Anonymous at Sun, 27 Oct 2024 15:41:52 UTC No. 16452299

>>16452279
Pi is one quarter of the surface area of the unit sphere

Anonymous at Sun, 27 Oct 2024 15:55:55 UTC No. 16452317

>>16452279
pi is the square of the area under the Gaussian

Anonymous at Sun, 27 Oct 2024 16:11:52 UTC No. 16452337

>>16452299
3d
>>16452317
2d

Anonymous at Sun, 27 Oct 2024 16:15:53 UTC No. 16452342

>>16452337
How the fuck is a Gaussian 2D? It's a single-variable function.

Anonymous at Sun, 27 Oct 2024 16:25:58 UTC No. 16452360

>>16452279
All identities involving phi are like this desu. You get phi from phi.

Anonymous at Sun, 27 Oct 2024 16:27:46 UTC No. 16452364

>>16452360
You can get any number or constant from anything. Pi phi aren't special

Anonymous at Sun, 27 Oct 2024 16:30:18 UTC No. 16452368

>>16452299
interesting. and checked

I have yet to consider surface area in my studies,
much obliged anon

Anonymous at Sun, 27 Oct 2024 16:35:50 UTC No. 16452376

>>16452360
fascinating, any other examples worth mentioning? i'm still spellbound by this result

Anonymous at Sun, 27 Oct 2024 22:36:12 UTC No. 16452891

>>16452368
Can someone tell me the name of this book? Thanks

Anonymous at Sun, 27 Oct 2024 23:16:09 UTC No. 16452933

the universe is flat

Anonymous at Mon, 28 Oct 2024 00:27:10 UTC No. 16452989

>>16452891
Yessir
https://archive.org/details/TheRhindPapyrusVolume1

Anonymous at Mon, 28 Oct 2024 00:31:29 UTC No. 16452994

>>16452933
Perhaps, but is it spinning
Does it have a precession?

Anonymous at Mon, 28 Oct 2024 01:24:24 UTC No. 16453051

>>16452342
(x,f(x))

Anonymous at Mon, 28 Oct 2024 01:25:26 UTC No. 16453054

>>16452279
No shit. Pi emerges from geometry and is an irrational number, so not obtained by a mere sequence of one dimentional arithmetic operations

Anonymous at Mon, 28 Oct 2024 01:30:39 UTC No. 16453061

>>16452376
If x is a solution of x^2+ax+b=0 then

sum((cx)^n,0,oo) = mx^2

where

c=a/(b-a^2)
m=-1/(b+abc)

For x=phi a=b=-1 so c=1/2 and m=2. Find other a,b that make c,m come out to "nice" numbers to get more identities like this.

Anonymous at Mon, 28 Oct 2024 11:48:15 UTC No. 16453435

>>16452376
These basic ones always seemed rather amazing to me, despite how simple they are

Anonymous at Mon, 28 Oct 2024 13:41:55 UTC No. 16453511

>>16453435
indeed, they are interesting to me as well.

though, i'd never expect to see such a series converge back to the square.

"Measure of the land for lower egypt"

"The regal measure (lower egypt)"

Anonymous at Mon, 28 Oct 2024 13:47:48 UTC No. 16453517

>>16453435
They're literally the definition...

Anonymous at Mon, 28 Oct 2024 14:25:07 UTC No. 16453553

>>16452368
Would making cuts thru a 3d sphere increase it's surface area? Like a sphere with tunnels in it? Or maybe a hollow sphere? Houses twice as many ppl in a hollow earth surface dweller but that's impossible becuz the center of gravity would be in a crust with no mass not like a solid planet would have it but is the gravity from spinning? Or just mass?

Anonymous at Mon, 28 Oct 2024 14:28:55 UTC No. 16453560

Honestly if they're never going to tell us in my lifetime I really don't even give a f***

E.t. fone home

Anonymous at Mon, 28 Oct 2024 15:24:49 UTC No. 16453612

>>16453553
Hmm interesting perspectives
https://en.m.wikipedia.org/wiki/Sphere_eversion

And holography

I suppose the donut needs an inner star to serve as the center of mass.

The connecting of dimensions seems a common theme with the cubit, these are natural extensions, thanks

Anonymous at Mon, 28 Oct 2024 15:40:02 UTC No. 16453623

>>16452279
I don't get it.
Isn't it just
>1/(1-x/2) =2x^2
>1=2x^2-x^3
>x^3-2x^2+1=x^2(x-1)-(x-1)(x+1)=(x-1)(x^2-x-1)
?

Anonymous at Mon, 28 Oct 2024 17:13:13 UTC No. 16453745

>>16453623
if the equivalence is obvious, you must be a greater mathematician than I

https://www.wolframalpha.com/input?i2d=true&i=1%3D+2Power%5Bx%2C2%5D+-+Power%5Bx%2C3%5D

is a beautiful cubic equation, though i don't see how it is equivalent to

https://www.wolframalpha.com/input?i=%28sum+0..inf+goldenratio%5En%2F2%5En%29%2F10+%3D+goldenratio%5E2+%2F5


img src

https://arxiv.org/pdf/1009.1711

Anonymous at Mon, 28 Oct 2024 17:34:39 UTC No. 16453775

>>16452279
>>16452279
phi=1

Anonymous at Mon, 28 Oct 2024 17:40:06 UTC No. 16453780

Anonymous at Mon, 28 Oct 2024 17:42:23 UTC No. 16453783

>>16453775
https://www.wolframalpha.com/input?i=%28sum+0..inf+1%5En%2F2%5En%29%2F10+%3D+1%5E2%2F5

>true

the plot thickens, thank you anon,
may no transit police prevent the enjoyment of your bratwursts

Anonymous at Mon, 28 Oct 2024 23:43:33 UTC No. 16454158

>>16452279
Place a coin on its edge and spin it and take a long exposure picture

Anonymous at Mon, 28 Oct 2024 23:49:57 UTC No. 16454167

>>16454158
The 2d, spinning quickly, creates a 3d image

Coupd one call this a form of holography?

Anonymous at Tue, 29 Oct 2024 17:33:21 UTC No. 16454960

>>16453783
https://www.wolframalpha.com/input?i=%28sum+n%3D0..inf+x%5En+%2F+2%5En+%29%2F10%3D+x%5E2+%2F+5+

only 3 solutions
- 1/phi, 1 , phi

Anonymous at Tue, 29 Oct 2024 18:41:27 UTC No. 16455056

>>16452279
You misspelled
>in flat spacetimes

Anonymous at Tue, 29 Oct 2024 18:44:46 UTC No. 16455064

>>16453435
>x + 1 has the same decimal part as x - 1
wow such magic
very cool

Anonymous at Tue, 29 Oct 2024 18:47:59 UTC No. 16455071

>>16454960
There's also phi = 0
>inb4 but muh 0^0 = 1
no

Anonymous at Tue, 29 Oct 2024 19:26:42 UTC No. 16455115

>>16455056
thats just it anon,
how I arrived at this equation..

Anonymous at Tue, 29 Oct 2024 19:28:09 UTC No. 16455118

>>16455115
>Some symbol equals a ratio equals something I'll never explain
Oof, schizo post.

Anonymous at Tue, 29 Oct 2024 19:28:57 UTC No. 16455119

>>16455071
https://www.wolframalpha.com/input?i=sum+n%3D0..inf+0%5En%2F2%5En
hmm wolfram raising an issue in the infinite series

Anonymous at Tue, 29 Oct 2024 19:35:34 UTC No. 16455130

>>16455118
apologies, I thought we were being witty. as I never added the "in flat spacetimes" to the phrase "pi only belongs in 2D"

warning, it may be more schizo now, (as it rests on playing with a mathematical coincidence that the volume of 3-sphere 1 has a radius close to the 1/phi . and the blackbody radiation curve looks like the volume of n-ball per dimension
https://en.wikipedia.org/wiki/Volume_of_an_n-ball


I find it quite intriguing it stemmed from following a simple mathematical coincidence in pic related, in the amusement of a mathematician, i would assert this https://www.wolframalpha.com/input?i=radius+of+ball+of+volume+1 indeed have a radius exactly equivalent to [math]\phi^{-1}[/math] and let [math]\pi[/math] float.

The intensity of the procedure deepened and lo and behold does a n-dimensional series emerge. Lets consider the 2-d case.

[math]A = \pi r^2[/math]
[math]1^2 = \pi r^2[/math]
[math] \sqrt{\frac{1}{\pi}}= r[/math]
now, setting r to [math]\frac{1}{\phi}[/math] we infer pi from this mathematical coincidence
[math] \sqrt{\frac{1}{\pi_{new}}}= \frac{1}{\phi}[/math]
we find that, this new pi for 2d is
[math]pi_{new} = \phi^2[/math]

now,
[math]V_n = \phi^n r^n[/math]. Its rather cathartic, we see in the n-series rather arcane coefficients, for example why [math]\frac{4 \pi}{3}[/math] in [math]V_{3-sphere}=\frac{4 \pi}{3}r^3[/math] or [math]\frac{\pi^2}{2}[/math] in R^4 . Performing this trick with of setting [math]\pi_{n}[/math] with the unity volume and the inverse golden radius, cancels them all out, for example [math]\pi_4=\frac{\phi^2}{2}[/math] and [math]V_{3-sphere}=\frac{3}{4 \pi}[/math]
which infers perhaps a unique perspective on what the n-sphere series perhaps aught to be, or how to interpret volumes of 1d and 0d objects.


[math]\odot = \frac{V_{n-sphere}}{V_{n-cube}} = \sum_{n=0}^{\infty} \frac{\phi^n}{2^n}[/math] which goes right back to [math]\frac{\phi^2}{5}[/math]

Anonymous at Tue, 29 Oct 2024 19:43:49 UTC No. 16455138

>>16455119
Well that's wrong though. It's a series of all zeroes, of course it converges to zero.
>>16455130
>blackbody radiation curve looks like the volume of n-ball per dimension
That's no coincidence.

Anonymous at Tue, 29 Oct 2024 19:49:25 UTC No. 16455143

>>16455138
so many computation edge cases with the nothingness

>That's no coincidence.
the insinuation to me is rather striking, could it be that the the egyptian interpretation of the ultraviolet catastrophe was that radiation would poor into these unseen spatial dimensia?

Anonymous at Tue, 29 Oct 2024 20:13:44 UTC No. 16455169

>>16453612
Whats the video?

Anonymous at Tue, 29 Oct 2024 20:46:56 UTC No. 16455207

>>16455169
the solar lens pyramid movie
https://youtu.be/KMAtkjy_YK4?t=3024

Anonymous at Wed, 30 Oct 2024 02:19:57 UTC No. 16455494

>>16455130
https://oeis.org/A229780

Anonymous at Wed, 30 Oct 2024 09:56:45 UTC No. 16455722

>>16455494
Nice, can we add imhotep 2500bc to this list with this formula?