title

Group theory, abstraction, and the 196,883-dimensional monster

description

An introduction to group theory (Minor error corrections below)
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: https://3b1b.co/monster-thanks
Timestamps:
0:00 - The size of the monster
0:50 - What is a group?
7:06 - What is an abstract group?
13:27 - Classifying groups
18:31 - About the monster
Errors:
*Typo on the "hard problem" at 14:11, it should be a/(b+c) + b/(a+c) + c/(a+b) = 4
*Typo-turned-speako: The classification of quasithin groups is 1221 pages long, not 12,000. The full collection of papers proving the CFSG theorem do comprise tens of thousands of pages, but no one paper was quite that crazy.
Thanks to Richard Borcherds for his helpful comments while putting this video together. He has a wonderful hidden gem of a channel: https://youtu.be/a9k_QmZbwX8
You may also enjoy this brief article giving an overview of this monster:
http://www.ams.org/notices/200209/what-is.pdf
If you want to learn more about group theory, check out the expository papers here:
https://kconrad.math.uconn.edu/blurbs/
Videos with John Conway talking about the Monster:
https://youtu.be/jsSeoGpiWsw
https://youtu.be/lbN8EMcOH5o
More on Noether's Theorem:
https://youtu.be/CxlHLqJ9I0A
https://youtu.be/04ERSb06dOg
The symmetry ambigram was designed by Punya Mishra:
https://punyamishra.com/2013/05/31/symmetry-new-ambigram/
The Monster image comes from the Noun Project, via Nicky Knicky
This video is part of the #MegaFavNumbers project: https://www.youtube.com/playlist?list=PLar4u0v66vIodqt3KSZPsYyuULD5meoAo
To join the gang, upload your own video on your own favorite number over 1,000,000 with the hashtag #MegaFavNumbers, and the word MegaFavNumbers in the title by September 2nd, 2020, and it'll be added to the playlist above.
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These animations are largely made using manim, a scrappy open-source python library: https://github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
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.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe
Various social media stuffs:
Website: https://www.3blue1brown.com
Twitter: https://twitter.com/3blue1brown
Reddit: https://www.reddit.com/r/3blue1brown
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Patreon: https://patreon.com/3blue1brown
Facebook: https://www.facebook.com/3blue1brown

detail

{'title': 'Group theory, abstraction, and the 196,883-dimensional monster', 'heatmap': [{'end': 872.433, 'start': 851.259, 'weight': 0.938}, {'end': 1318, 'start': 1304.82, 'weight': 1}], 'summary': "Delves into the youtube math community's focus on numbers over 1 million, the vast size of permutation groups with the largest known group having around 9×10^159 elements, the significance of symmetry in s5 and its relevance to physics, the abstraction of groups and their role in understanding numbers, and the enigmatic nature of the monster group with its colossal size of 8×10^53, whimsically named sporadic groups, and relevance to string theory and physics.", 'chapters': [{'end': 42.768, 'segs': [{'end': 42.768, 'src': 'embed', 'start': 4.532, 'weight': 0, 'content': [{'end': 12.078, 'text': "Today, many members of the YouTube math community are getting together to make videos about their favourite numbers over 1 million and we're encouraging you,", 'start': 4.532, 'duration': 7.546}, {'end': 13.379, 'text': 'the viewers, to do the same.', 'start': 12.078, 'duration': 1.301}, {'end': 15.48, 'text': 'Take a look at the description for details.', 'start': 13.879, 'duration': 1.601}, {'end': 21.164, 'text': 'My own choice is considerably larger than a million, roughly 8 times 10 to the 53.', 'start': 16.221, 'duration': 4.943}, {'end': 24.545, 'text': "For a sense of scale, that's around the number of atoms in the planet Jupiter.", 'start': 21.164, 'duration': 3.381}, {'end': 26.745, 'text': 'So it might seem completely arbitrary.', 'start': 25.025, 'duration': 1.72}, {'end': 33.846, 'text': 'But what I love is that if you were to talk with an alien civilization or a super intelligent AI that invented math for itself,', 'start': 27.305, 'duration': 6.541}, {'end': 37.967, 'text': 'without any connection to our particular culture or experiences,', 'start': 33.846, 'duration': 4.121}, {'end': 42.768, 'text': 'I think both would agree that this number is something very peculiar and that it reflects something fundamental.', 'start': 37.967, 'duration': 4.801}], 'summary': 'Youtube math community making videos about favorite numbers over 1 million; choice is 8x10^53, comparable to atoms in jupiter, reflecting something fundamental.', 'duration': 38.236, 'max_score': 4.532, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE4532.jpg'}], 'start': 4.532, 'title': 'Youtube math community', 'summary': "Discusses the youtube math community creating videos about their favorite numbers over 1 million with a highlight on the speaker's choice of 8 times 10 to the 53, approximately the number of atoms in jupiter.", 'chapters': [{'end': 42.768, 'start': 4.532, 'title': 'Youtube math community celebrates favorite numbers', 'summary': 'Discusses the youtube math community coming together to create videos about their favorite numbers over 1 million, with the speaker choosing 8 times 10 to the 53, approximately the number of atoms in jupiter, highlighting the universal intrigue of this significant number.', 'duration': 38.236, 'highlights': ["The speaker discusses the YouTube math community's initiative to create videos about favorite numbers over 1 million, encouraging viewers to participate.", "The speaker's own choice of a favorite number is 8 times 10 to the 53, roughly the number of atoms in Jupiter, emphasizing the magnitude and scale of this chosen number.", 'The speaker highlights the universal intrigue of the chosen number, suggesting that both an alien civilization and a super intelligent AI would find it peculiar and reflective of something fundamental.']}], 'duration': 38.236, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE4532.jpg', 'highlights': ["The speaker's own choice of a favorite number is 8 times 10 to the 53, roughly the number of atoms in Jupiter, emphasizing the magnitude and scale of this chosen number.", "The speaker discusses the YouTube math community's initiative to create videos about favorite numbers over 1 million, encouraging viewers to participate.", 'The speaker highlights the universal intrigue of the chosen number, suggesting that both an alien civilization and a super intelligent AI would find it peculiar and reflective of something fundamental.']}, {'end': 364.06, 'segs': [{'end': 115.17, 'src': 'embed', 'start': 86.995, 'weight': 3, 'content': [{'end': 90.436, 'text': "Groups are typically defined a little more abstractly than this, but we'll get to that later.", 'start': 86.995, 'duration': 3.441}, {'end': 91.797, 'text': 'Take note!', 'start': 91.417, 'duration': 0.38}, {'end': 100.86, 'text': 'The fact that mathematicians have co-opted such an otherwise generic word for this seemingly specific kind of collection should give you some sense of how fundamental they find it.', 'start': 92.017, 'duration': 8.843}, {'end': 105.925, 'text': 'Also take note, we always consider the action of doing nothing to be part of the group.', 'start': 101.963, 'duration': 3.962}, {'end': 112.448, 'text': 'So if we include that do-nothing action, the group of symmetries of a snowflake includes 12 distinct actions.', 'start': 106.525, 'duration': 5.923}, {'end': 115.17, 'text': 'It even has a fancy name, D6.', 'start': 113.069, 'duration': 2.101}], 'summary': 'Mathematicians find symmetries fundamental, snowflake group includes 12 actions', 'duration': 28.175, 'max_score': 86.995, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE86995.jpg'}, {'end': 157.508, 'src': 'embed', 'start': 128.036, 'weight': 2, 'content': [{'end': 131.318, 'text': 'categorizing the many different ways that something can be symmetric.', 'start': 128.036, 'duration': 3.282}, {'end': 137.035, 'text': "When we describe these sorts of actions, there's always an implicit structure being preserved.", 'start': 132.472, 'duration': 4.563}, {'end': 142.398, 'text': 'For example, there are 24 rotations that I can apply to a cube that leave it looking the same.', 'start': 137.655, 'duration': 4.743}, {'end': 146.761, 'text': 'And those 24 actions taken together do indeed constitute a group.', 'start': 143.019, 'duration': 3.742}, {'end': 157.508, 'text': 'But if we allow for reflections which is a kind of way of saying that the orientation of the cube is not part of the structure we intend to preserve you get a bigger group with 48 actions in total.', 'start': 147.481, 'duration': 10.027}], 'summary': '24 rotations preserve cube symmetry, reflections expand group to 48 actions.', 'duration': 29.472, 'max_score': 128.036, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE128036.jpg'}, {'end': 276.609, 'src': 'embed', 'start': 239.929, 'weight': 0, 'content': [{'end': 245.792, 'text': 'bump the number of points up to 12, and the number of permutations grows to about 479 million.', 'start': 239.929, 'duration': 5.863}, {'end': 249.294, 'text': "The monster that we'll get to is rather large,", 'start': 246.733, 'duration': 2.561}, {'end': 254.016, 'text': "but it's important to understand that largeness in and of itself is not that interesting when it comes to groups.", 'start': 249.294, 'duration': 4.722}, {'end': 256.678, 'text': 'The permutation groups already make that easy to see.', 'start': 254.557, 'duration': 2.121}, {'end': 264.062, 'text': 'If we were shuffling 101 objects, for example with the 101 factorial different actions that can do this.', 'start': 257.517, 'duration': 6.545}, {'end': 267.764, 'text': 'we have a group with a size of around 9 times 10 to the 159..', 'start': 264.062, 'duration': 3.702}, {'end': 276.609, 'text': 'If every atom in the observable universe had a copy of that universe inside itself, this is roughly how many subatoms there would be.', 'start': 267.764, 'duration': 8.845}], 'summary': 'Permutations grow to 479 million with 12 points, reaching 9x10^159 in larger groups.', 'duration': 36.68, 'max_score': 239.929, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE239929.jpg'}, {'end': 317.562, 'src': 'embed', 'start': 293.538, 'weight': 5, 'content': [{'end': 302.391, 'text': 'One of the earliest applications of group theory came when mathematicians realized that the structure of these permutation groups tells us something about solutions to polynomial equations.', 'start': 293.538, 'duration': 8.853}, {'end': 305.029, 'text': 'You know how?', 'start': 304.468, 'duration': 0.561}, {'end': 309.974, 'text': 'in order to find the two roots of a quadratic equation, everyone learns a certain formula in school?', 'start': 305.029, 'duration': 4.945}, {'end': 317.562, 'text': "Slightly lesser known is the fact that there's also a cubic formula, one that involves nesting cube roots with square roots in a larger expression.", 'start': 310.755, 'duration': 6.807}], 'summary': 'Group theory reveals insights into polynomial equations, including quadratic and cubic formulas.', 'duration': 24.024, 'max_score': 293.538, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE293538.jpg'}], 'start': 43.588, 'title': 'Symmetry, group theory, and practical applications', 'summary': 'Delves into the fundamental concept of symmetry and group theory, emphasizing the categorization of symmetries into different groups and the practical applications of group theory in solving polynomial equations. it also highlights the vast size of permutation groups, with the largest known group having around 9 × 10^159 elements, and discusses the limitations of finding formulas for degree-five polynomials.', 'chapters': [{'end': 105.925, 'start': 43.588, 'title': 'Understanding symmetry and group theory', 'summary': 'Explains the concept of symmetry and group theory, emphasizing how mathematicians define groups as collections of actions that leave an object looking the same, and how the concept is fundamental in mathematics.', 'duration': 62.337, 'highlights': ['Mathematicians define groups as collections of actions that leave an object looking the same, emphasizing the fundamental nature of this concept.', 'Symmetry is codified through group theory, where actions such as reflection and rotation are considered part of a group, illustrating the diverse ways symmetry can be represented.', "Group theory is fundamental in mathematics, as evidenced by the co-opting of a generic word like 'group' to represent a specific kind of collection."]}, {'end': 364.06, 'start': 106.525, 'title': 'Group theory and symmetry', 'summary': 'Explores the concept of symmetries and permutation groups, demonstrating that the symmetries of objects can be categorized into different groups, and the size of permutation groups can be quite large, with the largest known group having around 9 times 10 to the 159 elements. furthermore, it highlights the practical applications of group theory in solving polynomial equations and the limitations of finding formulas for degree-five polynomials.', 'duration': 257.535, 'highlights': ['The largest known permutation group has around 9 times 10 to the 159 elements, which is approximately the number of subatoms if every atom in the observable universe had a copy of that universe inside itself.', 'The number of permutations for 12 points is approximately 479 million, showcasing the exponential growth in permutations as the number of points increases.', 'The concept of symmetry and permutation groups plays a crucial role in understanding solutions to polynomial equations, as the structure of these groups reveals limitations in finding formulas for certain degree polynomials.']}], 'duration': 320.472, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE43588.jpg', 'highlights': ['The largest known permutation group has around 9 times 10 to the 159 elements, which is approximately the number of subatoms if every atom in the observable universe had a copy of that universe inside itself.', 'The number of permutations for 12 points is approximately 479 million, showcasing the exponential growth in permutations as the number of points increases.', 'Symmetry is codified through group theory, where actions such as reflection and rotation are considered part of a group, illustrating the diverse ways symmetry can be represented.', 'Mathematicians define groups as collections of actions that leave an object looking the same, emphasizing the fundamental nature of this concept.', "Group theory is fundamental in mathematics, as evidenced by the co-opting of a generic word like 'group' to represent a specific kind of collection.", 'The concept of symmetry and permutation groups plays a crucial role in understanding solutions to polynomial equations, as the structure of these groups reveals limitations in finding formulas for certain degree polynomials.']}, {'end': 572.082, 'segs': [{'end': 478.339, 'src': 'embed', 'start': 364.66, 'weight': 0, 'content': [{'end': 369.948, 'text': 'And that impossibility has everything to do with the inner structure of the permutation group S5.', 'start': 364.66, 'duration': 5.288}, {'end': 382.714, 'text': 'A theme in math through the last two centuries has been that the nature of symmetry in and of itself can show us all sorts of non-obvious facts about the other objects that we study.', 'start': 373.368, 'duration': 9.346}, {'end': 389.839, 'text': "To give just a hint of the many, many ways that this applies to physics, there's a beautiful fact known as Noether's theorem,", 'start': 383.475, 'duration': 6.364}, {'end': 394.562, 'text': 'saying that every conservation law corresponds to some kind of symmetry, a certain group.', 'start': 389.839, 'duration': 4.723}, {'end': 401.007, 'text': 'So all those fundamental laws like conservation of momentum and conservation of energy each correspond to a group.', 'start': 395.363, 'duration': 5.644}, {'end': 407.029, 'text': "More specifically, the actions we should be able to apply to a setup such that the laws of physics don't change.", 'start': 401.787, 'duration': 5.242}, {'end': 411.05, 'text': 'All of this is to say that groups really are fundamental,', 'start': 408.109, 'duration': 2.941}, {'end': 415.772, 'text': 'and the one thing I want you to recognize right now is that they are one of the most natural things that you could study.', 'start': 411.05, 'duration': 4.722}, {'end': 417.993, 'text': 'What could be more universal than symmetry?', 'start': 416.312, 'duration': 1.681}, {'end': 423.835, 'text': 'So you might think that the patterns among groups themselves would somehow be very beautiful and symmetric.', 'start': 418.813, 'duration': 5.022}, {'end': 426.476, 'text': 'The monster, however, tells a different story.', 'start': 424.575, 'duration': 1.901}, {'end': 433.959, 'text': "Before we get to the monster, though at this point some mathematicians might complain that what I've described so far are not groups exactly,", 'start': 427.176, 'duration': 6.783}, {'end': 437.52, 'text': 'but group actions, and that groups are something slightly more abstract.', 'start': 433.959, 'duration': 3.561}, {'end': 443.903, 'text': "By way of analogy, if I mention the number 3, you probably don't think about a specific triplet of things.", 'start': 438.241, 'duration': 5.662}, {'end': 449.986, 'text': 'You probably think about 3 as an object in and of itself, an abstraction, maybe represented with a symbol.', 'start': 444.423, 'duration': 5.563}, {'end': 458.071, 'text': "In much the same way, when mathematicians discuss the elements of a group, they don't necessarily think about specific actions on specific objects.", 'start': 450.726, 'duration': 7.345}, {'end': 463.075, 'text': 'They might think of these elements as a kind of thing in and of itself, maybe represented with a symbol.', 'start': 458.491, 'duration': 4.584}, {'end': 470.34, 'text': 'For something like the number 3, the abstract symbol does us very little good unless we define its relation with other numbers.', 'start': 464.115, 'duration': 6.225}, {'end': 473.322, 'text': 'For example, the way that it adds or that it multiplies with them.', 'start': 470.78, 'duration': 2.542}, {'end': 478.339, 'text': 'For each of these, you could think of a literal triplet of something.', 'start': 474.229, 'duration': 4.11}], 'summary': 'The nature of symmetry in math reveals non-obvious facts, with conservation laws corresponding to specific groups.', 'duration': 113.679, 'max_score': 364.66, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE364660.jpg'}, {'end': 551.854, 'src': 'embed', 'start': 498.448, 'weight': 6, 'content': [{'end': 503.473, 'text': 'If you flip a snowflake about the x-axis, then rotate it 60 degrees counterclockwise.', 'start': 498.448, 'duration': 5.025}, {'end': 507.457, 'text': 'the overall action is the same as if you had flipped it about a diagonal line.', 'start': 503.473, 'duration': 3.984}, {'end': 517.208, 'text': 'All possible ways that you can combine two elements of a group like this defines a kind of multiplication.', 'start': 511.983, 'duration': 5.225}, {'end': 520.131, 'text': 'That is what really gives a group its structure.', 'start': 517.929, 'duration': 2.202}, {'end': 524.697, 'text': "Here, I'm drawing out the full 8x8 table of the symmetries of a square.", 'start': 521.073, 'duration': 3.624}, {'end': 529.883, 'text': 'If you apply an action from the top row and follow it by an action from the left column,', 'start': 525.318, 'duration': 4.565}, {'end': 532.886, 'text': "it'll be the same as the action in the corresponding grid square.", 'start': 529.883, 'duration': 3.003}, {'end': 543.783, 'text': 'But if we replace each one of these symmetric actions with something purely symbolic, well,', 'start': 538.956, 'duration': 4.827}, {'end': 551.854, 'text': "the multiplication table still captures the inner structure of the group, but now it's abstracted away from any specific object that it might act on,", 'start': 543.783, 'duration': 8.071}], 'summary': 'Symmetries of a square form a group with an 8x8 multiplication table.', 'duration': 53.406, 'max_score': 498.448, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE498448.jpg'}], 'start': 364.66, 'title': 'Symmetry and group structures in math and physics', 'summary': "Explores the significance of symmetry in s5, its relevance to physics via noether's theorem, the abstract nature of elements in mathematics, and the relationship between symmetries and group structure, emphasizing the unique nature of the monster group.", 'chapters': [{'end': 437.52, 'start': 364.66, 'title': 'The nature of symmetry in math and physics', 'summary': "Discusses the significance of symmetry in the permutation group s5, its relevance to physics through noether's theorem, and the fundamental nature of groups in studying universal symmetry and patterns, leading up to the unique nature of the monster group.", 'duration': 72.86, 'highlights': ['The significance of symmetry in the permutation group S5 and its impact on the study of non-obvious facts about other objects in math has been a theme in mathematics over the last two centuries.', "Noether's theorem states that every conservation law in physics corresponds to a certain group, demonstrating the deep connection between symmetry and fundamental laws like conservation of momentum and energy.", 'Groups are fundamental and universal, playing a crucial role in the study of symmetry, making them one of the most natural subjects to study in mathematics.', 'The discussion leads to the revelation that the patterns among groups themselves may not be as beautiful and symmetric as expected, as exemplified by the unique nature of the monster group.']}, {'end': 497.888, 'start': 438.241, 'title': 'Abstract elements in mathematics', 'summary': 'Discusses the abstract nature of elements in mathematics, drawing an analogy with the number 3 and highlighting the importance of defining relations and combinations between elements within a group.', 'duration': 59.647, 'highlights': ['The importance of defining relations between numbers, such as addition or multiplication, is emphasized. (Relevance Score: 3)', 'The concept of elements in a group is likened to the abstract nature of the number 3, represented with symbols and independent of specific actions. (Relevance Score: 2)', 'The significance of how elements combine with each other within a group is highlighted, particularly in the context of actions. (Relevance Score: 1)']}, {'end': 572.082, 'start': 498.448, 'title': 'Group structure and symmetries', 'summary': 'Explains how the combination of symmetries defines a kind of multiplication, giving a group its structure. the multiplication table captures the inner structure abstracted away from specific objects, analogous to the usual multiplication table written symbolically.', 'duration': 73.634, 'highlights': ['The combination of symmetries defines a kind of multiplication, giving a group its structure.', 'The multiplication table captures the inner structure abstracted away from specific objects.', 'The overall action of flipping a snowflake about the x-axis and rotating it 60 degrees counterclockwise is the same as flipping it about a diagonal line.', "Drawing out the full 8x8 table of the symmetries of a square demonstrates the group's structure.", 'Replacing symmetric actions with something purely symbolic still captures the inner structure of the group.']}], 'duration': 207.422, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE364660.jpg', 'highlights': ["Noether's theorem links conservation laws to groups in physics.", "Symmetry's significance in S5 impacts non-obvious math facts.", "Groups' fundamental role in studying symmetry in mathematics.", 'The unique nature of the monster group exemplifies group patterns.', 'The abstract nature of elements in mathematics, akin to the number 3.', 'Defining relations between numbers, such as addition or multiplication.', 'The combination of symmetries defines a kind of multiplication in a group.', 'The multiplication table captures the inner structure of a group.', "Replacing symmetric actions with something purely symbolic captures group's inner structure.", "The overall action of flipping a snowflake demonstrates group's structure."]}, {'end': 995.933, 'segs': [{'end': 597.979, 'src': 'embed', 'start': 572.082, 'weight': 0, 'content': [{'end': 577.703, 'text': 'they free us to think about more complicated numbers and they also free us to think about numbers in new and very different ways.', 'start': 572.082, 'duration': 5.621}, {'end': 585.188, 'text': 'All of this is true of groups as well, which are best understood as abstractions above the idea of symmetry actions.', 'start': 578.823, 'duration': 6.365}, {'end': 587.65, 'text': "I'm emphasizing this for two reasons.", 'start': 586.029, 'duration': 1.621}, {'end': 592.274, 'text': 'One is that understanding what groups really are gives a better appreciation for the monster.', 'start': 588.191, 'duration': 4.083}, {'end': 597.979, 'text': 'And the other is that many students learning about groups for the first time can find them frustratingly opaque.', 'start': 592.835, 'duration': 5.144}], 'summary': 'Understanding groups as abstractions aids in appreciating the monster and can be frustrating for students.', 'duration': 25.897, 'max_score': 572.082, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE572082.jpg'}, {'end': 773.458, 'src': 'embed', 'start': 747.712, 'weight': 2, 'content': [{'end': 753.936, 'text': 'This particular isomorphism between cube rotations and permutations of four objects is a bit subtle.', 'start': 747.712, 'duration': 6.224}, {'end': 760.36, 'text': 'but for the curious among you, you may enjoy taking a moment to think hard about how the rotations of a cube permute its four diagonals.', 'start': 753.936, 'duration': 6.424}, {'end': 770.074, 'text': "In your mathematical life, you'll see more examples of a given group arising from seemingly unrelated situations.", 'start': 764.747, 'duration': 5.327}, {'end': 773.458, 'text': "And as you do, you'll get a better sense for what group theory is all about.", 'start': 770.554, 'duration': 2.904}], 'summary': 'Isomorphism between cube rotations and permutations, explore group theory in math.', 'duration': 25.746, 'max_score': 747.712, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE747712.jpg'}, {'end': 813.485, 'src': 'embed', 'start': 789.07, 'weight': 4, 'content': [{'end': 794.813, 'text': 'There are even plenty of situations where groups come up in a way that does not feel like a set of symmetric actions at all,', 'start': 789.07, 'duration': 5.743}, {'end': 796.773, 'text': 'just as numbers can do a lot more than count.', 'start': 794.813, 'duration': 1.96}, {'end': 804.677, 'text': 'In fact, seeing the same group come up in different situations is a great way to reveal unexpected connections between distinct objects.', 'start': 797.714, 'duration': 6.963}, {'end': 807.158, 'text': "That's a very common theme in modern math.", 'start': 805.217, 'duration': 1.941}, {'end': 813.485, 'text': 'And once you understand this about groups, it leads you to a natural question, which will eventually lead to the monster.', 'start': 808.218, 'duration': 5.267}], 'summary': "Groups in math reveal unexpected connections between objects, leading to the 'monster.'", 'duration': 24.415, 'max_score': 789.07, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE789070.jpg'}, {'end': 877.415, 'src': 'heatmap', 'start': 851.259, 'weight': 0.938, 'content': [{'end': 858.266, 'text': "there's the division between infinite groups, for example the ones describing the symmetries of a line or a circle, and finite groups,", 'start': 851.259, 'duration': 7.007}, {'end': 859.887, 'text': "like the ones we've looked at up to this point.", 'start': 858.266, 'duration': 1.621}, {'end': 863.83, 'text': "To maintain some hope of sanity, let's limit our view to finite groups.", 'start': 860.608, 'duration': 3.222}, {'end': 872.433, 'text': 'In the same way that numbers can be broken down into their prime factorization, or molecules can be described based on the atoms within them.', 'start': 865.011, 'duration': 7.422}, {'end': 877.415, 'text': "there's a certain way that finite groups can be broken down into a kind of composition of smaller groups.", 'start': 872.433, 'duration': 4.982}], 'summary': 'Finite groups can be broken down into smaller groups, like prime factorization for numbers or atoms for molecules.', 'duration': 26.156, 'max_score': 851.259, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE851259.jpg'}, {'end': 980.041, 'src': 'embed', 'start': 954.963, 'weight': 1, 'content': [{'end': 960.327, 'text': 'The first question is like finding the periodic table, and the second is a bit like doing all of chemistry thereafter.', 'start': 954.963, 'duration': 5.364}, {'end': 965.09, 'text': 'The good news is that mathematicians have found all of the finite simple groups.', 'start': 961.267, 'duration': 3.823}, {'end': 971.275, 'text': 'Well, more pertinent is that they proved that the ones that they found are, in fact, all the ones out there.', 'start': 966.131, 'duration': 5.144}, {'end': 976.378, 'text': 'It took many decades, tens of thousands of dense pages of advanced math,', 'start': 972.095, 'duration': 4.283}, {'end': 980.041, 'text': 'hundreds of some of the smartest minds out there and significant help from computers.', 'start': 976.378, 'duration': 3.663}], 'summary': 'Mathematicians found all finite simple groups after decades of work involving tens of thousands of dense pages and hundreds of smart minds, with significant computer assistance.', 'duration': 25.078, 'max_score': 954.963, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE954963.jpg'}], 'start': 572.082, 'title': 'Group symmetry', 'summary': 'Explores the relationship between groups and symmetric actions, emphasizing the abstraction of groups and their relevance in understanding numbers. it also delves into the isomorphism between cube rotations and permutations and its significance in group theory, including the categorization of all finite groups.', 'chapters': [{'end': 679.334, 'start': 572.082, 'title': 'Understanding groups and symmetry', 'summary': 'Discusses how understanding the relationship between groups and symmetric actions can provide a better appreciation for the concepts, with an emphasis on the abstraction of groups above the idea of symmetry actions and their relevance to new ways of thinking about numbers and complicated numbers.', 'duration': 107.252, 'highlights': ['Understanding the relationship between groups and symmetric actions can provide a better appreciation for the concepts, with an emphasis on the abstraction of groups above the idea of symmetry actions and their relevance to new ways of thinking about numbers and complicated numbers.', 'The symmetries of a cube and the permutation group of four objects, although initially feeling very different, are actually the same in the sense that their multiplication tables will look identical.', 'Emphasizing the abstraction of groups above the idea of symmetry actions is important for understanding what groups really are and can provide a better appreciation for the concepts.']}, {'end': 995.933, 'start': 679.334, 'title': 'Isomorphism between cube rotations and permutations', 'summary': 'Discusses the isomorphism between cube rotations and permutations of four elements, the significance of this isomorphism in group theory, and the monumental achievement of categorizing all finite groups by finding all the finite simple groups.', 'duration': 316.599, 'highlights': ['The task of categorizing all finite groups involves finding all the simple groups, which is akin to finding the periodic table, and then finding all the ways to combine them, resembling the study of chemistry thereafter. Mathematicians have found and proved that all the finite simple groups have been discovered, a monumental achievement in the history of math, requiring decades, tens of thousands of dense pages of advanced math, and significant computational assistance.', 'Understanding the nature of simple groups, or atoms, is crucial outside of group theory, as seen in the example where the atomic structure of a certain group is essential in solving a different part of math.', 'Group theory is used to prove that there is no formula for a degree five polynomial, as it involves showing that a mythical quintic formula implies the decomposition of the permutation group on five elements into a special kind of simple group, known as the cyclic groups of prime order.', 'A group is not just about symmetries of a particular object, but an abstract way that things can be symmetric, and it can arise in various situations that do not feel like a set of symmetric actions at all, revealing unexpected connections between distinct objects, which is a common theme in modern math.', 'The isomorphism between cube rotations and permutations of four elements is a fundamental idea in group theory, as it preserves composition and leads to the natural question of what are all the groups up to isomorphism, a question that is exceedingly hard to answer due to the division between infinite and finite groups and the complexity of finite group categorization.', 'The correspondence between the rotations of a cube and permutations of four objects remains true for all products, leading to the concept of an isomorphism, one of the most important ideas in group theory.']}], 'duration': 423.851, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE572082.jpg', 'highlights': ['Understanding the relationship between groups and symmetric actions can provide a better appreciation for the concepts, with an emphasis on the abstraction of groups above the idea of symmetry actions and their relevance to new ways of thinking about numbers and complicated numbers.', 'The task of categorizing all finite groups involves finding all the simple groups, which is akin to finding the periodic table, and then finding all the ways to combine them, resembling the study of chemistry thereafter. Mathematicians have found and proved that all the finite simple groups have been discovered, a monumental achievement in the history of math, requiring decades, tens of thousands of dense pages of advanced math, and significant computational assistance.', 'The symmetries of a cube and the permutation group of four objects, although initially feeling very different, are actually the same in the sense that their multiplication tables will look identical.', 'The isomorphism between cube rotations and permutations of four elements is a fundamental idea in group theory, as it preserves composition and leads to the natural question of what are all the groups up to isomorphism, a question that is exceedingly hard to answer due to the division between infinite and finite groups and the complexity of finite group categorization.', 'A group is not just about symmetries of a particular object, but an abstract way that things can be symmetric, and it can arise in various situations that do not feel like a set of symmetric actions at all, revealing unexpected connections between distinct objects, which is a common theme in modern math.']}, {'end': 1285.392, 'segs': [{'end': 1019.906, 'src': 'embed', 'start': 996.413, 'weight': 0, 'content': [{'end': 1003.297, 'text': 'There are 18 distinct infinite families of simple groups, which makes it really tempting to lean into the whole periodic table analogy.', 'start': 996.413, 'duration': 6.884}, {'end': 1009.18, 'text': 'But groups are stranger than chemistry, because there are also these 26 simple groups that are just left over.', 'start': 1003.857, 'duration': 5.323}, {'end': 1010.781, 'text': "They don't fit the other patterns.", 'start': 1009.48, 'duration': 1.301}, {'end': 1013.702, 'text': 'These 26 are known as the sporadic groups.', 'start': 1011.421, 'duration': 2.281}, {'end': 1019.906, 'text': 'That a field of study rooted in symmetry itself has such a patched together fundamental structure is.', 'start': 1014.343, 'duration': 5.563}], 'summary': 'There are 18 families of simple groups, with 26 sporadic groups standing apart, highlighting the complexity of group theory.', 'duration': 23.493, 'max_score': 996.413, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE996413.jpg'}, {'end': 1093.142, 'src': 'embed', 'start': 1033.833, 'weight': 1, 'content': [{'end': 1040.537, 'text': "These are essentially the symmetries of a regular polygon with a prime number of sides, but where you're not allowed to flip the polygon over.", 'start': 1033.833, 'duration': 6.704}, {'end': 1046.5, 'text': 'Another of these infinite families is very similar to the permutation groups that we saw earlier,', 'start': 1041.497, 'duration': 5.003}, {'end': 1050.062, 'text': "but there's the tiniest constraint on how they're allowed to shuffle n items.", 'start': 1046.5, 'duration': 3.562}, {'end': 1054.5, 'text': 'If they act on 5 or more elements, these groups are simple.', 'start': 1051.698, 'duration': 2.802}, {'end': 1063.024, 'text': "Which incidentally is heavily related to why polynomials with degree 5 or more have solutions that can't be written down using radicals.", 'start': 1056.02, 'duration': 7.004}, {'end': 1066.986, 'text': 'The other 16 families are notably more complicated,', 'start': 1064.064, 'duration': 2.922}, {'end': 1072.829, 'text': "and I'm told that there's at least a little ambiguity in how to organize them into cleanly distinct families without overlap,", 'start': 1066.986, 'duration': 5.843}, {'end': 1077.912, 'text': 'but what everybody agrees on is that the 26 sporadic groups stand out as something very different.', 'start': 1072.829, 'duration': 5.083}, {'end': 1085.997, 'text': 'The largest of these sporadic groups is known, thanks to John Conway, as the monster group, and its size is the number I mentioned at the start.', 'start': 1078.732, 'duration': 7.265}, {'end': 1091, 'text': "The second largest, and I promise this isn't a joke, is known as the baby monster group.", 'start': 1086.798, 'duration': 4.202}, {'end': 1093.142, 'text': 'Together with the baby monster.', 'start': 1091.941, 'duration': 1.201}], 'summary': 'Groups of symmetries, including 26 sporadic groups, with the largest known as the monster group, and the second largest as the baby monster group.', 'duration': 59.309, 'max_score': 1033.833, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE1033833.jpg'}, {'end': 1279.727, 'src': 'embed', 'start': 1231.482, 'weight': 2, 'content': [{'end': 1237.585, 'text': 'But after more numerical coincidences like this were noticed, it gave rise to what became known as the monstrous moonshine conjecture.', 'start': 1231.482, 'duration': 6.103}, {'end': 1239.566, 'text': "Whimsical names just don't stop.", 'start': 1238.105, 'duration': 1.461}, {'end': 1248.051, 'text': 'This was proved by Richard Borchards in 1992, solidifying a connection between very different parts of math that at first glance seemed crazy.', 'start': 1240.447, 'duration': 7.604}, {'end': 1253.114, 'text': 'Six years later, by the way, he won the Fields Medal, in part for the significance of this proof.', 'start': 1248.911, 'duration': 4.203}, {'end': 1257.476, 'text': 'And related to this moonshine is a connection between the monster and string theory.', 'start': 1253.934, 'duration': 3.542}, {'end': 1263.14, 'text': "Maybe it shouldn't come as a surprise that something that arises from symmetry itself is relevant to physics,", 'start': 1258.077, 'duration': 5.063}, {'end': 1268.063, 'text': 'but in light of just how random the monster seems at first glance, this connection still elicits a double take.', 'start': 1263.14, 'duration': 4.923}, {'end': 1276.004, 'text': 'To me, the monster and its absurd size is a nice reminder that fundamental objects are not necessarily simple.', 'start': 1269.778, 'duration': 6.226}, {'end': 1279.727, 'text': "The universe doesn't really care if its final answers look clean.", 'start': 1276.764, 'duration': 2.963}], 'summary': 'Monstrous moonshine conjecture connects math and physics, proven by richard borcherds in 1992, leading to fields medal.', 'duration': 48.245, 'max_score': 1231.482, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE1231482.jpg'}], 'start': 996.413, 'title': 'Simple and monster groups', 'summary': 'Discusses 18 infinite families and 26 sporadic groups, their analogy to the periodic table, constraints on shuffling items, and the complexity in organizing the 16 families. it also explores the enigmatic nature of the monster group, its colossal size of 8 x 10^53, whimsically named sporadic groups, connection to moonshine conjecture, and relevance to string theory and physics.', 'chapters': [{'end': 1072.829, 'start': 996.413, 'title': 'Simple groups and sporadic groups', 'summary': 'Discusses the 18 distinct infinite families and 26 sporadic groups in simple groups, highlighting the analogy to the periodic table and providing examples of infinite families with prime order and constraints on shuffling items, as well as the complexity in organizing the 16 families into distinct groups.', 'duration': 76.416, 'highlights': ['The 18 distinct infinite families of simple groups and the 26 sporadic groups are discussed, revealing the complexity in the fundamental structure of simple groups, analogous to the periodic table. (Relevance: 5)', 'Examples of infinite families are provided, such as cyclic groups with prime order, which are the symmetries of regular polygons, and permutation groups with constraints on shuffling items, particularly related to solutions of polynomials with degree 5 or more. (Relevance: 4)', 'The complexity in organizing the other 16 families into cleanly distinct families without overlap is mentioned, indicating the intricacy in classifying these simple groups. (Relevance: 3)']}, {'end': 1285.392, 'start': 1072.829, 'title': 'The monster group and its mathematical significance', 'summary': 'Explores the enigmatic nature of the monster group, its colossal size of 8 x 10^53, its whimsically named sporadic groups, the connection to moonshine conjecture, and its relevance to string theory and physics.', 'duration': 212.563, 'highlights': ['The monster group, with its size of 8 x 10^53, stands out as an enigmatic fundamental object in math, along with its whimsically named sporadic groups.', 'The connection between the monster group and the moonshine conjecture, proved by Richard Borchards in 1992, solidified a connection between very different parts of math and won him the Fields Medal.', 'The monster group also has a connection to string theory, emphasizing the relevance of symmetry in physics, despite its seemingly random nature.', 'The enigmatic nature of the monster group serves as a reminder that fundamental objects in the universe are not necessarily simple and do not adhere to human notions of simplicity.']}], 'duration': 288.979, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/mH0oCDa74tE/pics/mH0oCDa74tE996413.jpg', 'highlights': ['The 18 distinct infinite families of simple groups and the 26 sporadic groups are discussed, revealing the complexity in the fundamental structure of simple groups, analogous to the periodic table. (Relevance: 5)', 'The monster group, with its size of 8 x 10^53, stands out as an enigmatic fundamental object in math, along with its whimsically named sporadic groups.', 'The connection between the monster group and the moonshine conjecture, proved by Richard Borchards in 1992, solidified a connection between very different parts of math and won him the Fields Medal.', 'Examples of infinite families are provided, such as cyclic groups with prime order, which are the symmetries of regular polygons, and permutation groups with constraints on shuffling items, particularly related to solutions of polynomials with degree 5 or more. (Relevance: 4)', 'The complexity in organizing the other 16 families into cleanly distinct families without overlap is mentioned, indicating the intricacy in classifying these simple groups. (Relevance: 3)', 'The enigmatic nature of the monster group serves as a reminder that fundamental objects in the universe are not necessarily simple and do not adhere to human notions of simplicity.', 'The monster group also has a connection to string theory, emphasizing the relevance of symmetry in physics, despite its seemingly random nature.']}], 'highlights': ['The largest known permutation group has around 9 times 10 to the 159 elements, which is approximately the number of subatoms if every atom in the observable universe had a copy of that universe inside itself.', "The speaker's own choice of a favorite number is 8 times 10 to the 53, roughly the number of atoms in Jupiter, emphasizing the magnitude and scale of this chosen number.", 'The monster group, with its size of 8 x 10^53, stands out as an enigmatic fundamental object in math, along with its whimsically named sporadic groups.', 'The connection between the monster group and the moonshine conjecture, proved by Richard Borchards in 1992, solidified a connection between very different parts of math and won him the Fields Medal.', "The speaker discusses the YouTube math community's initiative to create videos about favorite numbers over 1 million, encouraging viewers to participate."]}