It’s not necessarily the most efficient, but it’s the best guess we have. This is largely done by trial and error. There is no hard proof or surefire way to calculate optimal arrangements; this is just the best that anyone’s come up with so far.
It’s sort of like chess. Using computers, we can analyze moves and games at a very advanced level, but we still haven’t “solved” chess, and we can’t determine whether a game or move is perfect in general. There’s no formula to solve it without exhaustively searching through every possible move, which would take more time than the universe has existed, even with our most powerful computers.
Perhaps someday, someone will figure out a way to prove this mathematically.
tiramichu@sh.itjust.works 3 weeks ago
These categories of geometric problem are ridiculously difficult to find the definitive perfect solution for, which is exactly why people have been grinding on them for decades, and mathematicians can’t say any more than “it’s the best one found so far”
For this particular problem the diagram isn’t answering “the most efficient way to pack some particular square” but “the smallest square that can fit 17 unit-sized (1x1) squares inside it.” - with the answer here being 4.675 unit length per side.
Trivially for 16 squares they would fit inside a grid of 4x4 perfectly, with four squares on each row, nice and tidy. To fit just one more square we could size up to 5x5, and it would remain nice and tidy, but there is then obviously a lot of empty space, which suggests the solution is in-between. But if the solution is in between, then some squares must start going slanted to enable reduction in size, as it is only by doing this we can utilise the unfilled gaps and start poking corners in there.
So, we can’t answer what the optimal solution is going to look like, but we can certainly demonstrate that it’s going to be very ugly and messy.
Another similar (but less ugly) geometric problem is the moving sofa problem which has again seen small iterations over a long period of time.
DozensOfDonner@mander.xyz 3 weeks ago
Lol, the ambidextrous sofa. It’s a butt plug.
ouRKaoS@lemmy.today 3 weeks ago
For two!
Rusty@lemmy.ca 3 weeks ago
Now I want to rewatch Requiem for a dream.
CascadianGiraffe@lemmy.world 3 weeks ago
It’s also a great name for a cover band.
Butt rock covers of gospel songs perhaps?
blackbrook@mander.xyz 3 weeks ago
All this should tell us is that we have a strong irrational preference for right angles being aligned with each other.
DominatorX1@thelemmy.club 3 weeks ago
We have an interpreter in our head. It maps and makes sense of the mysterious whatever. Some of it cultural, some biological. It is vast. There might not even be things and space.
blackbrook@mander.xyz 3 weeks ago
Well yes, and what it means for “there to be things” is a whole discussion in itself. But the concepts of space and time are rather deep and fundamental (to our mental models regardless of how or if that maps to objective reality). The preference for right angles is much less fundamental and we can see past and get over it.
cyrano@lemmy.dbzer0.com 3 weeks ago
Thanks for the explanation
DominatorX1@thelemmy.club 3 weeks ago
For A problem like this. If I was going to do it with an algorithm I would just place shapes at random locations and orientations a trillion times.
It would be much easier with a discreet tile type system of course