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dekhn 2 minutes ago [-]
One of the greatest classes I ever took was "Cybernetics", taught by David Huffman ("the" Huffman). he started out the very first day talking about information theory, into sphere packing, and on to applications of sphere packing to communications.
I distinctly remember him concluded with something like "Sphere packing is hard, except in 11 dimenions" or something like that, but when I look at the history, I can't see how he knew that in 1994?
Can someone explain why 83 and 87 can't get any smaller?
sheept 17 minutes ago [-]
It is possible they can; it’s not yet proven that the listed packings for 83 and 87 are optimal.
entropicdrifter 32 minutes ago [-]
Because the outer perimeter must be a square. 83 and 87 could shrink the outer perimeter in one dimension, but not in both at the same time.
danbruc 37 minutes ago [-]
Which of the blocks do you think you could move to shrink the solution? Or are you thinking of a completely different arrangement?
nemomarx 37 minutes ago [-]
They got updated to be smaller this year, so maybe there's still more gains to be had?
Buttons840 1 hours ago [-]
"God is dead and the optimal packing of squares killed him." I will never not think of this meme when looking at these horrors. I see it, but I don't like it. ;)
It’s unintuitive that a messy configuration of squares can be more optimal than neatly arranging them aligned. And by looking at all the current best solutions it does appear that the neat configurations are usually the best , but not always. How does one explain the messy cases?? Is that about how division can result in irrational numbers, and when the number of optimal squares approach one you end up with the messy squares?
1 hours ago [-]
coppercrisp62 2 hours ago [-]
Did an interval-arithmetic branch and bound once, getting the rounding modes right took me weeks.
Isn't this obvious? Why do you need a proof for it, just stack the cubes next to each other? If we're talking infinitesimally thin squares, then stack them on top of each other? Am I missing something?
raincole 1 hours ago [-]
It takes less time for you to try to read about the question than to type this comment. I know the link doesn't contain visualization, but... come on.
233mhz 1 hours ago [-]
The whole point is that you can fit more than by naively stacking them...
AlexandrB 1 hours ago [-]
Look at some of the other links people have posted for optimal packings. The optimal 11 square packing looks nothing like what you're describing ("just stack the cubes next to each other").
I distinctly remember him concluded with something like "Sphere packing is hard, except in 11 dimenions" or something like that, but when I look at the history, I can't see how he knew that in 1994?
The triangular view is most interesting. And a 20 minute video on this view is at https://youtu.be/uL5wuiy34rs
https://startupfortune.com/ai-models-formally-proved-walter-...
https://vplevris.medium.com/eleven-squares-one-tiny-gap-and-... (written just days before the new proof!)
https://jlevy.github.io/squares/cases/11.html
https://jlevy.github.io/squares/
For more packings (circles in circles, etc) check out this page: https://erich-friedman.github.io/packing/index.html
I really don't get it. If you think you've done something cool, why wouldn't you want to talk about it in your own words?
EDIT: I found it a few links down. https://jlevy.github.io/squares/cases/11.html