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Squares in Squares

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Re: Squares in Squares

#21

Post by zamadatix »

throawayonthe wrote: post #12
npodbielski wrote: post #6

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

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.

Re: Squares in Squares

#22

Post by charlesrice »

zamadatix wrote: post #20
javier_e06 wrote: post #13

Earlier quoted context omitted.

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?

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

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.

Re: Squares in Squares

#23

Post by npodbielski »

smrq wrote: post #8
npodbielski wrote: post #6

Why 4 is trivial but 6 had to be proved?

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.

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?

Re: Squares in Squares

#24

Post by zamadatix »

charlesrice wrote: post #22
zamadatix wrote: post #20

Earlier quoted context omitted.

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

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.

That's what I mean, I can't imagine why anyone would argue the same thing of graphs in general so I'm curious what the difference makes it so they find it so odd in this specific case.

Re: Squares in Squares

#26

Post by goodmythical »

fuzzythinker wrote: post #25
yzydserd wrote: post #14

Many squares in circles bests were found this month. https://erich-friedman.github.io/packing/squincir/

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

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.

Re: Squares in Squares

#27

Post by goodmythical »

onedognight wrote: post #15
NooneAtAll3 wrote: post #5

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.

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.

Re: Squares in Squares

#28

Post by goodmythical »

npodbielski wrote: post #23
smrq wrote: post #8

Earlier quoted context omitted.

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.

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?

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.
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