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For the uninitiated: this is the current most-efficient method found of packing 17 unit squares inside another square. You may not like it, but this is what peak efficiency looks like.
(Of course, 16 squares has a packing coefficient of 4, compared to this arrangement’s 4.675, so this is just what peak efficiency looks like for 17 squares)
Edit: For the record, since this blew up, a tiny nitpick in my own explanation above: a smaller value of the packing coefficient is not actually what makes it more efficient (as it is simply the ratio of the larger square’s side to the sides of the smaller squares). The optimal efficiency (zero interstitial space) is achieved when the packing coefficient is precisely equal to the square root of the number of smaller squares. Hence why the case of n=25, with a packing coefficient of 5, is actually more efficient than this packing of n=17, with a packing coefficient of 4.675. Since sqrt(25)=5, that case is a perfectly efficient packing, equal to the case of n=16 with coefficient of 4. Since sqrt(17)=4.123, this packing above is not perfectly efficient, leaving interstices. Obviously. This also means that we may yet find a packing for n=17 with a packing coefficient closer to sqrt(17), which would be an interesting breakthrough, but more important are the questions “is it possible to prove that a given packing is the most efficient possible packing for that value of n” and “does there exist a general rule which produces the most efficient possible packing for any given value of n unit squares?”
Isn’t this only true if the outer square’s size is not an integer multiple of the inner square’s size? Meaning, if you have to do this to your waffle iron, you simply chose the dimensions poorly.
Or maybe you just want waffles with 17 squares in them.
This makes me so angry for reasons I can’t articulate
What makes the lower suboptimal?
Since a link to a wiki article does not an explanation make:
The optimal efficiency (zero interstitial space) is achieved when the ratio of the side length of the larger square to the sides of the shorter squares (let’s call it the “packing coefficient”) is precisely equal to the square root of the number of smaller squares. Hence why the case of n=25, with a packing coefficient of 5, is actually more efficient than the packing of n=17 given in the waffle iron, with a packing coefficient of 4.675. Since sqrt(25)=5, that case is a perfectly efficient packing, equivalent to the case of n=16 with coefficient of 4. Since sqrt(17)=4.123, the waffle packing (represented by the orangutan) above is not perfectly efficient, leaving interstices. However, the packing coefficient of the suboptimal solution (represented by the girl) is actually 4.707, slightly further from sqrt(17), and thus less efficient, leaving greater wasted interstitial space.
Trying to understand what this actually means. Since these two diagrams have the same number of squares, does this mean the inefficient packing squares are actually slightly smaller in a way that’s difficult to observe?
Ah, no, it’s that the more efficient packing takes up less space, so the less efficient square is actually slightly larger than the other, compared to the smaller squares.
If the smaller squares are identical in both sets, then the larger square in the less-efficient set will be slightly bigger than the larger square in the more efficient set.
Related:
https://en.wikipedia.org/wiki/Square_packing
Nature is a lot more elegant with spheres:
https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres
It’s only more efficient when the containing square is large enough that there would be wasted space on the edges if the inner squares were lined up as a grid. The outer square of the waffle iron is almost but not quite large enough to fit a 4x5 grid. People losing their minds over this weird configuration being “more efficient” think it’s because it’s more efficient than a grid where all the space is used, which is not what this would be.
Yeah, there’s a lot of unused space there. Or just look at the gap in the middle of that row of 4. A slightly smaller square could have fit a 5x5, even.
It’s a novelty, not an optimization.
Yeah, if you have extra space but not enough for another row or column, just adjust the size of the inner squares.
Im a dipper. You put the syrup where you want it yourself. Do not rely on some fancy designed skillet to feed you the way you deserve.
The big perk of waffles is the surface area results in a lot of crispy with some fluffy. The fact that it holds syrup is just a perk
Thanks, I hate it!
To be honest I would love a waffle maker like this where some parts of the waffle are a little undercooked and other parts crispy.
Mathematicians: makes something with zero practical applications
Waffles:
The solution is to take a bite of waffle and then take a drink of syrup like it’s a chaser
TIHI
I’m pretty sure that waffle could easily fit 5 rows of 5, am I crazy?
It’s still funny
In the “optimal packing” scenario, it’s slightly too small - like 4.95x4.95
I am sad because these squares look very out of place, unlike hexagons which are beautiful and perfect and never cause problems whatsoever, ever ever!
Hexagons are the bestagons.
Pfft, let me know when “Big Waffle” develops its own proprietary 6-nanometer syrup squares. Until then I will defer to the Belgians and their superior waffle technology.
Those fat Belgian waffles have nothing on the Dutch stroopwafel technology coming out of asml
