"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?
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").
The triangular view is most interesting. And a 20 minute video on this view is at https://youtu.be/uL5wuiy34rs
edit: Or maybe something wrong with the way my browser (brave) is rendering it.
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/
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?
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