Squares in Squares

107

ccarlos-menezes 4 days ago 26 commentsRead Article on kingbird.myphotos.cc

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Discussion (26 Comments)Read Original on HackerNews

ebolyen•2 days ago

Very relevant comic:

https://thejenkinscomic.wordpress.com/2024/12/01/brady-bunch...

dmd•2 days ago

sestep•2 days ago

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.

yzydserd•2 days ago

Many squares in circles bests were found this month.

https://erich-friedman.github.io/packing/squincir/

fuzzythinker•1 day ago

Page doesn't say, but I'm guessing with help from AI

goodmythical•about 19 hours ago

Brief search reveals prior academic research related to sexual orientation and dating apps. Doesn't appear to have done anything in maths before.

But then, why were they the first to issue the correct series of prompts to produce these results?

This would lend credence to the efficacy of using LLMs as tools. If mathematicians in the packing field had used the tool before this liberal arts student, they'd have their names on the record page.

gus_massa•4 days ago

In case you want a challenge, 11 is the smaller that has a solution that has not been proven to be optimal.

amne•2 days ago

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.

javier_e06•2 days ago

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?

nuancebydefault•1 day ago

The total image size is scaled each time such that each solution takes up the same amount of space. It is easier to browse that way.

zamadatix•1 day ago

Would you also argue it's odd graphs don't all use the same scale as each other?

charlesrice•1 day ago

Do you want the graphs with 300 squares to be bigger than your screen, or do you want the graph with 1 square to be 30x30 px for no reason? They're just zoomed.

bradley13•2 days ago

Some of these are wild. You expect to see something systematic, but they have little gaps between oddly placed squares in the center.

NooneAtAll3•2 days ago

I love 130. "You thought I'm just a 2-wide strip? SIKE, here's 8-degree polynomial!"

onedognight•2 days ago

Unrelated squares in squares, I think the interjection is PSYCH.

goodmythical•about 19 hours ago

Both appear to be in use, the author succesfully communicated what they intended to communicate, and the audience (you) succesfully recieved the communication.

You've issued a distinction without a difference.

xnx•2 days ago

Awesome site. Slight peeve that arrangements with a prominent diagonal aren't all oriented in the same direction.

razorbeamz•2 days ago

Looks like Hiroshi Nagamochi did all the boring work.

npodbielski•2 days ago

Why 4 is trivial but 6 had to be proved?

smrq•2 days ago

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.

npodbielski•1 day ago

for me it is obvious. If I am reading s=3 as multiplier of side of smaller square to side of bigger square, which means that bigger square side is 3 times the side of smaller one, than it is obvious that it should be poossible to squeeze 9 small squares into bigger square. It is children puzzle after all. What is not obvious here?

goodmythical•about 19 hours ago

Because it could be possible, as we just saw with 5, that by some rotation of some number of cubes, you could fit six unit cubes in to box smaller than area 9. Since we just saw the unusual result in 5, it is worth verifying.

throawayonthe•2 days ago

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

zamadatix•1 day ago

It's two different trivial things. For each, it's just the case which doesn't require doing anything special.

One is trivial proofs, which are where 100% is covered. This doesn't really leave much to prove in terms of whether or not more area can be covered by a different layout.

The other is trivial packings, the very simple type without any tilting or need of gaps between squares. Trivial packings are only sometimes optimal. Of optimal trivial packings, only some can be shown optimal with an aforementioned trivial proof.

matthewfelgate•2 days ago

Sometimes nature is beautiful and sometimes it isn't.