[ Home ] [ wiz / dep / hob / lounge / jp / meta / games / music ] [ all ] [  Rules ] [  FAQ ] [  Search /  History ] [  Textboard ] [  Wiki ]

/hob/ - Hobbies

Video game related hobbies go on /games/
Email
Comment
File
Embed
Password (For file deletion.)

  [Go to bottom]   [Catalog]   [Return]   [Archive]

 No.30554[View All]

I liked the first math thread, but that hit the bump limit so I'm making another one.

Here is a neat tool posted in the previous thread that shows you how to do geometry the way the greeks did.

https://sciencevsmagic.net/geo/

Here are a series of MIT OCW courses that will help you learn calculus:

https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/

https://ocw.mit.edu/courses/mathematics/18-02sc-multivariable-calculus-fall-2010/

Full MIT OCW Mathematics catalog:

https://ocw.mit.edu/courses/mathematics/

Attached is the a Numberphile video about the seven bridges of Königsberg because I dunno what else to attach to this OP.
257 posts and 56 image replies omitted. Click reply to view.

 No.62364

Here's a small challenge. The solution has already been posted on another site, but if you've seen it please don't spoil it for others who might want to have a crack at solving it themselves.

Start with a 3x3 tile. At each level take the result of the previous level and place four copies of it in a 2x2 matrix, with a gap of one third of a tile between them. Then add one new tile in the center, which will overlap each of the four blocks for one third of a tile. Here's the result after two iterations:
┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐
│     │ │     │ │     │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │   ┌─┴─┴─┐   │ │     │
└─────┘ └───┤     ├───┘ └─────┘
┌─────┐ ┌───┤     ├───┐ ┌─────┐
│     │ │   └─┬─┬─┘   │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │     │ │     │ │     │
└─────┘ └─────┘ └─────┘ └─────┘

Repeat this process indefinitely. At each level consider the ratio of the area covered by tiles to the area of the minimal square enclosing that level. This enclosing square is simply the one determined by the outermost tiles.

The question is: does the series of ratios converge, and if so what is the limit?

 No.62365

Let's see if the third level fits into a wizchan post.
┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐ ┌─────┐
│     │ │     │ │     │ │     │ │     │ │     │ │     │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │   ┌─┴─┴─┐   │ │     │ │     │ │   ┌─┴─┴─┐   │ │     │
└─────┘ └───┤     ├───┘ └─────┘ └─────┘ └───┤     ├───┘ └─────┘
┌─────┐ ┌───┤     ├───┐ ┌─────┐ ┌─────┐ ┌───┤     ├───┐ ┌─────┐
│     │ │   └─┬─┬─┘   │ │     │ │     │ │   └─┬─┬─┘   │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │     │ │     │ │   ┌─┴─┴─┐   │ │     │ │     │ │     │
└─────┘ └─────┘ └─────┘ └───┤     ├───┘ └─────┘ └─────┘ └─────┘
┌─────┐ ┌─────┐ ┌─────┐ ┌───┤     ├───┐ ┌─────┐ ┌─────┐ ┌─────┐
│     │ │     │ │     │ │   └─┬─┬─┘   │ │     │ │     │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │   ┌─┴─┴─┐   │ │     │ │     │ │   ┌─┴─┴─┐   │ │     │
└─────┘ └───┤     ├───┘ └─────┘ └─────┘ └───┤     ├───┘ └─────┘
┌─────┐ ┌───┤     ├───┐ ┌─────┐ ┌─────┐ ┌───┤     ├───┐ ┌─────┐
│     │ │   └─┬─┬─┘   │ │     │ │     │ │   └─┬─┬─┘   │ │     │
│   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │ │   ┌─┴─┴─┐   │
└───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘ └───┤     ├───┘
┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐ ┌───┤     ├───┐
│   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │ │   └─┬─┬─┘   │
│     │ │     │ │     │ │     │ │     │ │     │ │     │ │     │
└─────┘ └─────┘ └─────┘ └─────┘ └─────┘ └─────┘ └─────┘ └─────┘

 No.62366

File: 1657512976955.png (3.41 KB, 386x257, 386:257, ClipboardImage.png) ImgOps iqdb

>>62364
is it 2/3

 No.62368

File: 1657533576404.png (113.13 KB, 1477x1499, 1477:1499, level4.png) ImgOps iqdb

>>62366
Correct. Do you have a proof sketch? Or did you eyeball the first few ratios?

 No.62378

File: 1657568103824.png (Spoiler Image, 4.06 KB, 348x359, 348:359, ClipboardImage.png) ImgOps iqdb

>>62368
just a guess, i drew them in paint and then selected the black pixels and looked at the areas

 No.62379

File: 1657568537700.png (Spoiler Image, 39.67 KB, 454x738, 227:369, ClipboardImage.png) ImgOps iqdb

>>62378
this is maybe a more acceptable method though

 No.62380

File: 1657570186144.png (398.7 KB, 2958x2971, 2958:2971, level5.png) ImgOps iqdb

>>62378 >>62379
That's pretty neat! Looking at the ratios and seeing them stabilize is a valid method for obtaining the value of the limit. (R's completeness gives us convergence of Cauchy series.)

 No.62399

File: 1657836872148.png (52.87 KB, 1035x3000, 69:200, ClipboardImage.png) ImgOps iqdb

i dont know the proper terms for this stuff but these shapes i will call racetracks

any configuration is allowed except a 2x2 filled in space, this keeps them looking like 'tracks' or paths. is there a rule that determines for any given racetrack whether or not you can visit all the tiles while avoiding tiles you already visited? and if not, how many tiles are left over that it can't visit using the most efficient path

im thinking it has something to do with the number of 'holes' or islands in the track, and the number of tiles with 1 neighbor, 2 neighbors, 3 neighbors, 4 nieghbors. like some kind of ratio between them? i also think by describing the shapes as # of holes, and # of neighbors, you can build different looking racetracks but as long as they have those in common they are functionally identical. maybe some neighbor amounts are more important then others, im not sure

 No.62400

>>62399
oh the colors of the stuff just visually reprsent tiles with 1 neighbor (yellow), 2 neighbors (green), etc. and the numbers below the shapes are for the neigbor tiles, holes, and bad tiles that cant be reached. i thought maybe by coloring them the solution would jump out, but it hasn't

 No.62403


 No.62405

>>62403
good lord i dont understand anything in those articles, time to scrap this idea

 No.62406

File: 1657913330189.jpeg (62.17 KB, 594x516, 99:86, images.jpeg) ImgOps iqdb

>>62399
Souds like the problem of the birdges of Königsberg [1], also related to pic which was all over the internet a few years ago.

[1] https://en.m.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg

 No.62407

>>62406
I solved this with wizards 8 years ago. It's possible, you just have to think "inside the box" and in terms of mirrors

 No.62408

File: 1657913883850.png (194.78 KB, 594x516, 99:86, ClipboardImage.png) ImgOps iqdb

>>62406
i remember than on old 4chan long ago, i recall the best meme answer was one gigantic line

 No.62409

File: 1657918029566.png (16 KB, 433x155, 433:155, IMG_20220715_154411.png) ImgOps iqdb

>>62364
I've been trying to hack this since it was posted, but I have to admit I'm a gigantic retard and I haven't got anywhere near. The most far I've gotten is a big complex expression (pic related). I thought O was getting near but that big square at the denominator is beyond my skill. Back to the drawing board I guess? If I keep going I'll either have to juggle polynomial coefficients and take the limit to infinity, ie a fucking mess.

 No.62410

File: 1657948833736-0.jpg (Spoiler Image, 1.15 MB, 1944x2592, 3:4, IMG_20220715_234246.jpg) ImgOps iqdb

File: 1657948833736-1.jpg (Spoiler Image, 1.04 MB, 2476x1858, 1238:929, IMG_20220716_000437.jpg) ImgOps iqdb

File: 1657948833736-2.png (Spoiler Image, 133.68 KB, 1222x759, 1222:759, IMG_20220716_001856.png) ImgOps iqdb

>>62409
Nevermind, I solved it. Turns out I was on the right path, it just took some number juggling. I also used the data from >>62379 to verify my formulas, and a bit of lisp to do the tedious work. Here's my proof, I rushed the last step, I just verified it with lisp.

 No.62411

File: 1657971859477.png (1.36 MB, 5919x5915, 5919:5915, level6.png) ImgOps iqdb

>>62409 >>62410
I am unable to read your handwriting / low contrast combo. The formula in >>62409 is very nearly correct as a starting point. What we seek is the limit at infinity. The minor problem is that your meanings of n in the numerator and denominator are out of step. When n=1 the numerator yields 41, but the denominator yields 9. The correct denominator for a numerator of 41 is 49. I assume this has been corrected in the handwritten notes and they proceed from a correct starting point, but I am unable to read those.

 No.62412

File: 1657978696517-0.png (Spoiler Image, 39.92 KB, 1269x439, 1269:439, equation1.1.png) ImgOps iqdb

File: 1657978696517-1.png (Spoiler Image, 40.53 KB, 1304x409, 1304:409, equation1.2.png) ImgOps iqdb

>>62411
Yeah, sorry about that, here is the updated equation and the pass to the limit substracting 1 from the top and adding 1 to the bottom.
After that, I just noted it would be easy to write it in base-2, yielding fractions of the form 101000/1000000 = 0.1010 ~= 1/3. as seen in the lisp output in >>62410.

 No.62413

>>62410
that's so cool i wish i understood math like that. i feel some sort of collective pride regardless being on the same website as people who can do stuff like this

 No.62414

>>62412
> equation1.1.png
This numerator is a beautiful reorganization of the numerator from >>62409 but the denominator is still off. When n=1 you get 41/36 instead of 41/49.

> equation1.2.png

The ratio on the right, wrapped in a limit, is correct. This is despite the denominator of equation1.1.png being off, because you rightly kept the dominant terms.

> After that, I just noted it would be easy to write it in base-2, yielding fractions of the form 101000/1000000 = 0.1010 ~= 1/3. as seen in the lisp output in >>62410.

This base-2 view is a very nice idea. Looking at the ratios and seeing a consistent pattern is a valid method for obtaining the value of the limit.

However, since the final part of the proof was done numerically, we have to consider the call for an analytic-only proof sketch to still be open.

╭─────╮ ╭─────╮ ╭─────╮ ╭─────╮
│     │ │     │ │     │ │     │
│   ╭─┴─┴─╮   │ │   ╭─┴─┴─╮   │
╰───┤     ├───╯ ╰───┤     ├───╯
╭───┤     ├───╮ ╭───┤     ├───╮
│   ╰─┬─┬─╯   │ │   ╰─┬─┬─╯   │
│     │ │   ╭─┴─┴─╮   │ │     │
╰─────╯ ╰───┤     ├───╯ ╰─────╯
╭─────╮ ╭───┤     ├───╮ ╭─────╮
│     │ │   ╰─┬─┬─╯   │ │     │
│   ╭─┴─┴─╮   │ │   ╭─┴─┴─╮   │
╰───┤     ├───╯ ╰───┤     ├───╯
╭───┤     ├───╮ ╭───┤     ├───╮
│   ╰─┬─┬─╯   │ │   ╰─┬─┬─╯   │
│     │ │     │ │     │ │     │
╰─────╯ ╰─────╯ ╰─────╯ ╰─────╯

(rounded corners)

 No.62491

File: 1659504814863.png (532.08 KB, 602x699, 602:699, Squid_Girl_Ikamusume_holds….png) ImgOps iqdb

I started doing pre-calculus, as well as physics. It's somewhat relaxing to do maths, I find it more interesting then the physics im doing. I wonder if I could just learn calculus in a couple of days, if I tried really hard. I kind of just want to learn to maths and get to the upper levels out of curiosity

 No.62497

>>62491
>precalculus
I've never understood what that's supposed to be. I recently read a calculus book and it doesn't seem to have much prerequisites beyond basic competence in symbolic manipulation (aka middle-school algebra) and passing familiarity with trig maybe.
Anyway you could very easily learn the essentials of calculus in just a few days, but it's a bit of a deep subject and you'll find you need to learn more in depth as you go.

 No.62500

>>62497
pre-calc is basically just a remedial/refresher course.

 No.62501

>>62497
It's basically highschool stuff lol. Im just a brainlet so I forgot. I pirated a bunch of textbooks to help me

 No.62587

File: 1661338994754.png (60.23 KB, 1358x468, 679:234, wizchan.png) ImgOps iqdb

>>35524
It's because putting things to the power of i like this create an image of a rotation.

Now that you have this knowledge try and solve this? (No googling this is a known result but its fun!!)

 No.62588

>>39808
Hey anon! You have gone very far in your journey congrats!!

you are at a cross roads however to go further into calculus you have to take two steps back before you can take a step forward.

I really recommend James Munkres topology. (http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf)

It's an advanced book however Go through the intro!! its 70 pages long and if you have gotten this far into calculus I reckon you can figure it out (use outside sources)

 No.62589

File: 1661427721898.png (Spoiler Image, 109.59 KB, 1280x996, 320:249, spoiler.png) ImgOps iqdb

> i^i

(Also, avoid the trolls who claim that f(x)=e^x is not a function.)

 No.62867

>>57937
Isnt this statement true for all x in the reals?

 No.62871

>>62867
It is.

(x-1)^2 + 1 != 0

 No.62873

(PEE * 2) * (POO * 2) = PEEPEEPOOPOO

 No.62992

I was amused by all the Novel AI/Diffusion AI stuff coming out recently, and started wondering about how automated proof solvers have developed since the development of machine learning and neural nets.
https://proverbot9001.ucsd.edu/
It's still really primitive, and I feel like the group isn't using the full complexity of machine learning that a lot of other machine learning projects have created, but:
https://proverbot9001.ucsd.edu/compare/
Proverbot9001: 2417 / 11729 proofs.
TacTok+ASTactic: 1377 / 11729 proofs.
ASTactic: 1142 / 11729 proofs.
They're getting a lot better a lot faster. Do any other anons think we may soon be reaching the point where you can just click a goddam button and a proverbot spits out a proof for you? Imagine if the Riemann Hypothesis got proved by a bot. What kind of shit would that storm up?

 No.63007

>>62992
The four color theorem was first proved by a computer and it took a long time until mathematicians accepted it.

 No.63008

>>63007
I think these are categorically different. The computer-aided proof of the 4 color theorem was proof by exhaustion as I understand it ("This isn't a proof, this is a phone book!"). These proofs being spit out by these neural nets are not necessarily that, as they're using actual proof strategies. Yes, they're way too detailed for publication since they pedantically show _every_ step, but it's possible to take the output from proverbot and rewrite it as a typical proof (i.e., use it as a REALLY good proof assistant). Granted it's tedious and a LOT of work, but it transforms work that requires Terry Tao/Ramanujan level genius into "keep your head to the grindstone"/"anyone can do this as long as they keep at it" work.

I also wonder if there's magic that would be able to use as a training set:
inputs - The low-level Coq proof output from proverbot.
outputs - The corresponding human written proofs.
And via some GPT-ish magic, generate a standard human proof. Call this "publish-or-perish" bot.

Take the inverse of publish-or-perish bot to feed proverbot a question, proverbot makes a proof, and then publish-or-perish bot outputs a human-readable proof.

If I were to go more into dreamland, something I wonder about whether proverbot would have this ability as it stands now is…
- Put the statement "There exists an algorithm with a (success rate>proverbot9001) and (runtime<proverbot9001)" into proverbot9001.
- Hopefully, the proof generated by proverbot9001 is constructive, in which case, take the algorithm generated by the output.
- Loop on the previous two items.
- Use this process to generate a theoretically optimal proverbot.

 No.63051

>>63008
When you do self-referencing statements like that you'll run into godel's first incompleteness (and halting problem, busy beaver, or komolgorov complexity by proxy which are all equivalent)

 No.63762

File: 1673208215471.png (516.3 KB, 586x512, 293:256, ClipboardImage.png) ImgOps iqdb

>>62406
Well that didn't take long

 No.63763

File: 1673208861383.jpg (83.36 KB, 586x512, 293:256, wizard_intelligence.jpg) ImgOps iqdb

>>63762
read instructions!!!!!!!!!!!!!!

 No.63788

>>63763
I need to stop multitasking..

 No.63814

>>62406
This is impossible:
Represent each room as a vertex, and also the outside with a vertex. Let each door be an edge between the vertices as depicted. The rooms in the top left, top right, bottom middle, and outside, then, are all odd degree vertices (5, 5, 5, and 9), thus there is no Eulerian path.

 No.63816

File: 1673510907137.png (501.92 KB, 594x516, 99:86, ClipboardImage.png) ImgOps iqdb

>>63814
just go through the doors more than 2 times and ur good

 No.63830

>>63814
Funny how it says "hard but possible!" even though it's a basic result in graph theory that it isn't.
And people still keep trying.
I do wonder why they keep trying.

 No.63831

>>63830
Good graph theory textbook? It wasnt obvious to me

 No.63833

File: 1673624329688.png (36.69 KB, 644x401, 644:401, Oekaki.png) ImgOps iqdb


 No.63834

>>63833
fantastic

 No.63836

>>63830
it's completely possible. the problem is sufficiently vague that you can interpret the rules and contraints in countless ways. if you don't understand what i mean, i could make a list of various assumptions and solutions using them

you might argue this is cheating, but you are doing the same interpretations and making equally baseless assumptions trying to turn it into a math problem

you could argue the real problem to solve isn't the original image, but the largely agreed upon mathematical intepretation, but no one has posted that.

 No.63838

>>63836
this is correct. The solution to the puzzle is as follows:

The original full image labels the puzzle as an autism test. The line-through-doors Autism test doesn't explicitly state that you can't cross lines, but we just assume so because that would be more of a challenge. Those with autism would take the instructions literally, without nuance, and not think of any potential rules that aren't written so they would solve it by crossing lines. So those who "beat" it are the real autists, while those who struggle by not crossing are normal

 No.63842

>>63831
My professor recommended "Introduction to Graph Theory" by Robin Wilson, it's free from the University of Edinburgh online. The proof I laid out depends on the chapter "Paths and cycles" where Eulerian paths are discussed.
>>63836
I like graph theory and wanted to give the proof I do not care about whether it's solvable within the bounds of the vague questioning.
>>63838
Can you show the solution with crossed lines then?

 No.63843

File: 1673651073358.gif (Spoiler Image, 1.9 MB, 1920x1669, 1920:1669, crossedgif.gif) ImgOps iqdb

>>63842
>Can you show the solution with crossed lines then?

Here. Try it for yourself first. the trick is to start somewhere going outward, then loop all the ways back around the whole puzzle.

 No.63856

>>63843
I stand corrected. Also nice job with the gif.
>>63833
I like that caricaturization of myself, I appreciate the effort you put into it, too.

 No.63899

funny, the video in the OP is just about the puzzle everybody is talking about

 No.63960

i was looking up jomon dogu earlier today and it is frustrating trying to see any patterns in how the dogu developed over time, across different regions of japan

is it crazy to use a map of japan as the ground, and then stack dogu above their region of japan, chronolgically? this way you can see where and when themes and patterns of the dogu emerge

i tried arranging stuff in 2d but there is no direct connection to the region of the map, you need the extra vertical axis to stack them i think


[View All]
[Go to top] [Catalog] [Return][Post a Reply]
Delete Post [ ]
[ Home ] [ wiz / dep / hob / lounge / jp / meta / games / music ] [ all ] [  Rules ] [  FAQ ] [  Search /  History ] [  Textboard ] [  Wiki ]