"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. ;)
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?
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?
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?
parent is changing the problem by suggesting to "pack in the 3rd dimension": lay all the squares on the same square footprint, resulting in always needing only a square with side length 1 on which all the needed "packed" unit squares are laid.
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").
It is not as arbitrary or ugly as it may seem at first - see the image here and the explanation: https://x.com/davidmbudden/status/2107646435659481548
Is there an explanation for this that isn't on X?
This is the paper that’s being referenced: https://pingyou.com/papers/eleven-squares.pdf
It’s not obvious to me that it has any deep significance.
Not clicking an x link, but good to hear
fwiw The prime site for square in square packing is at https://kingbird.myphotos.cc/packing/squares_in_squares.html
The triangular view is most interesting. And a 20 minute video on this view is at https://youtu.be/uL5wuiy34rs
This is cool! Something seems broken in the representation for 1850 and 1765, squares are strangely intersecting.
edit: Or maybe something wrong with the way my browser (brave) is rendering it.
Can someone explain why 83 and 87 can't get any smaller?
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.
It is possible they can; it’s not yet proven that the listed packings for 83 and 87 are optimal.
Which of the blocks do you think you could move to shrink the solution? Or are you thinking of a completely different arrangement?
They got updated to be smaller this year, so maybe there's still more gains to be had?
More on the 11-squares packing:
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
"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. ;)
A list of many square packings, with images:
https://jlevy.github.io/squares/
I like geometry. These packings show there are ugly numbers, like 51.
heh, 105 is a mess
The readme has no figures :( describing the packing?
There's a cool figure here: https://x.com/ojoshe/status/2107590622005924265
Any mirrors which don't require giving clicks to neo-Twitter, please?
https://kingbird.myphotos.cc/packing/squares_in_squares.html
For more packings (circles in circles, etc) check out this page: https://erich-friedman.github.io/packing/index.html
I had the same thought! Pics please.
EDIT: I found it a few links down. https://jlevy.github.io/squares/cases/11.html
<https://en.wikipedia.org/wiki/File:Packing_11_unit_squares_i...> from https://en.wikipedia.org/wiki/Square_packing
The readme also appears to be entirely LLM-written.
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?
Pics or it didn't happen
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?
Lot more pics here: https://jlevy.github.io/squares/
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?
Wow, I never would have imagined one could prove optimality for that accursed beautiful thing.
Did an interval-arithmetic branch and bound once, getting the rounding modes right took me weeks.
So this is a proof that the Walter Trump packing is the optimal packing?
Another interesting video related to these types of problems: https://youtu.be/mVH7OPx4QZU
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?
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.
The whole point is that you can fit more than by naively stacking them...
parent is changing the problem by suggesting to "pack in the 3rd dimension": lay all the squares on the same square footprint, resulting in always needing only a square with side length 1 on which all the needed "packed" unit squares are laid.
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").