Holomorphic Dynamics (early view)

I am having trouble reproducing your results. If I terminate the Newton solve after a large number of iterations (same approach to generating the Mandelbrot set), I can recreate your result. However, computing f'(z) = P(z)P''(z)/(P'(z))^2 always results in a zero when z = lambda/3. I get lambda/3 because z1 = 1 and z2 = -1 so (z1 + z2 + z3)/3 is z3/3.

You can now play with that side-by-side on the website: https://www.3blue1brown.com/lessons/holomorphic-dynamics To your first point, I may do a follow-up to address that. To be honest, it's something I'm only recently familiar with as of this month, but the relevant keyword search is "renormalization", and it's similar in spirit to why the Mandelbrot set has smaller copies of itself as you zoom in. To your second point, you are right that was insensitive, my apologies.

3blue1brown

Mind blowing! So many wonderful links. And I've wanted to see that side-by-side view of the Julia sets that arise from different points on the Mandelbrot set for a long time. Thank you Grant for another signature mix of beauty and depth! Just two thoughts for the final edit. Firstly I'd love to get any more insight into why it's a Mandelbrot of all things that shows up in that lambda fractal (especially as the title sets that up as the main teaser). And for my taste, poor Gaston Julia doesn't deserve to be ranked within the subset of mathematicians with similar unfortunate injuries...

Steven Siddals

-i is mis-labeled as "1i" on many (not all) of the axes.

Oh and seeing the familiar Mandelbrot beetle show up in such an unexpected place... I'm not easily spooked but that was almost a religious experience. Also obviously reinforces point 2 above. I'm just astounded that it shows up in such form.

PeetieGonzalez

2 major epiphanies from me upon seeing these videos. 1) The boundary is fractal in nature because there is no part of the boundary that isn't a boundary between ALL the roots of the polynomial. Already mind=blown. 2) The Mandelbrot set, if we think of it as the BOUNDARY between the area with finite cycles and the infinity trajectory, is the boundary between infinity and ALL number of finite cycles. And hence that boundary is also fractal in nature. This is the most informative and mindblowingly clear video on maths I have ever seen. I'm going to have to relegate your video on Euler's Identity (edit... actually the fourier transform, by extention) to my #3 spot (or #2 if I count this as one topic in two parts). So glad you're back.

PeetieGonzalez

Watching this video, and the last one, I’m reminded of your video on fractal dimension where you declared that fractals are about roughness, not self-similarity. This set of videos, I think, is a really good way to explain that statement. I think that message could be made stronger in this video.

Sachin Shukla

That was a very interesting video, I reallly enjoyed it. There were two bits though that felt a little less well defined than usual when watching your finished videos. At 21:27 where you say "it (the madelbrot set) seems it has a more general relationship to parameter spaces" I would love to know more about what that relationship is (is it known, and just too complicated to explore in the video, or we genuinunly don't know why?). I would also be very curious to know how it change things in your function to paramter space example if the other two roots are elsewhere than 1 and -1. Again really awesome video. I love the way you explain things with visualisations, if that kind of thing had been around when I was in uni, I would probably still be in pure math (and not computer science ), as it is you are really drawing me back to my roots :-).

James Matheson

same

Glad to have you on board, thanks!

3blue1brown

You've just played with my math addiction and made me a Patreon T_T No, but more seriously, you're a true example to me, I often have tears while watching what you do.

very nice and interesting - I like these two videos a lot! Maybe stress a bit more that the (black) Mandelbrot set appears only in the "left" fixed seed case (and not the fixed function).

Edith Dubiner

It is mind boggling to realize where Mandelbrot Sets are hiding! At 16:34 - the hint regarding how to calculate the derivative disappears a bit too fast [at least for non-native english readers] to read.

Uri Agassi

15:16:  Ooooo . . . you really kicked a hornet's nest THERE.  Doubling down, trolling for trolls, daring the teeming hordes of lightly-informed armchair philosophers to reflexively howl their pontifications about the improbability of the "probability being zero."  Gutsy move Mav.  Is your asbestos armor at hand?  Prepare for an onslaught of indignant rage:  Just sayin'.

Don Sanderson

Nice Video! At 4:09 the z^2 + c / 1 disappears and then reappears a couple seconds later.

You're not the first to suggest renormalization as a new video :) That could definitely be a fun one. After all, forget the Newton maps, just explaining why the Mandelbrot set itself has tiny (and deformed) mandelbrot sets inside it would be fun.

3blue1brown

Thanks for the notes, good catches! Not sure how the calculus chapter was misnamed, but that's fixed now. For the z^2 + c example, my personal taste is that it's easier to see that f'(0) = 0 by display f'(z) = 2z, rather than showing the roots, but I suppose to each their own. To 24:22, it's not always the point at infinity, but one easy case is that where f(z) is a polynomial, one of those exceptional points definitely will be the point at infinity.

3blue1brown

It was actually one of the Patrons who pointed me to this Mandelbrot connection, which I had not known anything about. Benefits of the early releases!

3blue1brown

Good catch, thanks!

3blue1brown

You're correct, this video only explains the case with 3+ stable attractors. And in fact, the case with 2 attractors can have a non-fractal boundary (e.g. Newton's method on quadratic, or f(z) = z^2). What's the right fix here, being extra clear about what is and isn't explained?

3blue1brown

I think it's this paper but can't find an english translation: http://www.numdam.org/item/BSMF_1919__47__161_0/

Awesome stuff, could you include a reference to Fatou's 1919 Theorem at 18:24 in the description?

Small suggestion: When you explain how the derivative is locally a linear transformation around 11:00 - 11:30, you might include a local coordinate system so that the rotation aspect is visible.

Rion Boom Crabhands Keon

Grant these two videos on fractals are amazing! I did my maths master thesis on holomorphic dynamics. I did it because I had questions about the mandelbrot set that popular expositions did not answer. Your video now answers so many of them in a clear and compelling way, without hiding the complexity away! I am amazed that you managed to digest the material to such an extent for the benefit of posterity. Kudos! I only wish you'd spend a little more time on the mandelbrot-like shapes in the parameter space for cubic polynomials, because I wished you'd answer some of the questions I didn't get to in my master's, like: - in the z^2+c mandelbrot set, the main cardiod represents attracting points, but in the cubic case, it represents cycles (i guess 2-cycles). What to the other parts of the set represent? - why are the mandelbrot sets in the cubic parameter space a bit "deformed"? what does it mean? - are there more mandelbrot sets around the same parameter space? how are they related? These are complex questions of course... related to renormalisation. Perhaps a topic for a new video? ;)

I loved this video!! I have a couple of nits I'd like to address, and it looks like these haven't been pointed out by anyone else yet: - 3:59 visual glitch where the size of the yellow box instantly changes - 4:22 the transition of the expression towards the left is a little choppy here - 10:00-11:55 it would probably be a good idea to connect the lessons here to the video you did a while back about stability (The other way to visualize derivatives) **NOTE: I just saw that you added the above-mentioned video to the "Essense of Calculus" series as Chapter 12, but in the title you call it "Chapter 2"! PLEASE FIX THIS!!** - 13:40 I think the word "minus" is confusing here - it sounds like you could be saying that it's set of solutions to [f(f(x)) - f(x)], though I know that's not your intention. Maybe you should either say "that are not" instead of "minus" here, or display the statement you're trying to make in set notation. - 18:50 "That one seed value is definitely going to find it". You're implying that 0 is always the centroid of the roots of the function z^2 + c, which is true, but some viewers may not know why so this might confuse them. You could probably also display the roots of z^2 + c, perhaps in red, alongside the yellow sequence points to mostly fix this. Or you could make proving this a mini-exercise, but you yourself mentioned feeling like you may be assigning too much homework here ;'). - 24:22 I'm just curious, what are the possible exceptions? Are they always randomly different depending on the Julia set, or are there particular points that are always of interest like the "point" at infinity or something?

Andrew Alvarez

At 4:09 the "z^2+c" switches back to the general expression for a couple of seconds. Not sure if that was intentional.

Hey Grant, I'm loving this mini-series! Around 3:40-3:50, the language of iterating rational functions is very central during the time that the boxed equations seem to be changing through various examples of rational functions. This is a little confusing in the sense that a verb you're using and the visual event on the screen are somewhat more incongruous than is usually the case. Thank you for this joyful visual experience of abstract mathematics! Daniel

7:12 OMG these visualizations are ridiculous (in the good way)

C.J. Smith

Some time ago I tried to visualize the behavior of holomorphic dynamics. Could be interesting for some of you: https://youtu.be/DzQHUkRGfFc

Daniel G

Thanks for another great video, Grant. I don't have near the math skills to critique our help out, so just thanks for great content.

RAD Donato

Beautiful as always, and very glad I was able to watch the followup immediately. I'm curious just how recent your discovery of all this material was. I know there was a bunch of points from the patreon comments on the Newton's Method video asking about Mandelbrot connections. Did you know how precise those connections were at the time people were asking that? From my end, it was immensely satisfying to have seen those questions along with the first video, and then to have this come along so soon after with such a gorgeous explanation.

Eric Severson

There is a typo at 16:27. Numerator should have z^3-2z+__2__. Love all the exercises btw. Great video!

Man that sent me to a very deep rabbit hole. Amazing job as usual! Minor error, at 16:30, exercise 2 has the numerator as z^3 -2z + z where as it should be z^3 -2z + 2

Interesting fact about Julia sets that (if I remember correctly) isn't mentioned: the Mandelbrot set is a map of all Julia sets, where points inside the Mandelbrot set correspond to Julia sets whose "interior" (black region) is a solid, continuous region (or at least composed of some), whereas points outside corresponds to Julia sets whose interior is a cantor dust (totally unconnected set).

Lionel Pöffel

Very nice. One point: the video *mentions* that the principle "every point is either not on a boundary at all or on a boundary between all colors / asymptotic behaviors" does not explain Julia sets being fractal and not smooth. But then it goes into this different direction, showing that every disk around any point in the Julia set grows arbitrarily. No reason for the fractalness is actually given. Is the complicated Montel's theorem needed to explain this as well? Or would there be a simpler explanation.

Lionel Pöffel

Could you put the video and article you reference in the description?

Christian Leichsenring

4:08, where you talk about z^2+c, your graphic messes up here.

Excellent, thanks for the catch.

3blue1brown

That's definitely very doable, and a good suggestion. In fact, in past videos I've interpolated between the identity and z -> z^2 with z -> z^(1 + alpha), which I can throw in here.

3blue1brown

Thanks for the fine comb!

3blue1brown

11:15 Grant: "It *looks* like multiplication by 2i" Me: "It *looks* like a brilliant demonstration about how to further improve manem's interpolation: what if instead of linearly moving every point to its destination, one moves the points along a path that also interpolates the derivative? In this case, that would include the points not shrinking prior to expanding in the zoomed in view, but just inexorably expanding. :)

Jesse Thompson

Fantastic, as always. Thanks. I think I found three minor typos: - At 3:20, there is a superfluous plus sign before the 2z^3 (rightmost expression). - At 12:54, exercise 1c, the "c" in "values c satisfying" should be typeset as $c$. - At 21:39, "contains 1 colors" -> "contains 1 color"

Thanks for posting! Watching now. I noticed one graphical glitch at about 4:09, where the rational function expression flickers between the general formulation and z^2+c. Edit: there's also a single frame glitch around 25:54.