- The triangular table view is fascinating. It looks like the periodic table. I wonder if there are number-theoretic lemmas (or at least conjectures?) about what "family" the optimal packing for a given number falls into (like diamond, diagonal strip, two blobs, etc). I didn't see anything when skimming the survey paper linked at the bottom of the site, but I'm sure there's a lot more literature here.
- Many squares in circles bests were found this month.
- Awesome site. Slight peeve that arrangements with a prominent diagonal aren't all oriented in the same direction.
- if, like me, you're a non-native english and speaker don't immediately understand what this is about: the page shows for each `n` what's the minimum `s` such that `n` squares with side of length 1 fit in a square with side of length `s`.
what I'm curious about though is what a proof for something like this looks like. and why does it need a proof? not to mention the randomness of some of the `n`s. Math is most of the time beatiful and whenever I see something like `n=11` I think "it looks wrong so it must be wrong" yet it has a proof.
- Same here. Non native English speaker. The first rule is that inner squares are of size 1. Always.
Yet, in each example the inner squares shrink. Uh?
It know it was a convention to better show the arrangement, normalizing, yadda yadda.
Yet, Uh?
- In case you want a challenge, 11 is the smaller that has a solution that has not been proven to be optimal.
- Some of these are wild. You expect to see something systematic, but they have little gaps between oddly placed squares in the center.
- I love 130. "You thought I'm just a 2-wide strip? SIKE, here's 8-degree polynomial!"
- Unrelated squares in squares, I think the interjection is PSYCH.
- Sometimes nature is beautiful and sometimes it isn't.
- Looks like Hiroshi Nagamochi did all the boring work.
- Why 4 is trivial but 6 had to be proved?
- i believe 4, 9, 16, 25 etc are just subdivisions of the unit square (they're perfect squares)
but the text also says "For the $n ≤ 324$ not pictured, the trivial packing (with no tilted squares) is the best known packing." applying 'trivial' to numbers that aren't perfect squares so iunno
- The 4 packing takes up 100% of its square; it's trivially optimal. The 6 packing only takes up 2/3 of it, so it's not necessarily obvious that you can't do better.
