Does anyone knows why the Leech lattice has 196560 closest vectors? I wrote this number in binary, and I noticed this was 101 111 111 111 111 010 000 Notice the insane amount of zeros, because it happens to be a multiple of 2¹²-1, to be exact: 196560=48(2¹²-1). This might be a coincidence though. Edit: This also works with E₈ lattice and the lattice in four dimensions, the general formula for 2n dimensions is 4n(2ⁿ-1). But there isn't an infinite amount of coincidences, so I can't really state a conjecture about this.
I came to the same result a while ago but was unable to go much further. I just want to point out that this also works for dimension 4 for which it gives 24 (the kissing number for the somewhat exceptional D4 lattice, associated with the Hurwitz quaternions) and in a way for dimension 2 for which it gives 4, the kissing number for the square lattice which is not maximal in terms of kissing number but is associated with the Gaussian integers and is self-dual. Edit, to recap: 2· 2·(2 ¹-1)= 4 2· 4·(2 ²-1)= 24 2· 8·(2 ⁴-1)= 240 2·24·(2¹²-1)=196560 Edit 2: With d as the dimension, 2*d is the number of semi-axis in that dimension, which seems to point out to the fact that around each of them cluster 2^(d/2)-1 points that are at minimum distance from the *0* point (any lattice point really). Edit 3: My working hypothesis is that 2^(d/2)-1 is half the number of adjacent points that any two mutually adjacent points share, so that the total number of shared points including the two central points is 2^(d/2+1), which is 2^(d/2+1)-2 excluding them, which in turn gives 2^(d/2)-1 adjacent points "each". My overarching goal here is to partition the total 2*d*(2^(d/2)-1) points into nice enough patterns that might give us an insight into why this formula works for certain dimensions but not for others. Edit 4: It is also worth noting that the kissing numbers form a sequence where each number divides the next, which might be a clue as to why the formula works only for certain dimensions, ando also ties in with the fact that "24 is the only number whose divisors - 1, 2, 3, 4, 6, 8, 12, 24 - are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1" (quote from enwp.org/24_(number) ).
@@TheSummoner Maybe 2n(√2ⁿ-1) is an upper bound for the kissing number with dimension n≥4? I checked it with Wikipedia, and these upper bounds check out if 4≤n≤24. Not only does this match if n=4, n=8, and n=24, the other kissing numbers come relatively close to this (at least 70% of this value), so it would be a great bound, that includes all of the significant results about kissing numbers.
Thank you Richard
Yep, that’s the refresher I needed.
Does anyone knows why the Leech lattice has 196560 closest vectors? I wrote this number in binary, and I noticed this was
101 111 111 111 111 010 000
Notice the insane amount of zeros, because it happens to be a multiple of 2¹²-1, to be exact: 196560=48(2¹²-1).
This might be a coincidence though.
Edit:
This also works with E₈ lattice and the lattice in four dimensions, the general formula for 2n dimensions is 4n(2ⁿ-1).
But there isn't an infinite amount of coincidences, so I can't really state a conjecture about this.
I came to the same result a while ago but was unable to go much further. I just want to point out that this also works for dimension 4 for which it gives 24 (the kissing number for the somewhat exceptional D4 lattice, associated with the Hurwitz quaternions) and in a way for dimension 2 for which it gives 4, the kissing number for the square lattice which is not maximal in terms of kissing number but is associated with the Gaussian integers and is self-dual.
Edit, to recap:
2· 2·(2 ¹-1)= 4
2· 4·(2 ²-1)= 24
2· 8·(2 ⁴-1)= 240
2·24·(2¹²-1)=196560
Edit 2:
With d as the dimension, 2*d is the number of semi-axis in that dimension, which seems to point out to the fact that around each of them cluster 2^(d/2)-1 points that are at minimum distance from the *0* point (any lattice point really).
Edit 3:
My working hypothesis is that 2^(d/2)-1 is half the number of adjacent points that any two mutually adjacent points share, so that the total number of shared points including the two central points is 2^(d/2+1), which is 2^(d/2+1)-2 excluding them, which in turn gives 2^(d/2)-1 adjacent points "each". My overarching goal here is to partition the total 2*d*(2^(d/2)-1) points into nice enough patterns that might give us an insight into why this formula works for certain dimensions but not for others.
Edit 4:
It is also worth noting that the kissing numbers form a sequence where each number divides the next, which might be a clue as to why the formula works only for certain dimensions, ando also ties in with the fact that "24 is the only number whose divisors - 1, 2, 3, 4, 6, 8, 12, 24 - are exactly those numbers n for which every invertible element of the commutative ring Z/nZ is a square root of 1" (quote from enwp.org/24_(number) ).
@@TheSummoner Maybe 2n(√2ⁿ-1) is an upper bound for the kissing number with dimension n≥4?
I checked it with Wikipedia, and these upper bounds check out if 4≤n≤24.
Not only does this match if n=4, n=8, and n=24, the other kissing numbers come relatively close to this (at least 70% of this value), so it would be a great bound, that includes all of the significant results about kissing numbers.
Thank you Prof. Borcherds!
Thank you.
Thank you sir
Thank you.