Newton's Fractal, the finalized verison

I wonder, would it produce any interesting looking pictures if you color dots not according to the corresponding root but according to the number of steps it took for the dot to converge to its root.

Eugene Pakhomov

What I really appreciate from all of your videos is how you connect ideas in a large context. The relationship with Voronoi diagrams blew me away. Thanks for such quality content!

For the thumbnails it looks like you colorized the boundaries with gradients? Is that mostly just photoshop, or was there a math equation you ran on the results?

Mark Tea

So cool! I played with the mandelbrot set in c# and winforms when I was young, and then when I got older I did it again but with a gpu shader. It was interesting to start running into the floating point precision issues there =) Saddly I never understood as detailedly as you present it :)

Mark Tea

The way this video went on a journey through several topics (and pointing out the many possible ways the story could have gone) I think is really pedagogical in showing how everything is connected through math

There are no current plans to cover those, but you're right they might make a fruitful side-mention in the quintic video.

3blue1brown

Wow, this is a real gem and a true joy. Thanks!

Edith Dubiner

Wow, thanks for introducing me to holomorphic dynamics! I have a general question about your thoughts on approximating roots: Do you have any plans to cover the Aberth Method or the QR Algorithm? I know the Aberth Method is basically an extension of Newton's method for finding all roots simultaneously, and the QR Algorithm is an eigenvalue-finding algorithm that's repurposed as a root-finder by Matlab's `roots()` and Python's `numpy.root()`. Both methods are supported by Wolfram's `NRoots[]` function. Since they both help you find all roots at once, they're more interesting to study for me personally. Of course, from a pure "only caring about getting all the roots" perspective and not from the perspective of building beautiful fractal imagery. It seems like you still intend to make a video tackling Abel-Ruffini eventually, so mentioning these algorithms as an aside might be pretty cool.

Andrew Alvarez

This video may be finalized, but just in case it's not _final_-finalized, there is a typo at 4:06: "How do we find theses?" where I think you mean "these?"

Karen

Thanks for all your comments and direction, immensely helpful.

3blue1brown

I love the "how it started" "how it's going" reference @24:00. ;-)

Gabe

Wow, such a stunning video and a great introduction to holomorphic dynamics! And I feel like all the patron remarks from the preview were addressed :) Thanks a lot, Grant!