It's important to note that while this seems counterintuitive, it's only the most efficient because the small squares' side length is not a perfect divisor of the large square's.
this post was submitted on 01 Jul 2025
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What? No. The divisibility of the side lengths have nothing to do with this.
The problem is what's the smallest square that can contain 17 identical squares. If there were 16 squares it would be simply 4x4.
He's saying the same thing. Because it's not an integer power of 2 you can't have a integer square solution. Thus the densest packing puts some boxes diagonally.
And the next perfect divisor one that would hold all the ones in the OP pic would be 5x5. 25 > 17, last I checked.
this is regardless of that. The meme explains it a bit wierdly, but we start with 17 squares, and try to find most efficient packing, and outer square's size is determined by this packing.
Initially I thought 4x4 square but this is a square of 4.675 sides. Reasonable. Clever maths though.
Now, canwe have fractals built from this?
Say hello to the creation! .-D
(Don't ask about the glowing thing, just don't let it touch your eyes.)
Good job. It'skinda what I expected, except for the glow. But I won't ask about that.
The glow is actually just a natural biproduct of the sheer power of the sq1ua7re
"fractal" just means "broken-looking" (as in "fracture"). see Benoît Mandelbrot's original book on this
I assume you mean "nice looking self-replicating pattern", which you can easily obtain by replacing each square by the whole picture over and over again
Fractal might have meant that when Mandelbrot coined the name, but that is not what it means now.
Why doesn’t he just make the square bigger? That’d be more efficient.
That's not more efficient because the big square is bigger
See, that’s the problem with people nowadays?They want to minimalise everything.
They should just slow down and breathe.
I think people have a hard time wrapping their heads around it because it's very rare to have this sort of problem in the real world. Typically you have a specific size container and need to arrange things in it. You usually don't get to pick an arbitrary size container or area for storage. Even if you for something like shipping, you'd probably want to break this into a 4x4 and a separate single box to better fit with other things being shipped as well. Or if it is storage you'd want to be able to see the sides or tops. Plus you have 3 dimensions to work with on the real world.
With straight diagonal lines.
Homophobe!
hey it's no longer June, homophobia is back on the menu
Why are there gaps on either side of the upper-right square? Seems like shoving those closed (like the OP image) would allow a little more twist on the center squares.
I think this diagram is less accurate. The original picture doesn’t have that gap
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Oh so you're telling me that my storage unit is actually incredibly well optimised for space efficiency?
Nice!
If there was a god, I'd imagine them designing the universe and giggling like an idiot when they made math.
You may not like it but this is what peak performance looks like.
Bees seeing this: "OK, screw it, we're making hexagons!"
Fun fact: Bees actually make round holes, the hexagon shape forms as the wax dries.
come on now, let them cook, trust the process
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Here's a much more elegant solution for 17
Can someone explain to me in layman's terms why this is the most efficient way?
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.
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
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Is this a hard limit we’ve proven or can we still keep trying?
We actually haven't found a universal packing algorithm, so it's on a case-by-case basis. This is the best we've found so far for this case (17 squares in a square).
It's kinda hilarious when the best formula only handles large numbers, not small. You'd think it would be the reverse, but sometimes it just isn't (something about the law of large numbers making it easier to approximate good solution, in many cases)
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It's the best we've found so far
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