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[Maximum Spin]

It's ridiculous to say "nobody looked at that". You didn't look at it, that doesn't mean industry professionals don't look at optimizing things all the time. The type of optimizations shown in those papers you showed are just idiotic from any sort of reasonable engineering standpoint, that's why people didn't bother with that.

He means "nobody's written any mathematical papers on it".

Which I don't think is true, but the problem is kind of trivial so there probably aren't a whole lot. Rectangle packing isn't really a growth field.

[Reelya]

Yeah sorry I realized my error.

BTW came across this big list of packing <thing> in <other thing> research.

https://web.archive.org/web/20160304034446/http://www2.stetson.edu/~efriedma/packing.html

It feels like packing the most <random thing> into <random other thing> mainly became a field after the 1970s. But it's one of those things that's almost meme-level; rather than general research it's about getting published as the guy who one-upped everyone else by getting slightly more of the thing into the other thing, so basically it's a type of game they came up with in the 1970s.

[bloop_bleep]

But it's one of those things that's almost meme-level; rather than general research it's about getting published as the guy who one-upped everyone else by getting slightly more of the thing into the other thing, so basically it's a type of game they came up with in the 1970s.

But that's a lot of math research today, isn't it? Like with the Twin Primes Conjecture. Everything's so developed nowadays even small improvements are rather notable.

[Maximum Spin]

But it's one of those things that's almost meme-level; rather than general research it's about getting published as the guy who one-upped everyone else by getting slightly more of the thing into the other thing, so basically it's a type of game they came up with in the 1970s.

But that's a lot of math research today, isn't it? Like with the Twin Primes Conjecture. Everything's so developed nowadays even small improvements are rather notable.

Math was always just a type of game. Geometry is just a type of game ancient Greek philosophers came up with to try to win prestige by making shapes using a compass and straightedge. Everything that actually serves a purpose is just an accident of the fact that the rules of math were based on physical observations of the real world.

[MaximumZero]

Ugh, I don't know enough about science or math or really much of anything to learn what I need to learn to help pick at the loose edges to help unravel any real mysteries, and that's kind of frustrating. Like, I want to help push the boundaries of knowledge, but I wouldn't even know what classes to take to get to the point of knowing what I need to learn. (It's not like I have anything else to do right now. I'm on crutches and stuck in a chair for the foreseeable future.)

[Reelya]

But it's one of those things that's almost meme-level; rather than general research it's about getting published as the guy who one-upped everyone else by getting slightly more of the thing into the other thing, so basically it's a type of game they came up with in the 1970s.

But that's a lot of math research today, isn't it? Like with the Twin Primes Conjecture. Everything's so developed nowadays even small improvements are rather notable.

Math was always just a type of game. Geometry is just a type of game ancient Greek philosophers came up with to try to win prestige by making shapes using a compass and straightedge. Everything that actually serves a purpose is just an accident of the fact that the rules of math were based on physical observations of the real world.

Yeah, but my point is that someone has to be the first to come up with a specific way of playing the game, and in this case working out the smallest shape of type X that N-number of things of type Y can fit in was that game. Every possible permutation of that is then a different way to play that specific game.

Things like tesselations and space-packings go back millenia. This thing about what's the smallest circle that 27 unit-squares can fit in isn't something that you really think of automatically as a problem you'd need to solve.

[TheSteppeWolf]

I'm going to try to learn French, any tips?

[Kagus]

I'm going to try to learn French, any tips?

My mom said to pucker your lips like you're going in for a kiss when speaking French.

...can't comment on how helpful that is 

[methylatedspirit]

Ugh, I don't know enough about science or math or really much of anything to learn what I need to learn to help pick at the loose edges to help unravel any real mysteries, and that's kind of frustrating. Like, I want to help push the boundaries of knowledge, but I wouldn't even know what classes to take to get to the point of knowing what I need to learn. (It's not like I have anything else to do right now. I'm on crutches and stuck in a chair for the foreseeable future.)

May I recommend 3Blue1Brown? He makes very intuitive videos on math, and they're amazing. I watched this guy, and I came away with a strong interest in math that only faded away because I had to focus on other things. Even if math isn't really your thing, his way of approaching problems leaves you curious about math, and dare I say, the world. To start, I'd suggest watching his video on the unexpectedly hard windmill question. This is one of those channels where you need to be paying attention to get anything from it, so do keep that in mind.

I'm sorry I don't have anything for science. I was more of a math person myself.

[MaximumZero]

Ugh, I don't know enough about science or math or really much of anything to learn what I need to learn to help pick at the loose edges to help unravel any real mysteries, and that's kind of frustrating. Like, I want to help push the boundaries of knowledge, but I wouldn't even know what classes to take to get to the point of knowing what I need to learn. (It's not like I have anything else to do right now. I'm on crutches and stuck in a chair for the foreseeable future.)

May I recommend 3Blue1Brown? He makes very intuitive videos on math, and they're amazing. I watched this guy, and I came away with a strong interest in math that only faded away because I had to focus on other things. Even if math isn't really your thing, his way of approaching problems leaves you curious about math, and dare I say, the world. To start, I'd suggest watching his video on the unexpectedly hard windmill question. This is one of those channels where you need to be paying attention to get anything from it, so do keep that in mind.

I'm sorry I don't have anything for science. I was more of a math person myself.

Hey, this is a great start, and is exactly what I was lamenting I didn't have. Thank you so much.

[Iduno]

I'm going to try to learn French, any tips?

I've heard that something like a video game or a comic makes it easier to learn things, because the entertainment value keeps you trying to learn.

[dragdeler]

-

[methylatedspirit]

I have this odd math question from my brain's archives that I keep thinking about, so I'd like to put this thing to rest. Start with:

Code: [Select]

            n 1 2 3 4 5 6 7 8 9

 n   mod 10 = 1 2 3 4 5 6 7 8 9
n^2  mod 10 = 1 4 9 6 5 6 9 4 1
n^3  mod 10 = 1 8 7 4 5 6 3 2 9
n^4  mod 10 = 1 6 1 6 5 6 1 6 1
n^5  mod 10 = 1 2 3 4 5 6 7 8 9
n^6  mod 10 = 1 4 9 6 5 6 9 4 1
n^7  mod 10 = 1 8 7 4 5 6 3 2 9
n^8  mod 10 = 1 6 1 6 5 6 1 6 1
n^9  mod 10 = 1 2 3 4 5 6 7 8 9
n^10 mod 10 = 1 4 9 6 5 6 9 4 1
n^11 mod 10 = 1 8 7 4 5 6 3 2 9
n^12 mod 10 = 1 6 1 6 5 6 1 6 1


Spoiler: Footnotes for this table (click to show/hide)

n = 0 not shown, since 0 mod 10 = 0

n > 9 not shown either; my argument for that goes:
n > 9 = 10a + c
= (10a + c) mod 10,
but 10a mod 10 = 0, so
(10a + c) mod 10 = c
Restricting domain of n to 0 <= n <= 9 does not lose generality; going further would just loop the same sequence.

If I were to restrict myself to, say, n = 3 and then go down the table, this sequence appears:
a(x) = 3^x mod 10 = 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1
(I'm underlining the repeating section)

If I change it to mod 7 and just restrict myself to n = 6 and going down a table similar to that one (so n, n^2, n^3....), this happens:
b(x) = 6^x mod 7 = 6, 1, 6, 1, 6, 1...
repeats (or to give the more formal term, "is periodic")

Then if I go to mod 13 and n = 11:
c(x) = 11^x mod 13 = 11, 4, 5, 3, 7, 12, 2, 9, 8, 10, 6, 1, 11, 4, 5, 3, 7, 12, 2, 9, 8, 10, 6, 1, 11...
is also periodic.

This family of sequences (which I'll define later) sometimes fails to be periodic if the values of n and the modulus aren't coprime:
d(x) = 6^x mod 12 = 6, 0, 0, 0, 0, 0, 0, 0...
It is eventually periodic with a period of 1 (just remove the first term), so that's something.

And as for "sometimes", I'll just steal from the table:
e(x) = 5^x mod 10 = 5, 5, 5, 5, 5...

Why does it seem to be the case that, given a sequence f(x) = n^x mod a (x is an integer => 0, a and n are integers => 1, a => n), f(x) is periodic if a and n are coprime for all a and n? And on top of that, the period of f(x) under those conditions seems to be at most equal to a. Are my suspicions true, and if so, how would you go about proving it? If they are false, give counterexamples.

(This feels like one of those problems that would be blindingly obvious with some knowledge of modular arithmetic, which I don't have.)

[wierd]

...

I see MOD, and think the MODULO operator.


This is the cognate of integer division;  Where with integer division you divide, then discard any remainder, with modulo, you discard the factor, and keep the remainder as the result.

EG, 3 MOD 2 = 1


I am pretty sure this is NOT what you are trying to do here.

[Bumber]

...

I see MOD, and think the MODULO operator.


This is the cognate of integer division;  Where with integer division you divide, then discard any remainder, with modulo, you discard the factor, and keep the remainder as the result.

EG, 3 MOD 2 = 1


I am pretty sure this is NOT what you are trying to do here.

Looks like it is? It's being used as part of a function to give a sequence.

See the line from the table:

Code: [Select]

n^2  mod 10 = 1 4 9 6 5 6 9 4 1
1² mod 10 = 1
2² mod 10 = 4
3² mod 10 = 9
4² mod 10 = 6
5² mod 10 = 5
...

The main table can also be traversed vertically to have a function from the exponent instead of the base.