3D Visualization and Animation of Euler's Formula using Python and Manim

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This video explores Euler's formula from a 3D visualization and animation using Python and Manim.
When visualized in 3D, Euler's formula can take on a variety of shapes and patterns depending on the angle of observation. When viewed from the side, the 3D visualization of Euler's Formula resembles a sinusoidal wave, with the imaginary component of the function representing the height of the wave.
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this has been on my mind for a while im glad somebody actually made it

JEE-oqme
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Gracias, muchísimas gracias. Que Dios los bendiga.

juancarlosvillasmilmolero
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This is great! though I would like to see, sometimes, theta(Or omega) being the reference axis (ofc Implicitly wrt time) while projecting the new axis, As it would help in understanding related topics such as convolution, Heat flow etc.

ravikiran
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My mind is tired these days from thinking about other problems—I haven't been able to solve this following one, but let's see what you have to say:
Let us solve for zᵖ = 1 for numerous natural numbers of p.
That zᵖ = 1 is like (1eⁱᶿ)ᵖ = 1.
If p = 1, by looking we know z = 1, but this means (1eⁱᶿ) = 1 and hence θ must be 0°, 360°...
(This is known by asking, ‘what value must θ be so that the clock (in your above video) points at 3.00 o'clock?’)
Now when we say z³ = 1 this means (1eⁱᶿ)³ = 1. If we take 120°, then 3×120° = 360° in other words 0°. It's as if it divides the circle into equal bits!
We can also shade along the helix different colours for each interval of θ that leads to 360°.
The practical application of zᵖ = 1 is we can form regular polygons by just setting ‘p’ to the number of sides we want. That, if we project onto the 2D Re/Im plane. But by considering the 3D extrusion, the answer is to divide the place from the 2D Re/Im plane's base to the extrusion reached at 360°, divide the sin/cos wave into 3.

Now.. what if we extend this system into fractional powers of z?
A naïve student thinks the number 1eⁱ⅔π is also re-written as 1eⁱ8/3π (i.e after adding 2π). In other words 1eⁱ¹²⁰, since both are projected onto the same complex number on the 2D Re/Im plane. When we cube both we get 1 (1eⁱ2π, 1eⁱ8π), but when I raise both to the power of 1.5, the second gives 1 (1eⁱ4π) but the first -1 (1eⁱπ).
*This is why your video matters.* A complex number necessitates the third dimension.
But then, how do we go about solving these fractional powers?
It seems when I have z³ = 1, I divide the wave/helix from the base to at 360° into 3, but when I have z¹˙⁵ = 1, I have to divide the length from the base to _720°_ into 3. Therefore the intervals become _longer!_ What the heck is going on and what is your take on this?

Also do you have a GitHub? Useful to post the code for this video there!

oosmanbeekawoo
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Your animation is beatiful. How can i get or learn the manim code ? I’m beginner en manim and I’ve tried several Ways and doesn’t work

andhreyuxmunoz
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Hi. Good job! Where can I download manim code?

Reincornator
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Can somebody explain why the imaginary unit is depicted equal in length as the real unit? Does it mean that i=sqrt(-1)=1?

pelasgeuspelasgeus
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Dr. Shinoda what program do you use to show this animated equation?

oscarortiz
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Cos theata grace horizontal and sine theta perpendicular spinning along imaginary plane contrary to what you are showing.

nandakumarcheiro
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Actually this is just trigonometry, although it is correct geometric visualization of complex numbers, it does not show e^(itheta) = 1. Also you should just write theta instead of t.

anywallsocket
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Emmm there is no Euler's formula in this visualization... it's just the definition of complex numbers...

yqisq