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 “what is 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 the container up to 5x5, and it would remain nice and tidy, but there is then obviously a lot of empty space, which suggests the solution must be in-between. But if the solution is in between, then some squares must start going slanted to enable the outer square to reduce in size, as it is only by doing this we can utilise unfilled gaps to save space by poking the corners of other squares into them.
So, we can’t answer what the optimal solution exactly is, or prove none is better than this, but we can certainly demonstrate that the solution is 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.
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.
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.
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 “what is 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 the container up to 5x5, and it would remain nice and tidy, but there is then obviously a lot of empty space, which suggests the solution must be in-between. But if the solution is in between, then some squares must start going slanted to enable the outer square to reduce in size, as it is only by doing this we can utilise unfilled gaps to save space by poking the corners of other squares into them.
So, we can’t answer what the optimal solution exactly is, or prove none is better than this, but we can certainly demonstrate that the solution is 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.
Lol, the ambidextrous sofa. It’s a butt plug.
For two!
Now I want to rewatch Requiem for a dream.
Requiem is the best movie that I’ve only wanted to watch once.
It’s also a great name for a cover band.
Butt rock covers of gospel songs perhaps?
All this should tell us is that we have a strong irrational preference for right angles being aligned with each other.
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.
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.
My point is, when we study our preference for right angles, what we’re studying is the interpreter. It has quirks.
Thanks for the explanation
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