Timeline for Sphere packings : what next after the recent breakthrough of Viazovska (et al.)?
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2020 at 15:39 | comment | added | Watson | Notice that $\Delta_{L}(n)^2$ is always a rational multiple of a power of $\pi$ (where $\Delta_L(n)$ denotes the best lattice sphere packing density in dimension $n$). It follows by Sawin's comment above, and by Prop 3.2.11 + 3.4.6 in Martinet's book "Perfect lattices" : an extremal lattice (i.e. achieves a local extremum of the density function) is a perfect lattice, which has to be similar to an integral lattice ! | |
| Apr 5, 2016 at 21:07 | vote | accept | Archie | ||
| Apr 5, 2016 at 18:19 | comment | added | usul | @Suvrit, I think there is some reason grounded in group theory and having to do with the Monster (perhaps someone knowledgeable can elaborate). | |
| Apr 5, 2016 at 18:03 | answer | added | Yufei Zhao | timeline score: 18 | |
| Apr 5, 2016 at 13:56 | comment | added | Suvrit | @StanleyYaoXiao Then there would have to be a reason that says why 24 is the final number (perhaps grounded in fixed point theory, no idea)... | |
| Apr 5, 2016 at 11:10 | comment | added | Stanley Yao Xiao | I think it is generally believed that the optimal packing in most dimensions is not a lattice packing... in fact, dimension 24 may be the last time where the optimal packing is a lattice packing. I believe in dimension 10 it is known that the best packing is not a lattice packing | |
| Apr 4, 2016 at 19:05 | comment | added | Will Sawin | Any integer lattice packing in dimension $n$ has a density of the form $\pi^{\lfloor \frac{n}{2} \rfloor} $ times the square root of a rational number, because of the formula for the volume of the sphere. Because the optimal packings in dimensions $1,2,3,8,24$ are integer lattices, this explains the occurrence of $\pi$. | |
| Apr 4, 2016 at 18:58 | comment | added | Archie | Yes, apologies, now edited I hope ! | |
| Apr 4, 2016 at 18:58 | history | edited | Archie | CC BY-SA 3.0 |
edited body; edited title
|
| Apr 4, 2016 at 18:48 | review | First posts | |||
| Apr 4, 2016 at 18:52 | |||||
| Apr 4, 2016 at 18:48 | history | asked | Archie | CC BY-SA 3.0 |