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.
@GiovannaIwishyou2 жыл бұрын
Congrats!
@abc-zq8yt2 жыл бұрын
Congratulations!!
@francotomatillo2 жыл бұрын
Congratulations!
@archenema67922 жыл бұрын
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?
@basqye92 жыл бұрын
@@archenema6792 yes
@larspos82642 жыл бұрын
When two of your favourite creators collaborate
@diribigal2 жыл бұрын
Thanks to Lyam Boylan for the fancy visuals
@yamsox2 жыл бұрын
A pleasure collaborating! Very interesting topic too, as I learned a lot in the process.
@jacksonstenger2 жыл бұрын
Great job on the animations! What program did you use out of curiosity?
@yamsox2 жыл бұрын
@@jacksonstenger Thanks! I used processing 3 for the bulk of the animations, and for the calculations I used python!
@jacksonstenger2 жыл бұрын
@@yamsox Awesome, thanks, I'll be sure to check out processing 3👍
@alexandersanchez91382 жыл бұрын
@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.
@arshsverma2 жыл бұрын
Awesome animations!
@DiracComb.75852 жыл бұрын
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.
@yamsox2 жыл бұрын
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
@jakebrowning23732 жыл бұрын
@@yamsox are you the editor? I like the visuals!
@yamsox2 жыл бұрын
@@jakebrowning2373 Thank you! I am indeed
@expchrist2 жыл бұрын
@@yamsox GREAT JOB!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
@matthewkemp69552 жыл бұрын
@@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.
@Helgabion2 жыл бұрын
Really cool colab!! I hope you make more videos like this in the future :)
@officialEricBG2 жыл бұрын
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.
@shubhankarkarn37472 жыл бұрын
what is almost-subtitling?
@yellow58762 жыл бұрын
What's up with the background music? Its too laud and disturbing
@vert48362 жыл бұрын
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.
@someperson90522 жыл бұрын
Where is it that you're quoting this from?
@maynardtrendle8202 жыл бұрын
Such beautiful concepts- and with equally beautiful presentations! 🌞
@lorinbenedict2 жыл бұрын
Professor Borcherds: Please, PLEASE lose the incredibly distracting "music". One of the (many!!) great things about your videos, up to this point, has been your conspicuous avoidance of such things. I think for someone as great as you are, such elements can only subtract. Hence, I respectfully recommend leaving the auditory window-dressing to the amateurs ;)
@NoNTr1v1aL2 жыл бұрын
Love the thumbnail and animations!
@AllenKnutson2 жыл бұрын
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.
@AllenKnutson2 жыл бұрын
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.
@agustinferrer6522 жыл бұрын
Please continue with the animated videos, great work!
@alan2here2 жыл бұрын
Presumably there's only one 1D lattice.
@reedfrey87452 жыл бұрын
I love the presentation, maybe skip the broken up equation fade-in, I keep wanting to see the whole equation while it's appearing.
@drakoz2542 жыл бұрын
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!
@cdellio2 жыл бұрын
Agreed, the music was too loud. One of my favorite videos from Dr. Borcherds to date, though - the visuals were incredibly helpful.
@alan2here2 жыл бұрын
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.
@aam502 жыл бұрын
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.
@chadgregory90372 жыл бұрын
tfw you feel smart cause you already knew about the 8e quasi crystalline lattice =]
@mbgdemon2 жыл бұрын
Keep making videos like this and this channel will be huge
@DavidVonR2 жыл бұрын
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).
@chadgregory90372 жыл бұрын
covid solved the kissing problem
@lepsza_wiadomosc2 жыл бұрын
@@chadgregory9037 im not sure if it was so called covid .. or pLandemic (?)
@melparadise73782 жыл бұрын
I'm not very good at maths, but I definitely love these crazy geometries that make up our existence.
@taydreash49242 жыл бұрын
I am confused, terrified, and intrigued
@angryscientist012 жыл бұрын
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.
@2Luke1002 жыл бұрын
"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"
@JimEadon5 күн бұрын
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?
@captainsnake85152 жыл бұрын
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.
@andrewxiwu2 жыл бұрын
Just a thought: For the animation is it possible to use white background. May be more aligned with Borcherds's "minimalist" presentation style
@yamsox2 жыл бұрын
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!
@oskuh.95772 жыл бұрын
These new animations make these videos even more fantastic! Brilliant.
@atomiccompiler94952 жыл бұрын
Amazing collaboration, thank you!
@svenreichard87262 жыл бұрын
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.
@queenappoline2 жыл бұрын
Beautiful and fascinating! Thank you!
@山山-y4q8 ай бұрын
⭐️💥 √6√ζ(2)=π⇔2×2×2×2×2×2×…, 😃🎉🎊 ⭐️⭐️⭐️ Here, the infinite product of 2 is troublesome, so I replaced it with S. All prime numbers of π function are shown in one P. ⭐️ √6√ζ(2)=π⇔2×2×2×2×2×2×…, π2^-1=2^(S-1), 2^-1=Π(1-1/P^2) 2^(S-1), 1=Π(1-1/P^2) 2^S, If 1 is γ/γ with the Euler constant γ, γ=γΠ(1-1/P^2)2^S, γ=(ζ(1)-1/S) Π(1-1/P^2)2^S, γ=(ζ(s)-1/S) Π(1-1/P^2)2^S, I tried to create a function display for play like that. I know it's funny, though. 🤪🤪🤪🤪🤪🤪🤪🤪🤪🤪🤪
@gunhasirac2 жыл бұрын
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.
@henryaudubon2 жыл бұрын
Wonderful combination of advanced mathematics with helpful visuals. Brilliant video!
@hex73292 жыл бұрын
I understand NOTHING of what I just saw, yet I am amazed anyway.
@gavintillman18842 жыл бұрын
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?
@xcl91892 жыл бұрын
the back ground music makes the video unnecessarily mysteries
@davideranieri5553 Жыл бұрын
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?
@robertjan002 Жыл бұрын
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.
@Ben-ls2ho2 жыл бұрын
You lost me at a sphere can be touched by 6 spheres in 2d, but more spheres can touch a sphere with more dimensions. I see only 6 spheres touching one sphere in 3d.
@AllenKnutson2 жыл бұрын
Stack oranges or cannonballs in the usual way.
@Ben-ls2ho2 жыл бұрын
@@AllenKnutson again, i don't see the oranges after six, touching the center orange. I'm sure I'm misunderstanding something.
@yetanotherjohn2 жыл бұрын
OK. You can have the first three dimensions, but after that, you must call them "Zegmorphs" or "Sqyrfs"; Calling them "dimensions" makes them seem almost useful.
@julesjacobs12 жыл бұрын
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.
@nachoijp2 жыл бұрын
Nobody can convince me that representations of higher scientific concepts don't look like magic symbology... You people are wizards, just say so! Lol
@AkamiChannel Жыл бұрын
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.
@fernandoperez-gonzalez2672 жыл бұрын
Music was so disturbing! Really bad choice. I got anxious watching the video because of this. Animations, on the other hand, were great.
@lauracampbellsykes75552 жыл бұрын
2 dim beings with 3 dim questions propelling into 8th dim thought appears to create something that looks like a galaxy ?
@patrickstetz79992 жыл бұрын
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)?
@WallyMahar Жыл бұрын
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?
@miguelandrade44392 жыл бұрын
It was not only interesting, but also a great cinematic experince!!! Loved it
@johnsavard75832 жыл бұрын
24 dimensions? Ah, you're talking about the Golay code.
@asmeurer2 жыл бұрын
The video replaced forward slash / with backslash \ in the notation, which confused me a bit.
@erickt26652 жыл бұрын
These visuals were very helpful!
@ectoplasm123452 жыл бұрын
Is there somewhere I can see an explanation of the visuals? I'm not familiar with these diagrams.
@cupass61792 жыл бұрын
this is beautiful!!! i wish i understood what i was seeing!!!!!
@OwenMaresh2 жыл бұрын
Would you be willing to go into some more detail about the holy constructions of the Niemeier lattices?
@quadricode2 жыл бұрын
I found the animations and music to mostly be distracting. It made it difficult to focus on what you were saying.
@MCNarret2 жыл бұрын
Something about emergent properties in higher dimensions make me feel left out.
@kwichmath57882 жыл бұрын
I'd love to see more videos in this style.
@amritsahota16412 жыл бұрын
An absolutely beautiful video
@rivkahlevi61172 жыл бұрын
Please lose the music. It's quite distracting at times.
@celestialbeas92142 жыл бұрын
How did they figure that out in the 1960s. thats insane.
@bbsonjohn2 жыл бұрын
Any discussion on leech lattice in string theory, and moonshine?
@양익서-g8j4 ай бұрын
우주너머에서는 서로의 우위가 없음을 알았어요.
@FrankBatistaElJibaro2 жыл бұрын
I was going to say the same thing.
@yphoenix90774 ай бұрын
With Mary Tyler Moore
@mister-86582 жыл бұрын
amazing
@hauthot2872 жыл бұрын
Who else came here from yamsox?
@einaeb5 Жыл бұрын
So amazing! ☺️✨✌🏼
@hengzhou45662 жыл бұрын
Another Conway. I only know the Conway who wrote "A Course in Functional Analysis". Fortunately I don't have to study this (hopefully).
@AllenKnutson2 жыл бұрын
That's John B Conway and this is John H Conway
@chevasit3 ай бұрын
Very good ⚡👍
@atol712 жыл бұрын
Space time curvature?
@dearmamajj Жыл бұрын
Yamsox brought me here 🥰
@singinginthedark27862 жыл бұрын
you completely left out quasicrystals which is extremely important when explaining about dimensional lattice. imagine if you could shine a light on a E8 lattice, the shadow would create a quasicrystal lattice. also a quasicrystal will always have a specific face facing a observer.
@singinginthedark27862 жыл бұрын
even easier way to explain, the only 2d thing that exist in our 3d world is a shadow of a 3d object, therefor 3d is a shadow of a 4d. this is not a truth, just a way to help people understand. dimensions are not made from light or lack of like a shadow, in reality is is more like a quantum shadow, and all dimensions are entangled.
@accountname10472 жыл бұрын
Fantastic video!
@lachlanperrier28512 жыл бұрын
That was amazing
@ophello2 жыл бұрын
How on earth can anyone conceptualize something in 24 dimensions?? I don’t honestly understand this.
@jespervalgreen64612 жыл бұрын
You can't possibly visualize 24 dimensions. But operationally we're fine: suppose you have 24 contacts on your phone, and they can all act independently of one another. And that's a twenty-four dimensional contact-space, or one way to conceptualize 24 dimensions. Now all you need is to learn a lot of rather difficult math.
@plxcxs58552 жыл бұрын
but how did i get here ?
@orthoplex642 жыл бұрын
I bought the sphere packings book just last year! I love the video but please fix Neil Sloane's name in the description :)
@dcterr1 Жыл бұрын
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!
@pattty8472 жыл бұрын
Here from yams
@charlieaydin13772 жыл бұрын
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
@julianwilson99192 жыл бұрын
Typo in the description: should be "Neil" instead of "Neal". Fantastic video!
@calvingoodall20652 жыл бұрын
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.
@thomaswatts65172 жыл бұрын
So cool!
@danielprovder2 жыл бұрын
There’s a Baez lecture here on yt that covers this construction, but doesn’t provide such captivating visuals
@furnace64412 жыл бұрын
The perfect video to answer all my questions about something I did not know existed ten minutes ago
@bigbluebuttonman1137 Жыл бұрын
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.
@杨远-r6d2 жыл бұрын
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.
@Past.Singularity2 жыл бұрын
Amazing.
@bryanbowen41932 жыл бұрын
You my friend are amazing at explaining and demonstrating higher concepts. Thank you.
@AdrianFacchi2 жыл бұрын
Ramping up the production value. Exciting!
@yugiohsc2 жыл бұрын
When did the production quality get this good??
@SaveSoilSaveSoil2 жыл бұрын
Thank you very much professor! The animation of this video is beautiful, too!
@authack97492 жыл бұрын
This is awesome even though I understand nothing of what's being said
@johncoates19232 жыл бұрын
Excellent! Presentation quality now worthy of the Professor.
@zubrz2 жыл бұрын
nice, thanks! there's a typo at 8:08 in the word "exceptional"