what's the most black magic shit you've seen in mathematics?

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and this one too

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>>9110245
I hear if you read Principia Mathematica by Russel and Whitehead aloud over a piece of graph paper a ghost will spin your pencil.

>>9110245


[math] \int\limits_{0}^{\pi} sin x dx = 2 [/math]

I refer specifically to the simple fact that the area under such a curve should turn out to be a natural number at all, and not yet another crazy, irrational number. And that such quantity in relation to such geometric object should be knowable.

>>9110245
for me its the TREE(3)

>>9110255

>tfw I have a hard copy of PM and also of the Simonomicon with its shit sigils on my shelf

get on my level, faggot. Though, I fear that the Reals, despite our sincere efforts to Banish them, are coming for me, and so I type in haste. I have been careful, but

>>9110259
wait what?
Isn't sin supposed to be the y coordinate of a unit circle?
if so the area underneath that y is 2.. that means that the area of a circle is 4?
Am I retarded?
I know it's pi * 1 in unit circle, but.. I don't see the error.

>>9110245
>>9110251
i realize the significance of the second one is that cool shapes form, but what is so significant about the first one

>>9110274
the area under [math] \sin x [/math] is not the area under a circle. The function that you're thinking is probably [math] \sqrt{1 - x^2} [/math]

>>9110274
>sin supposed to be the y coordinate of a unit circle?
y=1sina
x=1cosa
0<=a<2pi
Here y=sinx, x 0 to pi, it's not a unit circle
>Am I retarded?
try not to worry about it anon
>>9110259
I thought e^ipi=-1 was pretty spooky, and i^i=0.21 is just a plain real number.

>>9110315
afaik it is an example of a function that is continuous everywhere but differentiable nowhere

>>9110315
The first one is Weierstrass's function, it was the first example of a function that was continuous everywhere but differentiable nowhere. Before that it was assumed that it was assumed that every continuous function was differentiable, except at certain isolated points.

>>9110245
I've always though linearity of expected value is like black magic (particularly for dependent variables).

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Not the most black magic shit I've seen, but I thought it was a bit humorous.

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>>9110245
> This goddam sequence of numbers
> nature uses it as a rate of growth

The fact that there are two 1's at the start and no 0 really gets to me, but I understand that it must be the case for any growth to continue. I believe this holds a shitload or more secrets than we realize.

https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

Watch the inscribed rectangle one and space filling curves.

Mind blown

>>9110547

o shiiiiiiiiet

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One of my favorites:
https://en.wikipedia.org/wiki/Banach–Tarski_paradox

divide a sphere into four parts
rearrange them
here, two identical spheres!

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>>9110245

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>>9110245
[eqn]1^3+2^3+3^3+...+n^3=(1+2+3+...+n)^2[/eqn]

>>9110651
>AC
He said mathematics.

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>>9110697

>>9110811
This is comfy.

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>>9110533
Even better: Tupper's self-referential formula. This is cheating, though. Some would say it's ugly math, but still funny.

https://en.wikipedia.org/wiki/Tupper%27s_self-referential_formula

>>9110651
any undergrad can find an injection from one sphere to two spheres

it really is a very clever choice of subsets to only requires rotation

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>what's the most black magic shit you've seen in mathematics?

ahem...

>>9110811

Even without doing the induction, there's something very simple and pleasing in expressing the pattern for myself, writing it down, then later looking it up and seeing the exact same form having been used.

In the graphic, the "halving" on the even components' top layers is an enticing idea which admits of a variation, or elaboration.

>>9110830
Absolutely appalling. Of what importance is this inequality?

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>>9112217
Did you even look at the article...

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>>9110245
>>9110697
>>9110811
I give to you a fancier version.

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-1/12

>>9112217
absolutely none, it's just neat.

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>>9112543
Find a book.

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41 triple pendulums with very slightly different initial conditions.

>>9112561
How my life has been going, with t in 6-months periods.

>>9110533
I really liked book of proof. I learned a lot.

>>9110662
CA?

>>9112561
chaos is so neat

>>9112717

no

>>9112561
Just reading up on fractals/chaos theory right now.

Is this a cringe thread or am I plastered and on IFLS without knowing ?

>>9112497
And the original result follows from this by taking n=p^N for any prime p. That's neat.

>>9110352
What's even crazier is that now we know that there are in fact infinitely many more "ugly" functions (continuous everywhere, diff nowhere) than nice functions

>>9112770
/sci/ is the IFLS subreddit.

>>9110245

https://en.wikipedia.org/wiki/Borwein_integral

>>9112561
>science demonstrates the fine line between a good hair day and a bad one

>>9112198

And, as expected, although >>9110811 is manifestly a /geometric/ observation, its demonstration may nevertheless be effected by means of a completely straightforward induction proof, whose only trick is in knowing the simple algebraic representation of the nth triangular number.

We wish to show that

[eqn] \forall n \in \mathbb{N} \; , \;\;\; \sum\limits_{k=1}^{n} k^{3} = \bigg( \sum\limits_{k=1}^{n} k \bigg)^{2} [/eqn]

As 1=1, the basis case is inspected, and the above equation is taken as inductive hypothesis. Adding the cube of n+1 to both sides and recalling the simple algebraic form of the nth triangular number gives the following equalities, where the equality of the red items in particular completes the demonstration.

[eqn] \bigg( \sum\limits_{k=1}^{n} k^{3} \bigg) + (n+1)^{3} = \color{red}{ \sum\limits_{k=1}^{n+1} k^{3} } = \bigg( \frac{n(n+1)}{2} \bigg)^{2} + (n+1)^{3} = \bigg( \frac{(n+1)(n+2)}{2} \bigg)^{2} = \color{red}{ \bigg( \sum\limits_{k=1}^{n+1} k \bigg)^{2} } [/eqn]

>>9110245
(x% of x)/0.01 is x^2

>>9110245

1+2+3+4+5+... until the infinity, is equal to -1/12.

>>9110245
>what's the most black magic shit you've seen in mathematics?

"Our theories about gravity and the speed of the propagation of light tell us that the universe should be a billion times more massive than it appears to be! Must be Dark Energy!"

>>9110245
>what's the most black magic shit you've seen in mathematics?
https://en.wikipedia.org/wiki/Long_line_(topology)

This
https://www.youtube.com/watch?v=kbKtFN71Lfs&t=7s

>>9115142
can this be related to the space of (0,1) valued functions on the reals?

>>9110547
the universe is an infinte fractal that uses the fibbonaci sequence as a ratio for everything.

>>9112561
chaos is exponential and follows the golden ratio

>>9112561
>5'11 v. 6'0

>>9113139
>the basis case is inspected
Yuck

>>9115049
>mathematics
>gravity
>propagation of light
wat

>>9110245
cholesky decomposition is pretty nifty desu as is pca

>>9115561

Your disgust is unwarranted. The remark although banal, is also not inelegant, and above all is a clause in an express and completely explicated demonstration.

When I went over it, it felt vaguely familiar. Then I realized that it is the second exercise for the very first section of Andrews' Number Theory (Dover).

>>9110245

Brouwer's fixed point theorem trumps everything else that has been posted in this thread up to this point. "Dude patterns lmao". No, the above is where stuff starts getting weird and the walls start bleeding.

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fucking pic related

>>9110260
patricians unite

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>FRACTALS INTENSIFY

>>9115676
>Brouwer's fixed point theorem
why? its just the generalization of the fact that any continous function on the unit interval [0,1] intersects the identity function f(x) = x at least once.

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>>9112737

>>9114584
more like shitpost after shitpost amounts to nothing

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>>9115922

>>9115514
golden ratio?
I've seen the Feigenbaum constants as more fundamental

https://en.wikipedia.org/wiki/Feigenbaum_constants

>>9115676
>>9115922
You can actually use Brouwer's fixed point theorem to prove Hutchinson's theorem, which is a part of fractal geometry.
It can also be used in game theory to prove Nash's Theorem, so it's pretty cool

>>9110533
R by itself can be used to encode every picture ever made (as a picture is made out of discrete pixels), and it's a much smaller set than P(R^2)

>>9112561
Holy Macaroni

You have a source?

>>9116161
whats your point?

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Elliptic curves, obviously.

>>9115510
I KNOW DON'T REMIND ME. TELL ME YOUR SECRETS, DIELECTRIC INERTIA PLANE.

>>9116481
the fixed one

>>9115557
>tfw 5'9"

>>9116481
I think its a proof of that houses-and-water-lines graph theory problem's unsolvability via boiling it down to its core nature.

>>9115961
>those red lines on the left
WHERE HAVE I SEEN THOSE BEFORE AND WHY ARE THEY IN THE HIGHER-DIMENSIONAL MANDELBROT SET

>>9112561
I have a question. To me, it seems that the answer is obviously true, but I want to make sure.
Suppose I have two identical chaotic systems. Both start with nearly identical parameters, where the difference in parameters, [math]x[/math] ,is extremely small, and is measured both in absolute terms, and as a ratio of the two parameters' average. So, if one starts at 100,000 , and the other starts at 100,006 , you would measure [math]x[/math] as both 6, in absolute terms, and 3/100,003 (because the starting parameters are 100,003, give or take 3)
Then, let the systems run, and see how long it takes for them to deviate from one another by a predetermined amount, say, 1000. Also, graph the amount of deviance as a function of time, for good measure.
Given this information, is it possible to determine things such as "how long you can expect a system to run according to simulations, given the margin of error(deviance)" or "How far will a system deviate in t seconds assuming we know the starting deviance", etc.

>>9116924
laughed my ass off at this

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>>9117087

it resembles a particular plot of a simple process which quickly goes all qqqwrfhwlkjwlkjfwe; chaotic.

Checking the wiki for chaos theory quickly returns the pic you and I were (almost certainly) both thinking of.

>>9117087
It's the logistic map, plotted as a function of r.
https://en.wikipedia.org/wiki/Logistic_map

>>9117145
Holy cow, that's exactly what I was thinking of. The resemblance is uncanny. But, why does the bifurcation diagram of the logistic map show up in the Mandelbrot set when it's extrapolated to higher dimensions? What's the fundamental, underlying parallel that ties the two together, besides just both of them being fractals?

>>9117133
I don't have exact answers to your questions but a highly related concept are Lyapynov exponents, which you might enjoy reading about

>>9117169
the bifurcation diagram represents the cross sectional map of a strange attractor. the mandelbrot must be in higher dimensions an attractor of some kind.

>>9110262

>>9110533
The set of sets of R2 is at least aleph 1, no fucking shit it contains all possible information. your example is literally the autistic version of the library of Babel, which is actually fascinating precisely because it is finite.

>>9117169
check yourself, mate

this is how memes flow. This is how culture diverges. I experimented with it in highschool. Start an inside joke. Watch to see the first person who "doesn't get it" and fucks it up. Then, count how many people make different variations of the joke. By the time you hit the end of the graph, the original joke is an abstract idea that you can base other jokes around.

"knock knock"

>>9115961
Is this legit coming from the Mandlebrot set alone or is it some autistic "math art" made by someone who should be thrown from a helicopter?

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>>9110245
This is satanic as fuck

>>9110533
Looks like I'm going to jail for CP

>>9110245
When I was a CS undergrad taking a Formal Methods and Automata Theory class, our Professor showed us how to compute the nth Fibonacci number using the formula for the golden ratio. The trick was to use integer math instead of floating point math. And it worked.

I also was similarly impressed with the Gaussian series. Especially since Gauss came up with it as a child.

>>9118953
Mandelbrot is spoopier than life

I made a game of life in my first year of CS work

>>9119098
Also, I have an MSCS, but the toughest CS class I ever took was an undergraduate class on Automated Deduction. And this was one of the text books for that class:
Metalogic: An Introduction to the Metatheory of Standard First Order Logic
https://amzn.com/0520023560

That stuff is black magic math.

Covering spaces and their similarities to Galois groups

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>>9115217
wtf, is that wolfram thingy. But what if you use a hexagon instead of a tringle?

>>9118953
Jesus Christ. More info on this please

>>9119153
>>9119153
notice the game of life inside the game of life

>>9118635
https://en.wikipedia.org/wiki/Buddhabrot
a different way to color the mandelbrot set, basically the historgram of all intermediate steps of all points

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>>9119682

>>9110260
shape-god today!

>>9110245
[math] \sum_{n=1}^{+ \infty} n = - \frac{1}{12}[/math]

>>9110533
P(P(Reals)) contains every combination of the Real^2 planes?

>>9120010
Holy fucking shit didn't watch until the end

Doubt my rendition of life was turing complete

That's a fucking state machine in life, wtf

>>9118953
I got shivers.

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>>9110245
Grahams number and how its different from infinity:
https://youtu.be/8zE-IZynfKM

TREE(3) might also be interesting but I havent found an explanation yet that gets it into my brainlet head

>>9116459
Isn't R and R^2 the same cardinality? Doesn't this imply P(R) is the same cardinality as P(R^2)?

>>9121541
You're correct, but I think that Anon is talking about encoding it as a string of numbers after the decimal point.
E.g. taking a photo and breaking it down into whatever binary is used to encode it (please don't shoot me comp sci).

>>9115241
I'm not very trained in math, but I'm pretty sure you can create a bijection between the long line (L) and the real number line. And you can also create a bijection between (0,1) real valued functions and all possible real valued functions, so these two sets are mismatched by one powerset level of cardinality.

>>9121557
>>9116459
Ohhh I read it right now. Actually! Couldn't N encode all possible photos? Take the set of every possible finite binary sequence (which is the same cardinality as N) then order it in terms of increasing length alphabetically, like so...
0,1,00,01,10,11,000,001,010,......
Is our sequence. Then we simply enumerate it, so we are able to pick an integrr and get back an arbitrary binary sequence of finite length -which could encode any photo.

>>9121664
You can just use Gödel encoding.

>>9110245
Shanks transformation for extrapolation.

https://en.wikipedia.org/wiki/Shanks_transformation

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>>9121694
True! I didn't know this particular encoding but I wanted to give a more tangible answer than "both sets are countably infinite so you can create a bijection"

>>9118953
NO
DELETE THIS

>>9121734
wow, haven't seen this before

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I love the Dedekind cut but it always feels weird when I think about what its actually doing.

>>9110245
My favorite is the axiom that we can always make a choice, because it is incongruent with humanity's lack of free will. This is undeniable proof that maths have nothing to do with reality.

>>9122123
couldn't you choose not to use that axiom? what happens then?

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Gödel's incompleteness theorem is fucking weird

>>9122129
I don't believe any maths exists where the axiom of choice is not present.

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>>9122145
is it just something more akin to breaking down the first order logic behind fundamental theorems in math and realizing that as long as we invoke them we also invoke the axiom of choice, or is it more fundamental in that if you try to not have any reliance on axioms, you are still invoking the axiom of choice?

>>9122145
there are a few takes on axiomatic set theory that don't use choice, you're free to explore them