Mandelbrot Set: how it is generated

[onlyguitar1001]

oh my god, thankyou!!!

[danno321s]

The "parallel postulate" is wrong. That is why there is no right angle in the fractal world. Go out into nature and find a right angle. There are none. Hence, there are no parallel lines. Einstein showed that gravity is actually curved space, meaning all space is curved. 

[Mdeil20]

to be honest i dont get the math. i just like to look at the pictures :D

[danno321s]

@Mdeil20 complex numbers are mapped onto a Cartesian plane. Think of an infinite chess board.  Perform a step by step move on the chess board, take the result then use that as the input and repeat until bound or infinite.

[FredGandt]

Wow!

[nhmllr725]

You have a gift for explaining! Thank you so much!

[mkcactus]

How can i see those julia sets in fractal extreme?

[lurkern]

Thank you so very much for putting this out there :) <3

[VentedGlitchMaker]

You don't need to know all the specific mathematical details to be able to understand the concept. This is quite interesting I must say!

[WhatYouWantInSpadez]

yes, good stuff, this has helped me understand more. Thanks.

[deacumen]

In what mathmatics is this learned?

[danno321s]

@deacumen Learn Linear Algebra, Abstract Algebra, Set Theory, Real Analysis, Complex Analysis then jump in. Number Theory is complementary. These are all semester courses in college.

[danno321s]

@deacumen The classic text is "Fractals Everywhere" (now updated) by Michael F. Barnsley. I used this book in 1989.

[w4nt3d1000]

does this only works for f(z)=z²+c?

[danno321s]

@w4nt3d1000 No.

[time3space1]

Okay, I have fractal extream for a few months now, how do you get the real time trace of the iterations ?

[decayer]

Finally after years I can now understand! Great video series, thanks for time spend on this!

[danno321s]

@decayer Kudos!!!

[greatsoccerman96]

I think I love you!

[danno321s]

@greatsoccerman96 love? does it give?

[Kirklandkd13]

Ok what is the formula for the mandelbrot set? F(z)= z^2+ ?

[greatsoccerman96]

@Kirklandkd13 F(z) = z^2 + C I think :)

[Matthewsimplifier]

@Kirklandkd13

F(z)= z^2 + i

[JustSomePerson888]

great software, i used it on a 600MHz pentium about 10 years ago and it made great zoom videos, sadly it's near impossible to find a codec that'll play them right on anything these days..despite their being a then-standard avi....

[expandingvoid]

This is very helpful, thank you!!!

[vladissimo18]

Whit this explination we can say the univers it's trully a hologram?

[topbluffa1]

can my question please be answered

[Ozomness]

I'm going crazy for fractal images. I'm not really good at math but I love art and I respect math as a very fundamental code of our reality. Good video!!!

[Shalek]

That is so freaking cool. 

[lensip2]

@Shalek So is your comment

[UntameDAnger]

This shape replicates so many other shapes in nature, the discovery of the shape is the work of a genius and will be key to discovering how the universe has formed over the 13.5billion years because many of the shapes represent galaxial structures and the emptiness of space.

[topbluffa1]

so does a disconected only exist within a connected fractor but never the other way around.

[badslabber]

thanks for this video , i cant comprehend maths ,but i seem to have an understanding ,of this infinite reduction haha :)

[capefeather]

May he rest in peace...

[kingluke2]

thanks for this - always wanted to understand how fractals are generated, but didn't have the complex math background - now I do :)

[AleksHyper]

beardyman brought me here...I dont know math...

[trusta225]

@AleksHyper amen sir. amen

[fireluigi12]

how do you make the white lines, that show where the point s go?

[wordsandcolor]

fantastic video. well done.

[kizutsuku]

It isn't hard. Look at formula, play around, and enjoy.

[aNickzz]

So how do you work out the iterations for an (x, y) pair?

[JohnThePetru]

Also, F(z)=z^2 +C, if F(C)=C^2 +C remains bounded, also F(0) remains bounded, since F(0)= C, F^2(0)=C, F^3(0)= C^2+C. So F^n(C)=F^(n+2) (0). So not only in every connected Julia set 0 is connected but also C is connected... Maybe it's a good starting point for proofing that "in every connected Julia set 0 is connected". I'll try this way! :D

[JohnThePetru]

That's just... WOW

Thanks man!

[guyj247]

Great video man, so helpful!

[AndromedaChao2]

Is the Mandelbrot set the plot of all the C points that have connected Julia sets?

[badmephisto]

@AndromedaChao2 good way to summarize! yes!

[JohnThePetru]

@AndromedaChao2 I'm not sure of what plot means (in math at least (Yep, I'm a math student but I'm Italian :D))The Mandelbrot set is defined as Zc(n+1)=(Zn)^2 + C where C is the point where you apply this equation and Z0=0.

The Julia set is defined with the same equation F(Z)=Z^2 + C (you have to reiterate this) but in this one C is a given complex number and Z is the point where you apply this equation.

Now Z(0)=0, that means (using Zc(n+1)=(Zn)^2 + C) Zc1= C, Zc2= C^2 + C.

[JohnThePetru]

@AndromedaChao2 So we see that Zc2=F(C).So we see Zc(n+2)=F^n(C), this means that if Zc remains bounded (= is a black dot in the Mandelbrot set) F(C) also remains bounded, so his Julia set is connected. Basically it means that if you have a black dot in C=(a,b) in the Julia set generated by F(z)=z^2 +(a,b) the point (a,b) is a black dot.

[JohnThePetru]

@AndromedaChao2 So yes, I think the Mandelbrot set is the all the C points that have connected Julia sets.

Dunno if any of these make actually sense and if I have actually answered your question, but I hope I helped!

Lol, I just saw the comment of badmephisto -.-

BTW, does anyone know how color are chose?

[metallichika]

my brain hurts >_< trying to make sense of this

[birlibis2juggler]

One person has a Black and White computer screen! :)

[danno321s]

@birlibis2juggler use tones.

[lowed40]

Quick question for you. I've watched your entire series leading up to the Mandlebrot set and I think your videos are a great springboard into the discussion of the associated Julia Sets and how they lead to Mandelbrot. My only question is actually software related. How do you get the fractal eXtreme software to show both the Julia and Mandelbrot sets in the same window? Also, how do you get the cursor "chaser" that shows how the points escape to infinity or spiral to 0? many thanks!

[lowed40]

Quick question for you. I've watched your entire series leading up to the Mandlebrot set and I think your videos are a great springboard into the discussion of the associated Julia Sets and how they lead to Mandelbrot. My only question is actually software related. How do you get the fractal eXtreme software to show both the Julia and Mandelbrot sets in the same window? Also, how do you get the cursor "chaser" that shows how the points escape to infinity or spiral to 0? many thanks!

[johanhendriks]

This has been a very helpful video for me.

I've never known that there was such a relationship between mandelbrot and julia sets.

The concept of connected and disconnected is some food for thought :)

[Nevawake]

im 39 and i feel like a cheese cake.

[Zookhaarh]

not connected ones make me feel happy.... the connected ones make me frustrated... dont know why...

[Avgen14126]

i'm 17 and i feel incredibly stupid.

[SpiritualAtheist]

Great presentation. 

[QuantumLogic1080]

Could someone please explain 'zn+1 = zn2 + c' for me ??

[ben1996123]

Benoît Mandelbrot died 2 weeks ago -.-

[sivyig]

wow! just wow! thanks for the lecture.

[77wii77]

MAN I LOVE YOU!!! such an expert teacher

[johanhendriks]

@77wii77

I wonder if there's a relationship between a person's knowledge and your love for them :P

[dleddy14]

Great stuff. You are a very good teacher. 

[zeenon]

Didn't Matt Damon do a movie about this?

[TheAntiFascist2010]

fractal geometry is just the exhibition of infinite domain. You are really watching the exposition of chaos. Wow. So the first integer becomes compromised. This is what the narrator meas by saying fate. It is all bullshit.

[74andrewt]

ay?

[veeveevenn]

! ! ! ! !EXQUISITE! ! ! ! ! Thank you so much! I will be using your video as a homeschool project. Thank you, again.

[badmephisto]

@veeveevenn awesome! good to hear :) cheers

[m598lmr]

If you want a fractal generator, download Xaos, its free software and it generate many sets

[denelson83]

Actually, Fractal eXtreme isn't free. It's a shareware program.

[20secondsofawesome]

Indeed... indeed.

[fhellstrom]

please start uploading more vids both here and at badmephisto, seriously

[Leonchemist]

Wow great program! I tried to extract c values from Mandelbrot set and to generate Julia sets manually from them :wall:

[schenonef]

excellent video ! i understood everything you said, and i am very happy of having watched it.. keep it up!

[bcoolbrich]

I am a straight 20 student in math, on the 11th grade and i have no ideia of what you are talking about... is it worth for me to try understanding this or is it too complex for someone without a graduation?

[fractalmath]

@bcoolbrich its not complex, you would just have to watch all of my previous videos too. Sorry for late reply :p

[johanhendriks]

@fractalmath

It does use complex numbers though :P

but complex doesn't make it difficult, if that's what bcoolbrich meant

[kevrs2]

@bcoolbrich WELL if you had of taken things serious ,you my have noticed that its not "math" ,its called maths! ,well i guess this is from a country ,that dov ,in to the swimming pool ,as opposed ,dived , OH DEAR ,where does one even start :)

[Vivendium44]

@kevrs2 For a guy who's snobby about spelling differences between dialects, you sure suck at English.

[kevrs2]

@Vivendium44 who cares what you think!

[Vivendium44]

@kevrs2 Nobody, I'm sure. Nobody cares what anyone thinks on the Internet, so maybe you shouldn't be whining about people's grammar?

[kevrs2]

@Vivendium44 OK :)

[mpdp85]

@bcoolbrich if you dont even have a vague idea of what he is talking about you clearly dont know shit about maths.

[philipjterry]

@bcoolbrich I was terrible at Maths and I understand exactly what he is talking about. Maybe there is pattern there.

[betbetsukbet]

@bcoolbrich im a 13 yr old 7th grade student and i understand everything :P

[bcoolbrich]

@betbetsukbet this coment is old, i understand it very well now. i just didnt watch the first video. so stop replying to my comment please, thanks

[lpchiasson]

Thank you for these! Great videos, I finally have a clear idea how this all works together.

[TellerWest]

I really enjoyed these...and the notion that so much complexity flows seemingly out of nowhere is fascinating. Thumbs up!

[DettwilerPictures2]

Absolutely awesome ! I love maths and I saw all your videos about fractals. Please continue ! It's great. Your rubik's cube tips are CooOoOooOoOL too !

[JimmyGr90]

programm doesnt work on macs eh?

[z3125]

I read on wikipedia that if the series for a point z ever has radius greater than 2, it will diverge.

Is there a corresponding fact for a radius under which no points will diverge?

If not, is there a record for the smallest radius achieved by a divergent point?

[badmephisto]

Good questions!

Yes and Yes.

For certain numbers you can definitely show analytically that they must converge or escape. I dont know exactly what all of these regions are. But certainly this is NOT true in general. In particular it is not true for the regions we are most interested in ;) Complexity is elusive

[Tortak39]

Are you badmephisto?? :)

[byorda]

in person

[llamasownyou]

Is it possible to determine whether a C value escapes to infinity by taking a limit of the iteration rather than sampling it over and over again and guessing?

[badmephisto]

for some special cases you can do that analytically, but in general you must almost always go through the iteration process to figure out the answer. And you ARE correct in classifying it as "guessing", because that is what it is-- we iterate a million times, trillion times, but that still doesnt technically provide us with the right answer, FOR SURE, in some cases. So for some points, it is indeed only the best guess. Good question.

[Deluminator4]

Is there a way to set this up as a summation or series and test if it is convergent or divergent or is that totally unrelated? And if this were the case could it fail due to the complex numbers?

For example f(z) = Sum(X^2+C) from o to infinity

at f(0) and find if it is convergent or divergent could you see instead of guessing?

One last question, could you apply this to a three dimensional plane such as (x,y,i)? or even (x,y,z,i)?

[badmephisto]

I definitely saw 3d fractals... I just dont know if there would be a mandelbrot set in 3d. probably not.

and yes. The sequence is simply:

0, c, c^2 + c, (c^2+c)^2 + c, ....

does that converge or diverge? :) I think that for special cases of c you can do this. (For example for small c values.) But certainly not in general.

[badmephisto]

Sorry for the VERY late response on this. I just noticed it now heh. Good questions, both.

1: Sum(X^2 + c) is NOT what it is computing here. The sequence is recursively defined. What you would want instead is to look at the asymptotic behavior of the sequence c, c^2+c, (c^2+c)^2+c, .... And yes you can do this analytically for some special cases. For example for |c| > 2

2: The Mandelbrot set does not easy generalize to 3D and there is some ongoing work on that. Its not a solved problem

[badmephisto]

oops actually i just see that i DID answer. And pretty much the same thing too. Haha sorry about that :) cheers :p

[neurobact]

awesome math pattern. is the mandelbrot set defined by a function as well?

[Kakashinoor]

You have a very ornate way of speaking o.o who says z as zed if they are not from the UK o.O

[OrphanEater1]

Canadians

[jesseliston]

the usa is the only english speaking country who doesnt say z as zed

[danieldrehmer]

Pure awe

[xXdaveXsuperstarXx]

This is crazy.

[fona24]

wow! this is awesome!