Professor your work is already amazing, and this new concept for videos is interesting. Today I got highest grade at Mathematical Logic exam, for which I had a huge help from your ZFC videos. I just wanted to express gratitude.
So what's your opinion of the conclusion of Wittgenstein's Tractatus? Do you think he demonstrated that 2nd and 3rd order referents lead to inherently nonsensical conclusions? Do you think ideas about ideas are beyond the scope of symbolic representation?
@Lyam Boylan I think you have the slash the wrong way around at 6:45. When professor Borcherds says "kill off all copies of rho" he means quotient by the subspace spanned by rho. He doesn't mean deleting rho (or integer multiples of it) because that would still leave us in 25 dimensions.
Yes, that is correct. First take an orthogonal subspace to cannonball vector, to go from 26 dimensions to 25 dimensions, then take a translational quotient space to cannonball vector, to go from 25 dimensions to 24 dimensions. It still has to be proven that this takes us from Lorentzian space through "Parabolic" space to Euclidean space (relative to the inner product metric i.e. the isomorphism class of the orthogonal form aka symmetric bilinear form). I wonder if the Leech construction works in finite (non-zero) characteristic also, especially in char two, and in that case if we need to use a quadratic form with an orthogonal form derived through polarization? Probably only in a limited or degenerate way, since the cannonball vector doesn't make much sense in char p where p is a divisor of 70. So Leech lattice in char 11, 13, 17, 19, 23, or >= 29 would probably be more interesting.
It’s stuff like this that reminds me why I love math. For reasons we can’t explain, this suddenly works in 24 dimensions. Nothing could be more mysterious.
I feel like I am just being spoiled with a lecture so good I will dangerously assume all other math lectures are equally as good from this point onwards.
I don't know either, but 8 dimensions is probably connected with the octonions, 24 is twice the dimensionality of the standard model (actually maybe it is the actual dimensionality, bc I think you need two copies of SU(3) for the flavor and color symmetries of the strong force, I would argue you need two copies of SU(2) to account for the weak force and just orienting yourself in 3D, and then you need U(1) for the EM force and we can add another U(1) for time). And both of these are probably connected in some way by the octonions. Also note that if you take two copies of the octonions (which may be needed to construct an associative clifford algebra) the automorphisms groups are 14 + 14 = 28 dimensional, which is also the dimensions of the rotation space of an 8-dim vector space. I think of it as the octonion algebra being a sort of half-cover of SO(8). So if you take two copies of the standard model you have 24 dimensions. If you add in 4 degrees of freedom for compressing spacetime (aka gravity), you get 28 dimensions which matches the two sets of octonions. Just some dimensional numerology I've been thinking about lately.
I barely understood any of it - but stayed for the beauty of what you were describing even if my appreciation was superficial at best. I’d love to know how you go about producing the graphics for these presentations.
Dude i was ready to click away but i thought id give it a chance. About 25 seconds in i was so caught up i thought ‘i should get comfy for this’. This was so stimulating and well presented. Thank tou
Professor, the original paper by Conway&Sloane starts out not with Lorentzian Z^26 but with the even unimodular lattice in R^(25,1); that is, they also include half-integral points. Does your method still produce the Leech lattice up to isometry?
I love recognizing people cited in this video from Numberphile videos. It makes them seem so much more human, and in turn, the study of mathematics seem so much more approachable.
The number of uniform graphs is relatively small for prime node counts and has a big jump up compared to what you might expect at 12. I wouldn't be too surprised if there were relatively few 7D lattices as well. 24 is not as good as 12 but it's also fairly composite (factors: 2, 3, 4, 6, 12) so it can be constructed in lots of different ways with multiplication and grouping of things.
Wow, the Leech lattice is amazing and you gave a very good, clear explanation of it! I recall when I was trying to learn string theory back in the 80s that lots of physicists were excited about 26 dimensions, and I think the reason had something to do with the Leech lattice, though everything they said was way over my head at the time. Perhaps the Leech lattice is special in some way pertaining to the universe or the Multiverse, and perhaps we do live in 26 dimensions! I also suppose the 10 dimensions string theorists often refer to has something to do with E8, though I know next to nothing about any of this other than what I've already said here. If any of you can explain more of this to me, I'd greatly appreciate it!
"People spend hundreds of years studying lattices without noticing these dimensions are special" 25-dimensional being looking at the pattern on his carpet: "You guys are going to be mad, but"
Beautifully done, thank you. Did you know that the kissing problem, I.e., asking how many other spheres can touch a given sphere in n dimensions, goes back to the time of Newton? Newton argued that k(3) = 12, that is, a 3-sphere can be touched by 12 other spheres. Another mathematician argued that k(3) = 13, but of course Newton proved to be right. Modern results related to the kissing problem are achieved with dynamic programming methods, which give a range of values for what k(n) can be. As noted in the video, because of Leech lattices, we know the exact values of k(8) and k(24).
To be precise Co_0 is not simple -- it contains an obvious normal subgroup taking every vector to its negative. When you take the quotient, which acts on 98,280 pairs of spheres instead of 196,560 individual spheres, you get Co_1 which is simple. If you hold one pair in place, you get Co_2 which is simple. If you hold another pair in place, you get Co_3... which is simple. After that you stop getting new simple groups.
Also there's an even simpler (though much less concrete) construction of the Leech lattice in [Conway-Sloane]. Given a lattice L in n dimensions, one can try to make one in n+1 dimensions by looking for the "deepest holes", points in n-space farthest away from any lattice point in L, and putting another copy of L there but moved off some into the new dimension. (If you start with the unique lattice L in one dimension, you'll see that this "lamination" procedure will construct the triangular lattice in 2d.) Once you get to higher dimensions like 4 there are multiple deepest holes and so you have to make choices. But the choices wash out when you get to 8 and 24 dimensions -- while your choices matter along the way, you'll get redirected back through E8 and the Leech lattice.
My only critique is that sometimes the music is a bit overwhelming, making it hard to hear you speak. Beyond that, a beautiful construction combined with beautiful music and beautiful presentation.
Yeah that my fault, haha. Originally I had it too quiet so I bumped it up because I already knew what he was saying, having listened to it a dozen times in the process. Duly noted for the future
@@yamsox Very impressive visuals, some of the best I have ever seen. What do you use to make them, Manim? (edit): saw another comment where you say what you use to animate.
This takes me back to Part III in 87-88, when I did my essay on the LL. Interesting stuff. I still take an interest, but I left academic life in 88 and am very rusty! But you mentioned this was the easiest sporadic construction that you knew. Isn't M24 easier?
I really enjoyed that. I love those "bizarre" occurrences in mathematics. I believe that the 1..24, 70 coincidence is one of the "bizarre" coincidences that is key to constructing the Monster group, too. Does a similar computer-graphics type video exist for the Monster, too?
No. Richard Borcherds specifically stated that he explicitly constructed the Leech lattice in the video lecture because of the fact that it is easy to construct, unlike those for the other sporadic groups. As far as I understand, the monster group is by far the most complicated finite simple group.
@@newwaveinfantry8362 Yes, good point, I've read about the Monster, and lord knows what that would look like on the screen heh. But, I wonder if the high-level, greatly-simplified aspects of it could be visualised in a fun way.
@2:00 it is not a "joke played by the universe". It is a consequence of general relativity _with_ non-trivial local topology (so non-classical GR). The 8 and 24 arise naturally from the 4D Dirac spinors, via the Clifford algebra. You will not so easily see this if you use the matrix algebra rep. There are 8 disjoint 24-cells in the Standard Model of particle physics. An automorphism group of the whole lot is CPTt symmetry: CPT plus a triality "reflection".
Love the crossover video! You sound great edited together. My only complaint is that the piano is terribly trite for math, gives this an "air of majesty" that's maybe a bit much for me. Thanks for this video!
The animation was beautifully done. I think it suits topic that needs some concrete imagination to grab. To be honest I don’t find animation helpful most of the time in most of the math videos as the words already gives me good pictures of the topic, which however is perhaps extremely helpful for non mathematicians as they don’t have the background to translate the descriptions into concrete picture in their heads.
Is there a unified visual lattice like, consistent pattern alignment like the e8 visual? If not is it possible it is yet to be discovered by examining an approximate of the space viewed from a distance?
Good day Professor. As a layperson who recently discovered E8, I was struck by more than a passing similarity, with certain E8 projections, and rose windows of cathedrals, mosque dome patterns, and Buddhist mandalas. I’m neither a physicist nor qualified to speak on religion, but it seems to me these religious artworks are constructed on the lines (and vertices) of E8 projections. Does this interest you or do you have any comment on this? To me it seems clear they were accessing E8 (somehow), and working with different projections of E8, while also integrating this with their religious beliefs. Hopefully I have conveyed my point, though I probably have not worded it correctly in a technical sense.
This is more about Leech than E8. E8 is a root system lattice, but Leech lattice has no roots, only longer vectors of smallest norm. Also, most lower dimensional projections of lattices, that people use or are interested in, do not depend on the peculiar qualities of these higher dimensional lattices at all. Thus what you have seen might be more related to A1+A1, A2, B2, H2, G2, A3, B3, H3, A4, F4, H4 etcetera projections, or it might be totally unrelated to E8, having incompatible symmetries. That said, much of religious patterns are indeed based on mystical visions (drug induced or otherwise), connected to the properties of higher dimensional space, that these religious mystics have glimpsed through direct experience during altered states of human consciousness, a tribute to the mathematical nature of reality, even beyond its limited physical aspect. Such mathematics and deep numerology is ubiquitious in the critical minded Gnostic tradition (of all human cultures and religions), unlike the dumbed down merely belief based, uncritical and superstitious mass versions of religions. I still don't grasp what this has to do particularly with the Leech lattice, which is the topic of this video, so please stay on-topic will ya?
8:11 "This is by far the simplest way of explicitly constructing a sporadic simple group that I know of." With all due respect, do you really feel that this is simpler than constructing M12 from the set of squares in GF(11)? In any case, thanks for the nice presentation.
Beautiful video but the music is distracting, especially as it gets louder. I ordinarily don't have any trouble following along at 2x speed, but here I had to rewind a few times to understand what professor Borcherds said.
4π^2 All prime products are 4π^2= [4π^2]^(8/8)= ⇔ [4π^2]^8 × [4π^2]^(1/8) E8 ÷ E8=1 (E8)^24 ×(E8)^(1/24) 3×8=24 The 3rd generation of quark・lepton And it gives a hint to the ABC problem❗️
![BLACK BAG - Official Trailer [HD] - Only in Theaters March 14](http://i.ytimg.com/vi/Du0Xp8WX_7I/mqdefault.jpg)
Professor your work is already amazing, and this new concept for videos is interesting. Today I got highest grade at Mathematical Logic exam, for which I had a huge help from your ZFC videos. I just wanted to express gratitude.
Congrats!
Congratulations!!
Congratulations!
So what's your opinion of the conclusion of Wittgenstein's Tractatus? Do you think he demonstrated that 2nd and 3rd order referents lead to inherently nonsensical conclusions? Do you think ideas about ideas are beyond the scope of symbolic representation?
@@archenema6792 yes
A pleasure collaborating! Very interesting topic too, as I learned a lot in the process.
Great job on the animations! What program did you use out of curiosity?
@@jacksonstenger Thanks! I used processing 3 for the bulk of the animations, and for the calculations I used python!
@@yamsox Awesome, thanks, I'll be sure to check out processing 3👍
Thanks to Lyam Boylan for the fancy visuals
When two of your favourite creators collaborate
Awesome animations!
Really cool colab!! I hope you make more videos like this in the future :)
This is the highest production video of yours I’ve seen so far, very nice
Keep making videos like this and this channel will be huge
Love the thumbnail and animations!
These new animations make these videos even more fantastic! Brilliant.
Please continue with the animated videos, great work!
@Lyam Boylan I think you have the slash the wrong way around at 6:45. When professor Borcherds says "kill off all copies of rho" he means quotient by the subspace spanned by rho. He doesn't mean deleting rho (or integer multiples of it) because that would still leave us in 25 dimensions.
Yes, that is correct. First take an orthogonal subspace to cannonball vector, to go from 26 dimensions to 25 dimensions, then take a translational quotient space to cannonball vector, to go from 25 dimensions to 24 dimensions. It still has to be proven that this takes us from Lorentzian space through "Parabolic" space to Euclidean space (relative to the inner product metric i.e. the isomorphism class of the orthogonal form aka symmetric bilinear form).
I wonder if the Leech construction works in finite (non-zero) characteristic also, especially in char two, and in that case if we need to use a quadratic form with an orthogonal form derived through polarization? Probably only in a limited or degenerate way, since the cannonball vector doesn't make much sense in char p where p is a divisor of 70. So Leech lattice in char 11, 13, 17, 19, 23, or >= 29 would probably be more interesting.
Hmm, Leech lattice in char three might be interesting also, since 3 divides 24 but doesn't divide 70.
It’s stuff like this that reminds me why I love math. For reasons we can’t explain, this suddenly works in 24 dimensions. Nothing could be more mysterious.
I feel like I am just being spoiled with a lecture so good I will dangerously assume all other math lectures are equally as good from this point onwards.
Where is it that you're quoting this from?
Such beautiful concepts- and with equally beautiful presentations! 🌞
Amazing collaboration, thank you!
Wonderful combination of advanced mathematics with helpful visuals. Brilliant video!
I don't know either, but 8 dimensions is probably connected with the octonions, 24 is twice the dimensionality of the standard model (actually maybe it is the actual dimensionality, bc I think you need two copies of SU(3) for the flavor and color symmetries of the strong force, I would argue you need two copies of SU(2) to account for the weak force and just orienting yourself in 3D, and then you need U(1) for the EM force and we can add another U(1) for time). And both of these are probably connected in some way by the octonions. Also note that if you take two copies of the octonions (which may be needed to construct an associative clifford algebra) the automorphisms groups are 14 + 14 = 28 dimensional, which is also the dimensions of the rotation space of an 8-dim vector space. I think of it as the octonion algebra being a sort of half-cover of SO(8). So if you take two copies of the standard model you have 24 dimensions. If you add in 4 degrees of freedom for compressing spacetime (aka gravity), you get 28 dimensions which matches the two sets of octonions. Just some dimensional numerology I've been thinking about lately.
It was not only interesting, but also a great cinematic experince!!! Loved it
You my friend are amazing at explaining and demonstrating higher concepts. Thank you.
Professor this work is so amazing I've watched it over and over. Together with the RH video these two might be the best manim video's I've ever seen.
I'd love to see more videos in this style.
These visuals were very helpful!
I barely understood any of it - but stayed for the beauty of what you were describing even if my appreciation was superficial at best. I’d love to know how you go about producing the graphics for these presentations.
Beautiful and fascinating! Thank you!
This is beautiful. I'd be lying if I said I understood exactly what is going on here, but I am gonna keep coming back to this video until I do.
Excellent! Presentation quality now worthy of the Professor.
Thank you very much professor! The animation of this video is beautiful, too!
I'm not very good at maths, but I definitely love these crazy geometries that make up our existence.
Dude i was ready to click away but i thought id give it a chance. About 25 seconds in i was so caught up i thought ‘i should get comfy for this’.
This was so stimulating and well presented. Thank tou
An absolutely beautiful video
The perfect video to answer all my questions about something I did not know existed ten minutes ago
Excellent production! I'd call this experiment a success
Thanks to the both of you for such a wonderful video.
Professor, the original paper by Conway&Sloane starts out not with Lorentzian Z^26 but with the even unimodular lattice in R^(25,1); that is, they also include half-integral points. Does your method still produce the Leech lattice up to isometry?
This was so beautiful 🥰Thank you for your work and I hope you continue with that animations! ❤
Thank you for this beautiful video. Hope to see more!
I love recognizing people cited in this video from Numberphile videos. It makes them seem so much more human, and in turn, the study of mathematics seem so much more approachable.
The video replaced forward slash / with backslash \ in the notation, which confused me a bit.
Great presentation Professor.
This whole thing is fascinating as it is trippy.
It always boggles me how big of a field math really is and how weird it gets.
The number of uniform graphs is relatively small for prime node counts and has a big jump up compared to what you might expect at 12. I wouldn't be too surprised if there were relatively few 7D lattices as well. 24 is not as good as 12 but it's also fairly composite (factors: 2, 3, 4, 6, 12) so it can be constructed in lots of different ways with multiplication and grouping of things.
Wow, the Leech lattice is amazing and you gave a very good, clear explanation of it! I recall when I was trying to learn string theory back in the 80s that lots of physicists were excited about 26 dimensions, and I think the reason had something to do with the Leech lattice, though everything they said was way over my head at the time. Perhaps the Leech lattice is special in some way pertaining to the universe or the Multiverse, and perhaps we do live in 26 dimensions! I also suppose the 10 dimensions string theorists often refer to has something to do with E8, though I know next to nothing about any of this other than what I've already said here. If any of you can explain more of this to me, I'd greatly appreciate it!
WOW! Incredible! Please more like this!
Ramping up the production value. Exciting!
I like this; I found the almost-subtitling at the start a bit too much, and maybe the music a bit too loud, but otherwise the format is very pretty.
what is almost-subtitling?
"People spend hundreds of years studying lattices without noticing these dimensions are special"
25-dimensional being looking at the pattern on his carpet: "You guys are going to be mad, but"
Major pedagogical upgrade ! Nice !!
What an absolute LEGEND!
Beautifully done, thank you. Did you know that the kissing problem, I.e., asking how many other spheres can touch a given sphere in n dimensions, goes back to the time of Newton? Newton argued that k(3) = 12, that is, a 3-sphere can be touched by 12 other spheres. Another mathematician argued that k(3) = 13, but of course Newton proved to be right.
Modern results related to the kissing problem are achieved with dynamic programming methods, which give a range of values for what k(n) can be. As noted in the video, because of Leech lattices, we know the exact values of k(8) and k(24).
covid solved the kissing problem
@@chadgregory9037 im not sure if it was so called covid .. or pLandemic (?)
Excellent video!
To be precise Co_0 is not simple -- it contains an obvious normal subgroup taking every vector to its negative. When you take the quotient, which acts on 98,280 pairs of spheres instead of 196,560 individual spheres, you get Co_1 which is simple. If you hold one pair in place, you get Co_2 which is simple. If you hold another pair in place, you get Co_3... which is simple. After that you stop getting new simple groups.
Also there's an even simpler (though much less concrete) construction of the Leech lattice in [Conway-Sloane]. Given a lattice L in n dimensions, one can try to make one in n+1 dimensions by looking for the "deepest holes", points in n-space farthest away from any lattice point in L, and putting another copy of L there but moved off some into the new dimension. (If you start with the unique lattice L in one dimension, you'll see that this "lamination" procedure will construct the triangular lattice in 2d.) Once you get to higher dimensions like 4 there are multiple deepest holes and so you have to make choices. But the choices wash out when you get to 8 and 24 dimensions -- while your choices matter along the way, you'll get redirected back through E8 and the Leech lattice.
Thank you!
Really enjoyed this!
This was beautiful.
My only critique is that sometimes the music is a bit overwhelming, making it hard to hear you speak. Beyond that, a beautiful construction combined with beautiful music and beautiful presentation.
Yeah that my fault, haha. Originally I had it too quiet so I bumped it up because I already knew what he was saying, having listened to it a dozen times in the process. Duly noted for the future
@@yamsox are you the editor? I like the visuals!
@@jakebrowning2373 Thank you! I am indeed
@@yamsox GREAT JOB!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
@@yamsox Very impressive visuals, some of the best I have ever seen. What do you use to make them, Manim?
(edit): saw another comment where you say what you use to animate.
This is so amazing!
this is beautiful!!! i wish i understood what i was seeing!!!!!
This takes me back to Part III in 87-88, when I did my essay on the LL. Interesting stuff. I still take an interest, but I left academic life in 88 and am very rusty! But you mentioned this was the easiest sporadic construction that you knew. Isn't M24 easier?
I really enjoyed that. I love those "bizarre" occurrences in mathematics. I believe that the 1..24, 70 coincidence is one of the "bizarre" coincidences that is key to constructing the Monster group, too. Does a similar computer-graphics type video exist for the Monster, too?
No. Richard Borcherds specifically stated that he explicitly constructed the Leech lattice in the video lecture because of the fact that it is easy to construct, unlike those for the other sporadic groups. As far as I understand, the monster group is by far the most complicated finite simple group.
@@newwaveinfantry8362 Yes, good point, I've read about the Monster, and lord knows what that would look like on the screen heh. But, I wonder if the high-level, greatly-simplified aspects of it could be visualised in a fun way.
I am confused, terrified, and intrigued
the back ground music makes the video unnecessarily mysteries
So amazing! ☺️✨✌🏼
That was amazing
I understand NOTHING of what I just saw, yet I am amazed anyway.
amazing
So cool!
@2:00 it is not a "joke played by the universe". It is a consequence of general relativity _with_ non-trivial local topology (so non-classical GR). The 8 and 24 arise naturally from the 4D Dirac spinors, via the Clifford algebra. You will not so easily see this if you use the matrix algebra rep. There are 8 disjoint 24-cells in the Standard Model of particle physics. An automorphism group of the whole lot is CPTt symmetry: CPT plus a triality "reflection".
Wow... the visual.
When did the production quality get this good??
Amazing.
Love the crossover video! You sound great edited together. My only complaint is that the piano is terribly trite for math, gives this an "air of majesty" that's maybe a bit much for me. Thanks for this video!
Agreed, the music was too loud. One of my favorite videos from Dr. Borcherds to date, though - the visuals were incredibly helpful.
Great 👍
I love the presentation, maybe skip the broken up equation fade-in, I keep wanting to see the whole equation while it's appearing.
Would you be willing to go into some more detail about the holy constructions of the Niemeier lattices?
5:23 why is 70 chosen as the 't' variable? Is it just because it is the nearest square to sigma^24_0(n**2)?
Typo in the description: should be "Neil" instead of "Neal". Fantastic video!
Very good ⚡👍
Nice!
There’s a Baez lecture here on yt that covers this construction, but doesn’t provide such captivating visuals
The animation was beautifully done. I think it suits topic that needs some concrete imagination to grab. To be honest I don’t find animation helpful most of the time in most of the math videos as the words already gives me good pictures of the topic, which however is perhaps extremely helpful for non mathematicians as they don’t have the background to translate the descriptions into concrete picture in their heads.
Any discussion on leech lattice in string theory, and moonshine?
Is there a unified visual lattice like, consistent pattern alignment like the e8 visual? If not is it possible it is yet to be discovered by examining an approximate of the space viewed from a distance?
Just a thought: For the animation is it possible to use white background. May be more aligned with Borcherds's "minimalist" presentation style
That is certainly possible. One may argue you that black is more minimal that white (having it being the absence of light), but I see what you mean!
Is there somewhere I can see an explanation of the visuals? I'm not familiar with these diagrams.
Good day Professor. As a layperson who recently discovered E8, I was struck by more than a passing similarity, with certain E8 projections, and rose windows of cathedrals, mosque dome patterns, and Buddhist mandalas. I’m neither a physicist nor qualified to speak on religion, but it seems to me these religious artworks are constructed on the lines (and vertices) of E8 projections. Does this interest you or do you have any comment on this? To me it seems clear they were accessing E8 (somehow), and working with different projections of E8, while also integrating this with their religious beliefs. Hopefully I have conveyed my point, though I probably have not worded it correctly in a technical sense.
This is more about Leech than E8. E8 is a root system lattice, but Leech lattice has no roots, only longer vectors of smallest norm.
Also, most lower dimensional projections of lattices, that people use or are interested in, do not depend on the peculiar qualities of these higher dimensional lattices at all. Thus what you have seen might be more related to A1+A1, A2, B2, H2, G2, A3, B3, H3, A4, F4, H4 etcetera projections, or it might be totally unrelated to E8, having incompatible symmetries. That said, much of religious patterns are indeed based on mystical visions (drug induced or otherwise), connected to the properties of higher dimensional space, that these religious mystics have glimpsed through direct experience during altered states of human consciousness, a tribute to the mathematical nature of reality, even beyond its limited physical aspect. Such mathematics and deep numerology is ubiquitious in the critical minded Gnostic tradition (of all human cultures and religions), unlike the dumbed down merely belief based, uncritical and superstitious mass versions of religions. I still don't grasp what this has to do particularly with the Leech lattice, which is the topic of this video, so please stay on-topic will ya?
I bought the sphere packings book just last year!
I love the video but please fix Neil Sloane's name in the description :)
Space time curvature?
With Mary Tyler Moore
nice, thanks! there's a typo at 8:08 in the word "exceptional"
I am kind of curious why the music
Yamsox brought me here 🥰
How did they figure that out in the 1960s. thats insane.
Something about emergent properties in higher dimensions make me feel left out.
8:11 "This is by far the simplest way of explicitly constructing a sporadic simple group that I know of." With all due respect, do you really feel that this is simpler than constructing M12 from the set of squares in GF(11)? In any case, thanks for the nice presentation.
Presumably there's only one 1D lattice.
Beautiful video but the music is distracting, especially as it gets louder. I ordinarily don't have any trouble following along at 2x speed, but here I had to rewind a few times to understand what professor Borcherds said.
4π^2
All prime products are
4π^2=
[4π^2]^(8/8)=
⇔
[4π^2]^8 × [4π^2]^(1/8)
E8 ÷ E8=1
(E8)^24 ×(E8)^(1/24)
3×8=24
The 3rd generation of quark・lepton
And it gives a hint to the ABC problem❗️
24 dimensions? Ah, you're talking about the Golay code.
우주너머에서는 서로의 우위가 없음을 알았어요.