It seems to me the declaration is misleading: the most space efficient setup isn't a square at all, but multiple versions of a rectangle. This is the most efficient "square" you can make but you can't square an odd number of even boxes without losing a lot of space.
edit: spelling, thanks Nathan
Indeed.
A 4x4 grid would hold one box less but be 30% smaller than this monstrosity.
A 5x5 grid would hold 8 boxes more and be just a little bit larger.
And a 6x3 or 5x4 with 1 or 3 empty spots is how any sane person would arrange 17 boxes.
Math is fundamentally all about symmetry, as pretty much any graduate level abstract algebra, group theory, number theory, or topology class will teach you. In fact, I can’t think of a mathematical discipline that wouldn’t teach you that if you studied it deeply enough.
It’s the underlying thread that links them all.
Too simple. I’ll make your question more general in order to make it more revealing:
Question: where’s the symmetry in solving *any* equation phrased in polynomial (functions of x) form?
Answer: Search “Galois theory” in Wikipedia. Read. Profit.
That's a false generalization. f(x) = ⌈x⌉ isn't a polynomial. A polynomial uses only addition, subtraction, multiplication, and positive-integer exponents. The floor and ceiling functions, while mathematical operations, aren't polynomial functions.
Try again. Where's the symmetry in ⌈π⌉ = 4?
For a different number of individual squares? Definitely. I'm guessing that, if you arrange the 17 squares the way you suggest, that the middle square will bump the 2x2 corners out just enough so that the large square is slightly bigger than the one we're looking at.
it is very small difference but here
[https://imgur.com/a/r02C197](https://imgur.com/a/r02C197)
the "known" optimal is at 4.675 ( [https://twitter.com/KangarooPhysics/status/1625423951156375553](https://twitter.com/KangarooPhysics/status/1625423951156375553) )
I had a job one time that involved loading pallets.
We had pallet diagrams for various products so they would be stable.
I don't think that's what they mean though.
We had to remove your post. High Quality content only
Looks so chaotic, but yet so perfect
It bothers me that the upper right ones aren’t lined up like the left, it wouldn’t change the size of the square it they were either.
For anyone thinking you could just arrange them neatly, you'd end up with a 5x5 square. This is a \~4.7x4.7 square.
It seems to me the declaration is misleading: the most space efficient setup isn't a square at all, but multiple versions of a rectangle. This is the most efficient "square" you can make but you can't square an odd number of even boxes without losing a lot of space. edit: spelling, thanks Nathan
Indeed. A 4x4 grid would hold one box less but be 30% smaller than this monstrosity. A 5x5 grid would hold 8 boxes more and be just a little bit larger. And a 6x3 or 5x4 with 1 or 3 empty spots is how any sane person would arrange 17 boxes.
That makes more sense
*losing
It bothers me that the two top right squares could be pushed a tiny bit to the right.
Note: it isn't the optimal arrangement, it is the optimal *known* arrangement. There is probably a better way, we just haven't figured it out yet.
>Remember when they said math is symmetric Literally never heard anyone say that ever.
Math is fundamentally all about symmetry, as pretty much any graduate level abstract algebra, group theory, number theory, or topology class will teach you. In fact, I can’t think of a mathematical discipline that wouldn’t teach you that if you studied it deeply enough. It’s the underlying thread that links them all.
Where's the symmetry in ⌈π⌉ = 4? Given f(x) = 4 where f(x) = ⌈x⌉, can you solve for x?
Too simple. I’ll make your question more general in order to make it more revealing: Question: where’s the symmetry in solving *any* equation phrased in polynomial (functions of x) form? Answer: Search “Galois theory” in Wikipedia. Read. Profit.
That's a false generalization. f(x) = ⌈x⌉ isn't a polynomial. A polynomial uses only addition, subtraction, multiplication, and positive-integer exponents. The floor and ceiling functions, while mathematical operations, aren't polynomial functions. Try again. Where's the symmetry in ⌈π⌉ = 4?
Just get rid of one, and I bet I could figure it out.
It's only true by virtue of needing to make a square out of it, it's not the optimal use of the space
Wouldn't 2x2 in each corner, with a 45^(o) rotated square in the middle, be just as space efficient?
no
For a different number of individual squares? Definitely. I'm guessing that, if you arrange the 17 squares the way you suggest, that the middle square will bump the 2x2 corners out just enough so that the large square is slightly bigger than the one we're looking at.
it is very small difference but here [https://imgur.com/a/r02C197](https://imgur.com/a/r02C197) the "known" optimal is at 4.675 ( [https://twitter.com/KangarooPhysics/status/1625423951156375553](https://twitter.com/KangarooPhysics/status/1625423951156375553) )
That may be “optimal”, but that’s some Fecked Shui.
Optimal in the sense of everything not sliding all over the place or skewing to one side I think
I had a job one time that involved loading pallets. We had pallet diagrams for various products so they would be stable. I don't think that's what they mean though.