https://kingbird.myphotos.cc/packing/squares_in_squares.html
Mathematics has played us for absolute fools
If you can put the diagonal squares from the 17 solution in a 2-3-2 configuration, I can almost see a pattern. I wonder what other configurations between 17 and 132 have a similar solution?
Can someone explain to me in layman’s terms why this is the most efficient way?
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.
They proved it for n=5 and 10.
And the solutions we have for 5 or 10 appear elegant: perfect 45° angles, symmetry in the packed arrangement.
5 and 10 are interesting because they are one larger than a square number (2^2 and 3^2 respectively). So one might naively assume that the same category of solution could fit 4^2 + 1, where you just take the extra square and try to fit it in a vertical gap and a horizontal gap of exactly the right size to fit a square rotated 45°.
But no, 17 is 4^2 + 1 and this ugly abomination is proven to be more efficient.
It crams the most boxes into the given square. If you take the seven angled boxes out and put them back in an orderly fashion, I think you can fit six of them. The last one won’t fit. If you angle them, this is apparently the best solution.
What I wonder is if this has any practical applications.
yeah it vindicates my approach of packing stuff via just throwing it in there. no I’m not lazy and disorderly, this is optimal cargo space usage
There’s very likely applications in algorithms that try to maximize resource usage while minimizing cost
It’s a problem about minimizing the side length of the outer rectangle in order to fit rectangles of side length 1 into it.
It’s somehow the most efficient way for 17 rectangles because math.
These are the solutions for the numbers next to 17:
the line of man is straight ; the line of god is crooked
stop quoting Nietzsche you fucking fools
Do you know how inspiring documentaries describe maths are everywhere, telling us about the golden ratio in art and animal shells, and pi, and perfect circles and Euler’s number and natural growth, etc? Well, this, I can see it really happening in the world.
I love when I have to do research just to understand the question being asked.
Just kidding, I don’t really love that.
It is one prove more, why it is important to think literally out of the box. But too much people of this type
