2740: Square Packing
Title text: I also managed to improve the solution for n=1 to s<0.97, and with some upgrades I think I can hit 0.96.
The square packing problem is a type of geometry problem. The goal is to find the smallest possible "outer square" that will fit N "inner squares" that are each 1 unit wide and 1 unit tall. In the comic N=11, leading to its name of "The N=11 Square Packing Problem," and the value 's' is the length of the outer square's sides. (For example, with 16 squares arrayed in a 4x4 square, 's' would be 4. [an image would be helpful here])
This comic spoofs a common phenomenon for some values of N: sometimes the optimal solution looks very "sloppy" to human sensibilities. The lack of a uniform grid or pattern, where some squares look to be misaligned or a lot of space looks wasted, counterintuitively leads to a smaller value for 's' than something more "organized" might be. 'N=11' is one such "frustrating" solution (though it should be noted, the solution depicted has not yet been proven to be optimum).
A few days before this comic's post, a web page Squares in squares gained interest on social media platforms such as Twitter and Hacker News. For many values of N, that page depicts the best known solutions, some of them known to be optimum. The one for N=11 (best known but not proven to be optimum) is the one Randall uses for this comic; its general arrangement was found by Walter Trump in 1979 and slightly improved by Gensane et al. in 2004.
Munroe claims to have found a more efficient solution for this N=11 case, by physically deforming the squares involved in the best-known solution with a hydraulic press. The size of the resulting bounding square is indeed smaller, but the "solution" isn't actually one because the inner shapes have countless wrinkles and are no longer squares. Geometrical shapes in packing problems are not conventionally assumed to be deformable in this manner.
The title text mentions the same approach "improved" the solution for 1 unit square, whose optimum solution is obviously that unit square itself with s=1. Munroe remarks that if he had "some upgrades", presumably a more powerful hydraulic press, he could get the resulting square to be even smaller.
The humorous implication behind the comic and the title text is that rather than the shapes being mathematical, abstract shapes, they are actually physical squares, constructed of some extremely strong, but not completely incompressible material. It is not obvious what material that might be: even using a hydraulic press, its cross-sectional area can only be reduced to 92-94% of its original size. (The fact that the theoretical squares exist in a 2D universe in the problem statement, but here are seemingly 3D objects showing distortions in the sides facing the viewer after being presumably crushed from the top and sides in turn by the hydraulic press, is not more fully addressed.)
This is perhaps a related joke to 2706: Bendy, but now with squares and compressed areas instead of triangles and extended lengths. Unsolved packing problems also appear to be a long-standing interest of Randall, who shows himself pondering "the most efficient packing of round-cut diamonds of uniform size" in the What If? Expensive Shoebox, with the title text "A Google search for unsolved+packing+problems very nearly got me just now."
- Gensane, T., Ryckelynck, P. – Improved dense packings of congruent squares in a square. Discrete Comput Geom 34, pages 97–109 (2005). https://doi.org/10.1007/s00454-004-1129-z
- [11 squares are optimally packed inside a square arrangement. A grey square behind it shows the space that the squares take up. There are two squares in the top left and top right corner, then four in an L shape in the bottom left corner with the long side on the bottom. The remaining 5 squares are imperfectly formed into a larger square, rotated roughly 45 degrees, with the fifth square in the remaining space in the bottom right, also rotated. The squares are hand-drawn so the lines are imperfect.]
- Previous best
- (Gensane, 2004)
- [The same arrangement is shown, however the squares have been "crushed" as if they were real objects. The rotated squares have been most affected, with lines indicating crushing as well as rounded corners. The outer squares have been crushed into the inner squares, and many have cracks. A dashed line shows the old grey square while a new, smaller grey square marks the edges of the new arrangement.]
- New record
- [Caption below the panel:]
- I've significantly improved on the solution to the n=11 square packing problem by using a hydraulic press.
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