end': 1045.561, 'text': 'that gives a fair division between the thieves.', 'start': 1043.88, 'duration': 1.681}, {'end': 1049.046, 'text': 'That, my friends, is what beautiful math feels like.', 'start': 1046.403, 'duration': 2.643}], 'summary': 'Borsuk-ulam theorem ensures fair necklace division using two cuts.', 'duration': 26.309, 'max_score': 1022.737, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c1022737.jpg'}, {'end': 1093.408, 'src': 'embed', 'start': 1068.885, 'weight': 0, 'content': [{'end': 1075.892, 'text': 'The main thing to mention is that there is a more general version of the Borsuk-Ulam theorem, one that applies to higher dimensional spheres.', 'start': 1068.885, 'duration': 7.007}, {'end': 1081.977, 'text': 'As an example, Borsuk-Ulam applies to mapping hyperspheres in 4D space into three dimensions.', 'start': 1076.612, 'duration': 5.365}, {'end': 1089.645, 'text': 'And what I mean by a hypersphere is the set of all possible lists of four coordinates where the sum of their squares equals one.', 'start': 1082.758, 'duration': 6.887}, {'end': 1093.408, 'text': 'Those are the points in 4D space a distance one from the origin.', 'start': 1090.506, 'duration': 2.902}], 'summary': 'General version of borsuk-ulam theorem applies to higher dimensional spheres, e.g., mapping hyperspheres in 4d space to 3d.', 'duration': 24.523, 'max_score': 1068.885, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c1068885.jpg'}], 'start': 556.178, 'title': 'Sphere geometry and stolen necklace problem', 'summary': 'Discusses the borsuk-ulam theorem, connecting sphere geometry to algebraic concepts and explores the stolen necklace problem and its correlation to the theorem, generalizing it to higher dimensional spheres.', 'chapters': [{'end': 657.835, 'start': 556.178, 'title': 'Borsuk-ulam theorem and sphere geometry', 'summary': 'Discusses the borsuk-ulam theorem, relating sphere geometry to algebraic concepts and highlighting the connection between discrete necklace divisions and points on the sphere.', 'duration': 101.657, 'highlights': ["The Borsuk-Ulam theorem relates to a function that takes in points on the sphere and produces points in 2D space, with a guarantee that there will be an input such that flipping all its signs doesn't change the output.", 'Sphere geometry is represented with three coordinates in 3D space, and the standard unit sphere is the set of all points a distance 1 from the origin, related to the algebraic idea of a set of positive numbers that add up to 1.', 'Understanding the geometric idea of a sphere is crucial for connecting the Borsuk-Ulam theorem to the discrete necklace problem, seeking a continuous version of the problem to establish a link between necklace divisions and points on the sphere.']}, {'end': 1140.147, 'start': 660.12, 'title': 'The stolen necklace problem', 'summary': 'Discusses a continuous variant of the stolen necklace problem and its connection to the borsuk-ulam theorem, showing the correspondence between points on a sphere and necklace divisions, and generalizing the theorem to higher dimensional spheres.', 'duration': 480.027, 'highlights': ['The continuous variant of the puzzle aims to find two cuts on the necklace line to divide it into segments so that each thief has an equal length of each color, with a correspondence between points on a sphere and necklace divisions.', 'The Borsuk-Ulam theorem guarantees that some antipodal pair of points on the sphere land on each other in the plane, ensuring a fair division between the thieves.', 'The Borsuk-Ulam theorem has a more general version that applies to higher dimensional spheres, with examples in 4D space, and its implications for the general necklace problem.']}], 'duration': 583.969, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/yuVqxCSsE7c/pics/yuVqxCSsE7c556178.jpg', 'highlights': ["The Borsuk-Ulam theorem relates to a function guaranteeing an input such that flipping all its signs doesn't change the output.", 'Understanding the geometric idea of a sphere is crucial for connecting the Borsuk-Ulam theorem to the discrete necklace problem.', 'The continuous variant of the puzzle aims to find two cuts on the necklace line to divide it into segments so that each thief has an equal length of each color.', 'The Borsuk-Ulam theorem guarantees that some antipodal pair of points on the sphere land on each other in the plane, ensuring a fair division between the thieves.', 'The Borsuk-Ulam theorem has a more general version that applies to higher dimensional spheres, with examples in 4D space.']}], 'highlights': ["It's always possible to do fair division with only N cuts or fewer for N different jewel types.", 'The Borsuk-Ulam theorem guarantees a pair of antipodal points in a 3D sphere map onto a 2D plane', 'Understanding the geometric idea of a sphere is crucial for connecting the Borsuk-Ulam theorem to the discrete necklace problem.', 'The captivating interrelation between seemingly unrelated mathematical concepts, highlighting the appeal of discovering connections within math.', "The Borsuk-Ulam theorem relates to a function guaranteeing an input such that flipping all its signs doesn't change the output."]}