[ 3 / biz / cgl / ck / diy / fa / ic / jp / lit / sci / vr / vt ] [ index / top / reports ] [ become a patron ] [ status ]
2023-11: Warosu is now out of extended maintenance.

/sci/ - Science & Math

View post   
File: 1.25 MB, 2560x1920, Mandel_zoom_00_mandelbrot_set.jpg [View same] [iqdb] [saucenao] [google]
6426616 No.6426616[DELETED]  [Reply] [Original]

Can someone explain this to me in layman's terms? I read the intro in the Wikipedia article and didn't understand anything.

>> No.6426620

You zoom in and it looks the same

>> No.6426619

a mathematical set, which when you put its points onto a graph or something, turns into that shape, I think

>> No.6426624

>>6426616
For some coordinates of the complex plane Z = Z^2 + c blows up to infinity and for others it doesn't. The latter are represented in black and the former are colored based on how quickly they blow up. I think.

>> No.6426625

Solve an equation, the solution is the black shape

>> No.6426637

>>6426624
How does it blow up, or not blow up to infinity? If you replaced Z with a complex number, how would it change?

>> No.6426643

>>6426637
Well it's a recursive formula. Zn = Zn-1 ^2 + c, where c is the original complex number. After you apply the recursive formula a bunch of times you either get a big number indicating that it will probably blow up to infinity or a small number indicating that it probably won't.

>> No.6426666

>>6426616
the points for wich a certain sequence converge are blacks, the other are colored depending of their divergent speed.
The shape you obtain is pretty, that's abgout it, nothing really intersting from a mathematical point

>> No.6426669

>>6426666
>sets are uninteresting from a mathematical point
Good one ol' chap.

>> No.6426679

>>6426643
Thanks. What's Zn though?

>> No.6426683

>>6426679
Z subscript n. It's just the nth recursive term in the sequence, whereas Zn-1 is the one just before it.

>> No.6426684

>>6426616
To understand you need to know a bit about complex numbers.

Basically, this is a representation of complex numbers on the complex plane(horizontal axis is the real part and vertical axis the imaginary part)

Basically, for a complex number Z(n), the algorithm is Z(n+1) = Z(n) + c where c is a constant.

Now choose any complexe number, which we will call Z(0) and locate it on the complex plane.

Go through the algorithm, i.e. calculate Z(1), Z(2),... You will either notice that the number will tend to infinity(i.e. its real and imaginary parts will tend to infinity), or it won't. If it doesn't, colour the point in black.

That's it.

This simple algorithm produces the beautiful figure that you have posted.

Mind boggling isn't it?

>> No.6426686

>>6426684
oops, I meant the algorithm is Z(n+1) = Z(n)2+c

>> No.6426690

>>6426686
Z(n+1)=Z(n)^2+c

>> No.6426704

>>6426684
Cool. I don't get how the algorithm works though. I get complex numbers a bit, like i^2 = -1, but is Z(n) a function? Let's say I picked 1+2i as my number, how would the sequence go?

>> No.6426730

>>6426704
It's a function from integers to complex numbers, yeah. aka, a sequence

>> No.6426738

>>6426704
It's a recursively defined function.

>> No.6426740

>>6426704
I'm not gonna do the calculation, so lets just call it c.
So you start with the complex number Z0 = c
First iteration gives you Z1 = Z0^2 + c = c^2 + c
Second iteration is Z2 = Z1^2 + c
etc
If it tends to infinity, color the point c in blue, else in black