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Reply 120

Original post
by DerDracologe
Can i see the calculations to prove this? Let me guess… I wouldn’t understand them

I had to simplify it a bit for those who don't have a PhD but this is the basics:

Consider the generalized form of the Jacobian elliptic functions in the context of a doubly periodic lattice:

sn(u,k)2+cn(u,k)2=1anddn(u,k)2+k2⋅sn(u,k)2=1,\mathrm{sn}(u, k)^2 + \mathrm{cn}(u, k)^2 = 1 \quad \text{and} \quad \mathrm{dn}(u, k)^2 + k^2 \cdot \mathrm{sn}(u, k)^2 = 1,sn(u,k)2+cn(u,k)2=1anddn(u,k)2+k2⋅sn(u,k)2=1,
where sn(u,k)\mathrm{sn}(u, k)sn(u,k), cn(u,k)\mathrm{cn}(u, k)cn(u,k), and dn(u,k)\mathrm{dn}(u, k)dn(u,k) are solutions to the differential equation

(dsn(u,k)du)2=(1−sn(u,k)2)(1−k2sn(u,k)2).\left( \frac{d\mathrm{sn}(u, k)}{du} \right)^2 = (1 - \mathrm{sn}(u, k)^2)(1 - k^2 \mathrm{sn}(u, k)^2).(dudsn(u,k)​)2=(1−sn(u,k)2)(1−k2sn(u,k)2).

Now, combining this with the Fourier series expansion for complex exponential periodicities:

f(x)=∑n=−∞∞aneinωx,wherean=1T∫0Tf(x)e−inωx dx,f(x) = \sum_{n=-\infty}^\infty a_n e^{i n \omega x}, \quad \text{where} \quad a_n = \frac{1}{T} \int_0^T f(x) e^{-i n \omega x} \, dx,f(x)=n=−∞∑∞​an​einωx,wherean​=T1​∫0T​f(x)e−inωxdx,

we can represent non-trivial periodic solutions in terms of modular transformations τ=ω2ω1\tau = \frac{\omega_2}{\omega_1}τ=ω1​ω2​​, where ω1\omega_1ω1​ and ω2\omega_2ω2​ are the half-periods of the lattice satisfying ω1τ−ω2=2mπi\omega_1 \tau - \omega_2 = 2m\pi iω1​τ−ω2​=2mπi.

For hyperbolic analogs, consider the identity:

cosh⁡2(x)−sinh⁡2(x)=1,\cosh^2(x) - \sinh^2(x) = 1,cosh2(x)−sinh2(x)=1,

which directly corresponds to the imaginary projection of elliptic integrals:

∫0ϕdθ1−k2sin⁡2(θ)=u,where u=am−1(ϕ,k).\int_0^{\phi} \frac{d\theta}{\sqrt{1 - k^2 \sin^2(\theta)}} = u, \quad \text{where } u = \mathrm{am}^{-1}(\phi, k).∫0ϕ​1−k2sin2(θ)​dθ​=u,where u=am−1(ϕ,k).

Finally, invoking Ramanujan's Master Theorem, the qqq-series expansion of the Dedekind eta function:
η(τ)=eπiτ/12∏n=1∞(1−e2πinτ),\eta(\tau) = e^{\pi i \tau / 12} \prod_{n=1}^\infty (1 - e^{2 \pi i n \tau}),η(τ)=eπiτ/12n=1∏∞​(1−e2πinτ),

where τ=ω2ω1\tau = \frac{\omega_2}{\omega_1}τ=ω1​ω2​​ connects elliptic integrals with modular forms, providing a transcendental bridge between the realms of trigonometric and elliptic phenomena.
Original post
by iL1L
Did @Geo Lover 7 see this thread?
No.

Reply 122

Original post
by sdfj
Trigonometry said no........ so no
I have a PhD...

i dont believe in trigonometry?? its a bunch made up of crap about triangles by bored and mentally unstable people to torture students and that's it.
the earth is a pyramid and ur phd means nothing here.
Original post
by gileon2452
Is the earth flat?
No

Reply 124

Original post
by sdfj
I had to simplify it a bit for those who don't have a PhD but this is the basics:
Consider the generalized form of the Jacobian elliptic functions in the context of a doubly periodic lattice:
sn(u,k)2+cn(u,k)2=1anddn(u,k)2+k2⋅sn(u,k)2=1,\mathrm{sn}(u, k)^2 + \mathrm{cn}(u, k)^2 = 1 \quad \text{and} \quad \mathrm{dn}(u, k)^2 + k^2 \cdot \mathrm{sn}(u, k)^2 = 1,sn(u,k)2+cn(u,k)2=1anddn(u,k)2+k2⋅sn(u,k)2=1,
where sn(u,k)\mathrm{sn}(u, k)sn(u,k), cn(u,k)\mathrm{cn}(u, k)cn(u,k), and dn(u,k)\mathrm{dn}(u, k)dn(u,k) are solutions to the differential equation
(dsn(u,k)du)2=(1−sn(u,k)2)(1−k2sn(u,k)2).\left( \frac{d\mathrm{sn}(u, k)}{du} \right)^2 = (1 - \mathrm{sn}(u, k)^2)(1 - k^2 \mathrm{sn}(u, k)^2).(dudsn(u,k)​)2=(1−sn(u,k)2)(1−k2sn(u,k)2).
Now, combining this with the Fourier series expansion for complex exponential periodicities:
f(x)=∑n=−∞∞aneinωx,wherean=1T∫0Tf(x)e−inωx dx,f(x) = \sum_{n=-\infty}^\infty a_n e^{i n \omega x}, \quad \text{where} \quad a_n = \frac{1}{T} \int_0^T f(x) e^{-i n \omega x} \, dx,f(x)=n=−∞∑∞​an​einωx,wherean​=T1​∫0T​f(x)e−inωxdx,
we can represent non-trivial periodic solutions in terms of modular transformations τ=ω2ω1\tau = \frac{\omega_2}{\omega_1}τ=ω1​ω2​​, where ω1\omega_1ω1​ and ω2\omega_2ω2​ are the half-periods of the lattice satisfying ω1τ−ω2=2mπi\omega_1 \tau - \omega_2 = 2m\pi iω1​τ−ω2​=2mπi.
For hyperbolic analogs, consider the identity:
cosh⁡2(x)−sinh⁡2(x)=1,\cosh^2(x) - \sinh^2(x) = 1,cosh2(x)−sinh2(x)=1,
which directly corresponds to the imaginary projection of elliptic integrals:
∫0ϕdθ1−k2sin⁡2(θ)=u,where u=am−1(ϕ,k).\int_0^{\phi} \frac{d\theta}{\sqrt{1 - k^2 \sin^2(\theta)}} = u, \quad \text{where } u = \mathrm{am}^{-1}(\phi, k).∫0ϕ​1−k2sin2(θ)​dθ​=u,where u=am−1(ϕ,k).
Finally, invoking Ramanujan's Master Theorem, the qqq-series expansion of the Dedekind eta function:
η(τ)=eπiτ/12∏n=1∞(1−e2πinτ),\eta(\tau) = e^{\pi i \tau / 12} \prod_{n=1}^\infty (1 - e^{2 \pi i n \tau}),η(τ)=eπiτ/12n=1∏∞​(1−e2πinτ),
where τ=ω2ω1\tau = \frac{\omega_2}{\omega_1}τ=ω1​ω2​​ connects elliptic integrals with modular forms, providing a transcendental bridge between the realms of trigonometric and elliptic phenomena.

oh how mesmerizing..

Reply 125

Original post
by pretty_in_purple
oh how mesmerizing..

Well at least now you know that the earth cannot be a pyramid!

Reply 126

Original post
by sdfj
Well at least now you know that the earth cannot be a pyramid!

it can 100% be a pyramid. don't try to apply your made up concepts like 'math'(whatever tf that is) here.

Reply 127

Educate yourselves y'all @sdfj

Reply 128

Original post
by pretty_in_purple
it can 100% be a pyramid. don't try to apply your made up concepts like 'math'(whatever tf that is) here.

Has it been robbed and put in the british museum? If not, its not a pyramid. QED.

Reply 129

Original post
by pretty_in_purple
it can 100% be a pyramid. don't try to apply your made up concepts like 'math'(whatever tf that is) here.

Made up????? Here's the proof that the earth is flat - GCSE trigonometry...


Begin by considering the Earth's surface as a 2D Euclidean plane in which distances d between two points (x1,y1) and (x2, y2) are calculated using the standard Euclidean metric:

d= sqrt((x2−x1)2 +(y2−y1)2)

The sum of the angles (where A, B and C are each points on the earth):

α+β+γ=180∘

Trigonometric identities for the sine rule, valid only in flat geometry, are then applied:

alpha/sin(a) = beta/sin(b) = gamma/sin(c)



Using the formula for the curvature: K=1/L where K is the curvature of the earth and L is the longest distance between two points on the earth.

As L →∞, K→0

If K ~ 0, the curvature of the Earth is approximately equal to zero - therefore there are small bumps and hills on the earth as K is not equal to exactly zero but it doesn't curve! Therefore, the earth is flat!

@DerDracologe The flat earth proof is much more easy to understand...
(edited 9 months ago)

Reply 130

Original post
by pretty_in_purple
Educate yourselves y'all @sdfj

Videos can't beat maths :lolwut:

Reply 131

Original post
by sdfj
Videos can't beat maths :lolwut:


Where are you getting these from 😭

Reply 132

If earth is flat then whats underneath it? There must be a surface underneath the plane which we cant get too

Reply 133

Original post
by sdfj
Made up????? Here's the proof that the earth is flat - GCSE trigonometry...
Begin by considering the Earth's surface as a 2D Euclidean plane in which distances d between two points (x1,y1) and (x2, y2) are calculated using the standard Euclidean metric:
d= sqrt((x2−x1)2 +(y2−y1)2)
The sum of the angles (where A, B and C are each points on the earth):
α+β+γ=180∘
Trigonometric identities for the sine rule, valid only in flat geometry, are then applied:
alpha/sin(a) = beta/sin(b) = gamma/sin(c)
Using the formula for the curvature: K=1/L where K is the curvature of the earth and L is the longest distance between two points on the earth.
As L →∞, K→0
If K ~ 0, the curvature of the Earth is approximately equal to zero - therefore there are small bumps and hills on the earth as K is not equal to exactly zero but it doesn't curve! Therefore, the earth is flat!
@DerDracologe The flat earth proof is much more easy to understand...

all i read was 'bleh bleh bleh bleh bleh bleh'

ur wasting ur time buddy

Reply 134

Original post
by iL1L
Where are you getting these from 😭

I have a PhD (and a brain) :wink:

Reply 135

Original post
by sdfj
Videos can't beat maths :lolwut:

sure, wtv you say. you dont have a counter now bc my video made a point and is actually some proof unlike your proof which is a bunch of numbers and letters jumbled up together in the name of 'trigonometry'

Reply 136

Original post
by Anonymous
I have a PhD (and a brain) :wink:


Did you mean to reply anonymously?

Reply 137

Original post
by iL1L
Did you mean to reply anonymously?


Accident :hide:

I guess people with PhDs aren’t quite perfect (almost, but not quite… :colone: )

Reply 138

Original post
by sdfj
Accident :hide:
I guess people with PhDs aren’t quite perfect (almost, but not quite… :colone: )


:lol:

Reply 139

Shall we call you sDRfj now?

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