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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.


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



Full MIT OCW Mathematics catalog:


Attached is the a Numberphile video about the seven bridges of Königsberg because I dunno what else to attach to this OP.
231 posts and 51 image replies omitted. Click reply to view.


File: 1617801015185.pdf (79.21 KB, lebesgueMeasure.pdf)

I wrote up some notes on Lebesgue nonmeasurability. I think I tried to make a separate thread about it some time ago. I'm still stuck thinking about why it's not a bigger deal than it is. In my mind it should be up there with the other 'popular' math paradoxes–or at least up there in popularity with stuff like Banach-Tarski or the Klein bottle.


Thanks, that was useful. I'd been wondering for some time what Lebesgue integration was, and never bothered looking into it.


File: 1620689530102.png (67.57 KB, 808x414, 404:207, proof.png) ImgOps iqdb

Practicing writing proofs. Problem is from Mathematical Proofs: A Transition to Advanced Mathematics, Fourth edition, by Chartrand, Polimeni and Zhang.


File: 1620799939318.png (56.83 KB, 665x320, 133:64, proof.png) ImgOps iqdb


File: 1620845816976.png (37.01 KB, 578x294, 289:147, proof.png) ImgOps iqdb

Made it simpler.


This one is hardly readable, but it all boils down to a matter of style. I understand your phrasing in term of P(x) and Q(x) is a didactic device from te book, but you ought to state it more directly using prose.
The second step in your proof obscures the links between the data. Instead say x^2 lies in the interval (0,1) and -2x lies in (-2,0).
Lastly, on step 3, a double negation is always harder to read than a positive statement.
Actually, you can't use k for both numbers because you're constraining your proof for consecutive numbers (eg 4 and 5), so the right hand only
applies to such pairs of numbers.


Still, he's figured out how to use LaTeX and align things well, and he's actually got the idea of a proof down in that the proof is technically correct. For someone starting at this, he's starting out pretty good.

You're right about the second point on >>57945 , this proof is assuming consecutive integers.


(I just remember when I was first starting this and got shat on to high heaven and I pretty much just quit the field entirely, so I wanted to add some positive points)


Oh I do not mean to discourage anyone. But i myself found myself wanting someone to critique my proofs lest I miss anything important.


This isn't a proof as it does nothing to demonstrate that all even integers are equal to 2k where k is some integer, and the same for odd integers and 2k-1


This is 500 pages of chemical kinetics theory, examples, and data. I want to do my capstone on something similar, so I will dig through this to understand the basic theory, and retrace the author's steps in terms of math/computer stuff


It's a bit like me. I was also trying to study maths for a while. I didn't put enough effort and I had to quit.


The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy


>it took Gauss
wasnt that Lobachevsky?


Well, it's assumed that Gauss already knew about it since when Bolyai developed it and his father, Gauss's friend, sent a letter to Gauss to explain about his son's discovery and Gauss reply was not kind:
"To praise it would amount to praising myself. For the entire content of the work…coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years."
And that he already knew about it. Gauss was an arrogant person, btw, he didn't even wanted any of his sons to do mathematics because he knew they would never be better than him. Anyway, Lobachevsky is another person who developed non-euclidian geometry(around the same time as Bolyai) and it's believed that Gauss after learning Russian(at an old age), decided to read Lobachevsky's works which he became interested(it's said that he learned Russian to be able to read this very work, but I have read a russian paper that disagree with that and has valid arguments using Gauss's letters and journal).
Now, the interesting part is that by the time they accepted non-Euclidian Geometry, they realized that they have been using non-Euclidian Geometry for a long time in astronomy: Spherical Geometry. Which is why for hyperbolic Geometry we know who developed it: Bolyai and Lobachevsky; for Spherical Geometry, you are not going to find "the man who started it" since it has been developed since Ancient Greek, they just never realized it was non-Euclidean Geometry.


ig it's complicated like everything in life
btw are you this guy >>51927?


Nope, this >>60183 was my first post in this thread.


>The need for calculus arose mainly because physicists wanted to explore the notion of instantaneous rates of change
Not really. Calculus was already studied way before this, the concept of the area under a graph and the tangent to a point in a graph were both studied for many years. It just so happens that it could also be studied for instantaneous rates of change because it turned out that "finding the tangent to a point" and "the average velocity at a time t as t goes to zero" are the same thing.
>Likewise Hilbert spaces (generalizations for euclidean spaces) were developed so that QM could work.
Hilbert Spaces existed way before QM, it makes no sense that "it was developed so that it would work for a field was not even born yet".

It's not that the universe is based from mathematics, it's that using mathematics is the best way to calculate our observations of the world. Many fields of mathematics were(and are) developed without any relation to the real world, until someone almost 2 centuries later found out he could use groups in QM too, for example.

>there isn't much of a point to theorems that aren't useful

This is flawed since you never know when something is useful. Maybe we shouldn't have studied non-Euclidean Geometry since Euclidean Geometry worked quite well and there was no use for non-Euclidean for a long time.


I'm not an historian of mathematics, but I think your statement:
>Calculus was already studied way before this
only applies to integral calculus. As you pointed out, integral calculus (in a primitive form known as "the method of exhaustion") was already been studied by Eudoxus and Archimedes in ancient Greece, way before Newton and Leibniz. But before Newton and Leibniz, the problem of tangents was approached through very different and ad-hoc methods that have very little to do with our present-day notion of derivative. The idea of a difference quotient really is based on the definition of average velocity (the difference in the dependent variable represents distance, and it is divided by the difference in the independent variable that represents time) and as one takes the limit one obtains instantaneous velocity. It's really hard to deny that calculus was influenced by the study of the physical world. It is no coincidence that Newton worked both on calculus and physics.


I've just been realizing how much of group theory is applied in solving the Rubik's cube. So much so, that I really wonder why subgroups, generators, commutators, centers, and centralizers are not discussed in terms of the Rubik's cube.


Also, while I'm posting in this thread, I feel I should mention…I used to spend a large amount of time studying mathematics, and got burned out when I realize my return for the time I put in was so small. Now I feel this odd sense of jealousy (?) when I see others studying it. I've been trying to get over this, but don't know how.


Group theory goes well with many puzzles, anon. Many puzzles allows you to make a movement and another that gives you back the original position(so you have an inverse to every move), composition of moves are allowed and give the same result no matter the other, the identity is usually no move at all and so on. What is sad is the lack of books and professors teaching group theory in terms of puzzles. Mathematics is, in one way, solving puzzles and so it would go so well.


File: 1656704818843.png (67.53 KB, 480x500, 24:25, 480px-Rubik's_cube.svg.png) ImgOps iqdb

> composition […] give the same result no matter the other [sic]
If only that were true in the general case.
> The Rubik's Cube group is non-abelian as composition of cube moves is not commutative; doing two sequences of cube moves in a different order can result in a different configuration.


Yeah, it's jealousy. How do I get over this jealousy?


I spent years studying it too and got no return. However, I just see people as foolish for wasting their time on it since there's a 99% chance they will also get no return. Your jealousy is odd to me


Yes, I was wrong. I was thinking about abelian group for some reason. My mistake.


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?


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


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

is it 2/3


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

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


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

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


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

this is maybe a more acceptable method though


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.)


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


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



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


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

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


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


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

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


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

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.


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

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.


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.


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

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.


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


> 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)


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


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.


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


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

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