The secret π in the Mandelbrot Set

The mandelbrot set is probably the single most iconic picture in all of math. Yet, somehow, someway, there's always something about this fractal that I find myself scratching my head about. Today, let's look at one of those things :)

The source code for the animations can be found here:

Huge thanks to @alfcnz, @Bean_Piano for reviewing and helping me with the video!

A portion of this video was sponsored by Wren.

Music (In order)
Jujutsu Kaisen (but its okay if its lofi?)

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The unexpected pi hidden in the Mandelbrot Set
Some tags: vcubingx, v cubingx, vcubing x, v cubing x, mandelbrot set, pi, fractal,

[vcubingx]

Haven't been in a great headspace lately, sorry this took so long! If you enjoyed it, let me know and please consider subscribing :D

EDIT: I got a couple comments asking why I'm not entering it into SoME2, and my reasons are I don't feel like this is the best video I could've made in the 2-3 month period it was out, and I personally feel like it would be disingenuous and against the values of the contest for me to enter a sponsored video. And lastly, the contest is about publicizing lesser known creators, and I wouldn't want to take the spotlight from someone who could benefit a lot more from it than me.

[marcosgutman]

lets be honest its not even unexpected at this point, i could find my social security and credit card numbers in the mandelbrot set and it would just be like "huh"

[Aleph]

beautiful video vivek! the connection to pi was indeed mind-blowing :)

[AcamaroCutcher]

I just wanted to say that the quality of your videos has greatly improved since I last saw your channel. This is a great step up.

[onebronx]

The y=x becoming a tangent/secant line and limiting the process at the touch/intersection point was a nice geometrical reveal to me. So, I guess, for the full complex plane, we should consider a w=z hyperplane in a 4-dimentional complex space C² (with coordinates w and z analogous to y and x in R²), which limits the process when it becomes a tangent/secant to the hyperparabola w = z²+c, and boundaries of the Mandelbrot set are just values of the "offset" parameter "c" when it shifts the parabola so that w=z plane becomes exactly a tangential plane, right?

[georgenabraham]

I really appreciate that you put the source code of the animations.
You really helped me so much learn manim, thanks ❤️

[mathyland]

Definitely reminds me of 3blue1brown’s videos on block collisions counting digits of pi. This seems a little different cause the digits aren’t exact, and while tan(x) appears, the answer isn’t based on tan x being approximately x for small x like the block collision solution is.

Still I wonder if there is a connection.

[XxDoubleshotxX]

GOOD WORK! love to see the video learned a lot!!!

[Number_Cruncher]

It's nice to find inspirations that go a bit deeper than the math thought in school. I liked the way the ode appeared out of the blue.

[vinaymishra]

Fantastic, love to see your videos. I request you to please make one video also on fixed point techniques.

[TheHenrykH]

Route 113 music was an exquisite choice sir

[RickyMud]

I couldn’t of guessed it had to do with the poles of tan that’s so cool

[merijnbras]

Exactly this is what makes math beautiful

[Gust]

Hello! Just asking a doubt based on the installation of Manim : Can we also do it with the Anaconda Distribution of Python? Thanks!

[ZhanYiGe]

how the escape number of iteration N comes to the tangent.function? I just wonder.that when n approach N, the diffrential function is not smooth!

[oriontechtube]

your videos are extermly important. could you tel me how you add such beutiful animations you created

[johanngambolputty]

I don't know about opengl, but I've been playing around with opencl and I seem to be able to get a 1440p frame of an image orbit trap to render fast enough for around 24fps real time on a 6700xt gpu. I only have a basic pyqt script that renders that onto a qt label right now, but its nice to be able to play with image trap boundaries and see an image get warped around the mandelbrot set :) I wanted to also use gl (but not by itself) but it seems cl gl interop requires them to be compiled together :(

edit: now that I'm thinking about this again, it was pyopencl that needed to be built with interoperability, my actual ocl install did have khr_gl_sharing

[samwebber]

id be curious to learn how a circle plays into that appearance of pi

[echecheese]

bro the mandelbrot set kinda packing ngl. u know their number?

[alexbarnett]

can you explain why at 9:46 the distance between the two poles is pi/sqrt(epsilon), i found it as 2pi/sqrt(epsilon) because the asymptotes occur at pi radians, so nsqrt(epsilon) = pi, which implies n = pi/sqrt(epsilon). Then the distance between two poles is 2n, as the asymptotes are at n and -n? am i missing something?