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.62400
>>62399oh 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.62405
>>62403good lord i dont understand anything in those articles, time to scrap this idea
No.62406
>>62399Souds 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
>>62406I solved this with wizards 8 years ago. It's possible, you just have to think "inside the box" and in terms of mirrors
No.62410
>>62409Nevermind, 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
>>62409 >>62410I 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
>>62411Yeah, 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
>>62410that'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.pngThis 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.pngThe 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.62497
>>62491>precalculusI'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
>>62497pre-calc is basically just a remedial/refresher course.
No.62501
>>62497It's basically highschool stuff lol. Im just a brainlet so I forgot. I pirated a bunch of textbooks to help me
No.62587
>>35524It'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
>>39808Hey 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.62867
>>57937Isnt this statement true for all x in the reals?
No.62871
>>62867It 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
>>62992The four color theorem was first proved by a computer and it took a long time until mathematicians accepted it.
No.63008
>>63007I 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
>>63008When 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.63788
>>63763I need to stop multitasking..
No.63814
>>62406This 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.63830
>>63814Funny 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
>>63830Good graph theory textbook? It wasnt obvious to me
No.63836
>>63830it'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
>>63836this 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
>>63831My 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.
>>63836I 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.
>>63838Can you show the solution with crossed lines then?
No.63856
>>63843I stand corrected. Also nice job with the gif.
>>63833I 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
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