Filled Julia Set

Continues from here: • The Mandelbrot Set - N...
Featuring Dr Holly Krieger from MIT.
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Videos by Brady Haran
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[@n124lp]

As so many have observed, both the mathematics and the pictures are amazing. However, perhaps the most amazing fact about Julia sets, is that when Gaston Julia and others started exploring them, there were no computers around to make the beautiful pictures. They had to see the beauty of the mathematics in their heads.

[@culwin]

Math is great when you don't have to do any homework.

[@feralfoods]

more of Dr Krieger -- she has a true gift for explaining difficult mathematical concepts in such a simple and understandable manner.

[@skebess]

Personally, I can't concentrate, she's too beautiful.

[@skebess]

***** Good for you ;-)

[@TheOnlyMeta]

These videos are really interesting, and much more advanced than usual for numberphile. Personally, I'd like to see more of this style, and definitely more of Dr Krieger!

[@okphysics562]

Yes, more videos with Maths please! : )

[@danitimnev2102]

Green eyes, green dress, green marker, yay!

[@fabienmarsias9530]

It's kinda mandatory for redhead women to wear green clothes..

[@skebess]

I know, she's amazing. I'm a fan.

[@spaminbox]

watching these videos with complex mathematics is so far outside my knowledge base and daily experience when dr krieger is explaining, i am completely, completely transfixed. i watch repeatedly as if asking a magician to show me one more time. the mental stimuli is so invigorating, and so nourishing it's difficult to express my gratitude.. thank you dr krieger. thank you brady. in combination with your other channels (to use your language) my mind blows up.

[@unvergebeneid]

"If you haven't seen that, I suggest watching it before this one." Also, if you don't have a perfect memory for these kinds of things, you might want to watch it _again_ because of how much time passed before putting this one up.

[@Maarten_Bondzio]

And to see more of Dr. Krieger of course ;)

[@mpalin11]

She is so smart, graceful and beautiful! I could watch this forever ....

[@maxgoof8605]

I have yet to hear anyone describe the Mandelbrot/Julia set as a 4 dimensional object, where two of the dimensions represent the real and imaginary parts of C and two of them represent the real and imaginary parts of Z.

[@thebookofjoy]

I recall being lost in my Advanced Maths classes in Uni. The different in the quality of MIT professors is apparent.

[@andreaswendler368]

In response to this video, I made a short video showing different Julia Fractals morphing with the position of the complex number in the mandelbrot set: Julia Fractals Morphing (I know that the quality isn't the best, I wanted to get this done quick)

[@gasser5001]

yay for Dr Holly Krieger!

[@chopinmin]

What is a mathematical reason to the fact that I can see Julia sets inside of the mandelbrot set and the mandelbrot inside of Julia sets?

[@davidandrews1730]

Carry on! Help us learn. I was an advanced honors mathematics student in an american high school....I became a master plumber. As I grow long in tooth, I wish I had had teachers half as good as you. Later in life I got a Master Degree in Education specializing in K-6., the actual teaching experience was so impoverished that, I restart my company. I feel that the best people should spend part of their time with the youngest and most plastic minds.

[@honeypot11]

thank you SO MUCH for this

[@jamescook5617]

fantastic video, I'll be showing this to my complex analysis class tomorrow.

[@Thinkchronous]

Great Video. Really helps my thoughts along how to create local coordinate systems for fractals that navigate the structure.

[@hamburg1888]

more videos with Dr Holly Krieger pls

[@braxtonec]

Fascinating. I'm curious. In the infinite zoom-in animation of the Mandelbrot set there appear to be many changes in 'pattern'. I'm guessing that these are due to the seed value differences of the location upon which we are zooming. What I am wondering is whether the Julia set has some role in the change of pattern that can be seen in the outlying 'symmetries'. In my laymans terms: If we zoom in on a point in the Mandelbrot set and we could overlay an image of its corresponding Julia set, would we find these to images had patterns in common? Thanks for the great videos.

[@EmielNijhuis]

Dr Krieger mentions there are two simple filled Julia sets. The first is a disc: f(x) = x2. What / where is the other one? Great video by the way.

[@InfiniteMonkeysSA]

Phrasing

[@shaunchan9008]

i love how she thinks of the mathematical plane as some kind of map that she can zoom in and tell you what that portion of the map looks like. its like how people talk about the nazca lines

[@omp199]

Error at 2:33! That picture purports to be the filled-in Julia Set for _c_ = 1.1, but the set is connected, so it can't be. I know this because _c_ = 1.1 is not in the Mandelbrot Set. It looks like the filled-in Julia Set for _c_ = -1.1.

[@omp199]

SammYLightfooD Great minds think alike. ;)

[@NeoNoggie]

omp199 Excuse me, sirs, but this is KZbin. I'm going to need you two to start a slander war that is 69 comments long instead of acting like mature adults in the comment section.

[@I77AGIC]

I love her!

[@dandubs433]

That works like the dual-slit experiment. When the electrons are shot one at a time, a pattern of the waves appear slowly. Like the wave pattern is already there (entangled ?). The design at 1 seems to be the single shot pattern, entangled to the Mandelbrot Set. Still looks like 2 spheres collided creating electric copies of itself.

[@galbegun]

all of your numberphile vids are awesome but we barley get to see any with dr krieger and i love the videos which feature her.

[@ZeGiles]

Just a random question here: is there a value of c for which the Filled Julia Set actually looks like the Mandelbrot Set? That would be mind-blowing!

[@SammYLightfooD]

So one way to look at the Julia "Pictures" of some Mandelbrot points, is that they show you graphically how close a Mandelbrot Point is "near to the edge" as you notice the black area in Julia becoming smaller and smaller...

[@TIMS3O]

Is it analytically possible to calculate the area of the mandelbrot or the Julia set?

[@samueldravis7865]

More interestingly: the boundary has (Hausdorff) dimension 2!

[@antivanti]

Another interesting thing about Mandelbrot / Julia sets, and let's see if I remember this correctly. If you zoom in on the Mandelbrot set and then draw the Julia set for one of the coordinates you see and then do the exact same zoom on that Julia set they will look very similar.

[@1234s6]

This girl is sooo charming! :)

[@gyro5d]

Inside the Mandelbrot set, the Julia set outcome is a complete design (particle). Outside the Mandelbrot set, the Julia set is a wave/particle design? Probability Patterns? Does the

[@shanecook1993]

What percent of everything within the integer 2 is apart of the Mandelbrot Set?

[@Chancepia1]

this is so complex O.O

[@bruinflight1]

OK, this is destroying my mind. MORE PLEASE!!!

[@adamaenosh6728]

Is there a finite/countably infinite number of 'templates' that the Julia functions can take defined by areas inside and outside the Mandelbrot set?

[@ericsmith3554]

I am curious, does Dr. Krieger do research in chaos theory, nonlinear dynamics, or complex systems ?

[@CommanderdMtllca]

She is definitely my favorite

[@brentsykes1220]

How do you prove that the 2 ways of defining the Mandelbrot set result in the same set? Or I am missing something obvious?

[@cedv37]

Could we get also a fascinating visualization somehow in three dimension if c and z0 were varied together? I guess c and z0 together must be in 4 dimension, but could it be somehow projected to 3d? Does it make any sense? She showed at the end of the video that even without the visual aspects of the topic it has some interesting properties.

[@Shimmen]

You can sort of represent the mandelbrot in 3D (google mandelbulb) but that is through mapping the coordninates to spherical ones. There is no "true" 3D version of fractals because there is no true complex numbers with 3 dimensions.

[@KisoX]

What about a set where C is part of the julia set? For example we had C=0 and z0=0...are there any other numbers with this behaviour? What would a set of them look like?

[@aildoux]

"everything will make a lot more sense".... lol....

[@seven0929]

What about C = 1/4?

[@LuisMandel]

Suddenly this video reminded me Conway game of life. Is there any relationship between both? (I don't think so)

[@supersmaawgis]

Are the "small blobs" points or areas? and, "almost all the interesting stuff that happens for the funktion?f/)x= z^2 +c" cliffhanger?

[@msolec2000]

So, the circle is one of the two simple ones... what is the other one?

[@pavloskairis9994]

+Noel Goetowski ... named after French Mathematician, Gaston Julia.

[@StefanReich]

Where is the difference between Julia set and Filled Julia set?

[@fracturedreality88]

The ease at which you wield intelligence is humbling, When you start to apply the laws you have learned to everyday life is a day I look forward to.

[@berhcio]

why is there also Numberphile2?

[@mannaggiacristo]

Dr. Krieger is so cute ^-^

[@U014B]

In all seriousness, why is the Julia set named as such? Who is the eponymous Julia? Was Spike Spiegel really Mandlebrot this whole time?

[@davidcircuity6473]

Can you rotate it? ...does it replicate in 3-dimensions?

[@robkim55]

yes three dimensions is rather interesting.....

[@iliesgh26]

why do we start with 0 ?

[@FilipeJunqueira]

Another question: Any good recommendations of Textbooks about Complex dynamics (The filed of Maths which study dynamical systems defined by iteration of functions on complex number spaces, such as Julia's Sets).

[@feralfoods]

type this into google: "books chaos theory fractals" click on the top link that says "books about....". you will get a list of books ranging from $20 -- $200 on various mathematical applications of what you mentioned. hope this helps.

[@whauk]

I actually wrote a thesis about this stuff this year. I can recommend the following books: -Iteration of Rational Functions by Beardon (only talks about rational functions) -Fractal Geometry by Falconer (only polynomials in only one chapter, but very interesting links to fractals.) -Dynamics in One Complex Variable by Milnor (very theoretical mathematicl stuff. This is probably the way to go if you want to get to know about everything in the field.)

[@FilipeJunqueira]

Thanks Whauk. Just began to read Milnor's Book! : )

[@Snowflake_tv]

What's the difference btw Filled Julia's Set and just Julia Set?

[@entrophany]

how would this draw out in a 3 dimensional space?

[@ajiwibowo8736]

I love Dr Krieger 😍

[@Snowflake_tv]

03:43 I can't get it. If c = -1, and we are finding out what the filled Julia Set is going to be under the iteration of "f sub -1(z) = z²-1"..... Then the result is.... f(0) = -1 f(-1)= 0 f(0) = -1 .... it iterates. So within the domain, it never blows up. So that I cannot get the graph you drew for "f sub -1"

[@stevenhernandez8966]

Great series on the Mandelbrot & Julia Sets! Thank you Brady & Holly! As strange as it may sound, I've found a way to use this ma thematic phenomenon to describe some of the logic of Christian theology. Keep up the good work!

[@DOSTalks]

What is Holly's field of study called?

[@simonlanglois3219]

Complex dynamics

[@SomeMathematics]

Simon Langlois Yup, I'm learning dynamical systems on the real axis now! This map f_c is such an interesting case for such a simple definition. I'm also learning complex analysis to see how special holomorphic functions like f_c are.

[@FilipeJunqueira]

Question: Lets say I choose F(z) = g(z,c), and c = 1 (for instance) 1) Does my Filled-in Julia set is: all points z of function g(z,1) which the iteration of the function g(z,1) does not blowup, AND I start the iterations with 1; OR 2) My Filled-in Julia set is: all points z, which the iteration of the function g(z,1) does not blowup, AND I have to start the iterations for every complex number. Thanks

[@MadaxeMunkeee]

I'm not certain if I understood your question, but here's what's going on: First you choose a polynomial z^2 + c by choosing a value for c. In your case c = 1, so lets consider f(z) = z^2 + 1. The filled Julia set of this polynomial is the set of all points z in the complex plane for which repeated application of f does not eventually move the point to infinity. i.e. given a point, z_0 = 1 say, z_0 is in the filled Julia set if the sequence: 1, 1^2+1, 2^2+1, 5^2+1 ... is bounded. In this case, the sequence is clearly not bounded, so 1 is not in the filled julia set.

[@GriszaPlays]

the second one

[@dennismurphy6649]

Coolest ever...

[@dustinsc2023]

I wonder if there is more complex math then this that numberphile will show, this was fairly easy to understand. I agree with the below comments, I would like to see more complex math.

[@Submersed24]

Can the name sound any more wrong??

[@inside91]

I don't understand how just one point can become several points (that is a figure)?

[@10ozPoundCake]

In general, when generating a fractal you'll use a function that looks similar to f(z) = z*z + c where z is the point undergoing iteration (to see whether it "blows up" or not) and c is a constant. The difference between the Mandelbrot set and the Julia sets is in the constant term c that is added on. Mandelbrot set: f(z) = z*z + z0 where z0 is the original point that is under iteration (z gets smaller or larger while z0 stays at the original point z) Julia sets: f(z) = z*z + c where c is a constant term (that doesn't change throughout the whole process of generating the fractal) So now to answer your question simply, the points she refers to are values of c. At 3:20 she writes c = -.12 + .75i under the fractal she draws, so the fractal that she drew would be generated from the function f(z) = z*z - .12 + .75i As a side note, f(z) = z*z + c is just a common form. You could generate fractals from virtually any function you'd like. Hope this helps!

[@TheSentientCloud]

Chaos theory makes "me" all fractally and stuff. Basically, for a given point, we're testing whether or not that point escapes our bounds and tends off to infinity. If not, it's in the set. If so, it's not in the set. That's pretty much it.

[@skebess]

1 + 1 = 2

[@Oberon4278]

Dr. Krieger is great! She reminds me of John and Hank Green, or Emily Graslie -- people who are fun and engaging to watch, even when explaining things that would otherwise be incomprehensible or or dry. As an aside, your sound levels (i.e. volume) are not consistent across videos.

[@a_raw_bee]

watch the first one before this one he said It'll make more sense he said

[@wurnotantmlb]

this woman is magic!!!^^>

[@MarcoIblherRrvGreiffen]

I realy can understand the guy behind the cam in this video and that before.

[@seriousmax]

Amy Adams? :D

[@segamai]

Max Webster Right????

[@segamai]

Max Webster Even the voice is eerily similar..

[@Luden0906]

under 301 omg

[@helloitsme7553]

(z+c)(z-c) is also interesting try it

[@mr.soundguy968]

Since it involves analysis, why not say 'this video is an analytic continuation'

[@NathanTAK]

Question: Is there a point where the filled Julia set is also just the Mandelbrot set?

[@GeodesicBruh]

The way she chuckles is quite pretty

[@erbake]

It is fact, I'm stupid. I have no idea what she is talking about.

[@cosmicbuddhi8029]

Wow, she's pretty!

[@saparchitekt]

A widely held conjecture refuted!

[@hierro09]

Why is Amy Adams doing math?

[@Qwerty8]

I love you! Really, it is great. You are great. As far as i can see, but.. anyway. You know. Tremendous

[@vimalk78]

more smart amy adams

[@Prasen1729]

Math god, I adore this girl. She is something.

[@drkcaey]

Oh god. I can't take it. She's so attractive!

[@awolfcalledbed1232]

This too much brain for me.

[@horatiotrismegistus616]

Now I like math again! 1 + 1 = 3 if (1 man) + (1 woman) = (man, woman, child)

[@mathematics5573]

To you Americans, The full name is Mathematics. The abbreviation is Maths. What is Math? Math does not exist. Why not just call it M if you insist on abbreviating Maths to Math. I wish you Americans would stop abbreviating abbreviations. Maths, not Math.