Efficency

You got nothing on the 17 square packing

Can someone explain this?
- This is the most efficient (known) packing of 17 unit squares inside a square. If you're asking why it's like that, that's above my math proficiency level.
  
  https://en.wikipedia.org/wiki/Square_packing
  
  See also: https://kingbird.myphotos.cc/packing/squares_in_squares.html
- We've figured out optimal packing methods for any number of squares inside a big square. When a number is below and near a square number like 15, you just leave an empty box, but when it's far from the next square number, you'll be able to pack them more efficiently than just leaving empty squares around. Turns out this kind of stuff is hilariously hard to prove that it's the most efficient method.
https://xkcd.com/2740

?

https://en.m.wikipedia.org/wiki/Circle_packing
This is about the most efficient way to pack that number of circles. By looking at the bottom row of the 49, you can see that it's slightly less wide than 7 diameters, because it has 5 circles at the very bottom (taking up 5 diameters of width), but two are slightly raised, which also means they're slightly inward.

I mean it makes sense when you think about how the circles arrange in an infinte square and e.g. 4r square. There has to be some fuckery between the perfect packing and the small square packing. You can see a triangle of almost perfect packing in the middle of the 49 circle square, surrounded by fault lines in the structure and then some more good packing, and garbage in the bottom.

slightly related Steve Mould video

Or, they could do 6x8 with one obviously extra at the end. But this is a funny not a rational thing.
Yarr

Neat spacing leave much gap, patterned mess less space between.
Well-put. One perfect pattern at one scale, another perfect pattern at a different scale, and then there has to be a transition between them of optimal steps along the way. I like that.

How?

Yes, if you push the circles down a bit, it forms a 7 by 7 matrix. But if pushing the circles into a square matrix is not allowed: how?

Edit: I get it now. It is about (efficient) packing not about counting. I also get the 4th panel now...

7 by 7 matrix isn’t the optimal packing. The square shown is slightly smaller than 7 by 7.
- Thanks. I thought it was about counting. It all makes a lot more sense now. (And it also doesn't.)
Yeah it can fit almost 7 in a line in the last panel so theese definitely aren't the same squares(or circles)