Mandelbrot Set

mynamesmurph

Junior Member
Joined
Aug 10, 2014
Messages
51
Here are some screenshots of the problem I have.

RhrcwOo.jpg


h7sZstr.png


So for 91. a) I got
i, -1 + i, -i, -1 + i, -i, -1 + i

I'm pretty sure that's correct. But I'm not really sure what b is asking. What does it mean by bounded? Which complex numbers should I write? Do I pick two any two of the set there? Would the the absolute value of the terms be

1, √2, 1, √2, 1, √1

I think I'm close here to solving, but I'm not entirely sure what the Mandlebrot set is. This is all in the book that is dedicated to it. Any ideas?

Thanks yalls.
 
"Bounded" means just like the exercise says: there's a number that's bigger than all the numbers in the set, and there's another number that's smaller than all the numbers in the set, so the set of numbers is bounded by the bigger and smaller numbers. If all the numbers in the set were between, say, 3 and 7, then the set would be bounded. If the numbers, say, kept doubling, so they were always getting bigger and bigger with no top limit, then the set would not be bounded.

"The numbers" they'd be looking for, if the set is bounded, would be the bigger and smaller numbers. ;)
 
Ah, that was kind of simple, thank you.

As an aside, how does the Mandelbrot set of numbers correlate to the graph? I'm a bit lost as how it all works.
 
Ah, that was kind of simple, thank you.

As an aside, how does the Mandelbrot set of numbers correlate to the graph? I'm a bit lost as how it all works.
If the number belongs to the set it is 'colored' black, otherwise it is left as 'white' (the background color). For example the complex number 0 (0 + 0 i) would be colored black since the sequence of 'square the last number and add the first number' is bounded and, the complex number 1 (1 + 0 i) would be left at the background color since the sequence diverges to infinity.
 
The graph is the "Mandelbrot set" and the "set of Mandelbrot numbers" in the sense that each "Mandelbrot number" is a complex number, c, such that the sequence zn+1=zn2+c\displaystyle z_{n+1}= z_n^2+ c, z0=0\displaystyle z_0= 0, remains bounded. If we set up an xy-coordinate system with x-axis as the "real axis" and the y-axis as the "complex axis", so that the complex number z= x+ iy is represented on the graph by the point (x, y), then the "Mandelbrot set" is set of all Mandelbrot numbers represented as complex numbers on that graph.

If you are interested in this, you might also want to look up "Julia sets". The Julia set, Jc\displaystyle J_c, with c a complex number, represents, in the same way, the set of complex numbers, z0\displaystyle z_0, such that the sequence zn+1=zn+c\displaystyle z_{n+ 1}= z_n+ c with initial value z0\displaystyle z_0 converge. The single "Mandelbrot set" can be thought of as an index into the infinite number of Julia sets. If c\displaystyle c is deep in the interior of the Mandelbrot set, the Julia set, Jc\displaystyle J_c will look like a single "blob" and the more you move toward the boundary of the Mandelbrot set, the less symmetry you will see in the Julia set. As you move past the boundary of the Mandelbrot set, the corresponding Julia set will break into a number of oddly shaped parts and far away from the Mandelbrot set will be a scattered dust of single points.

By the way, while Mandelbrot worked for IBM and had all their computing power available (though admittedly not nearly as much as any laptop today!) Gustav Julia did his work on "iterations of rational functions" around 1920 and drew his graphs "by hand" with no computers or calculator available!
 
Forgive me, because I'm not even sure what I'm asking. :p I should say that I understand complex planes, and how to graph complex numbers.


In the two examples we have two complex numbers. Looking at the first z= i, are we saying the boundaries are 1, √2, 1, √2, 1, √1? or rather numbers between 1 and √2? So the starting complex number defines the boundary? So from there, any complex number is either within the set or it isn't right? What does it mean to approach infinity? The blurb in my textbook is so sparse I feel like it's not very thorough.

It would help to see the the points actually graphed. Can anyone recommend a video?

Thanks for the help folks.
 
Forgive me, because I'm not even sure what I'm asking. :p I should say that I understand complex planes, and how to graph complex numbers.


In the two examples we have two complex numbers. Looking at the first z= i, are we saying the boundaries are 1, √2, 1, √2, 1, √1? or rather numbers between 1 and √2? So the starting complex number defines the boundary? So from there, any complex number is either within the set or it isn't right? What does it mean to approach infinity? The blurb in my textbook is so sparse I feel like it's not very thorough.

It would help to see the the points actually graphed. Can anyone recommend a video?

Thanks for the help folks.

An interesting talk, IMO, is at
http://www.youtube.com/watch?v=3yE4d7O17wE
which may give you a better understanding of the set. The video makes the case for the repeated iteration. You actually don't know, in general, when you first start whether a point is in the Mandelbrot set or not. You choose a number to start, say c (a complex number) and iterate by squaring the number and adding c. So the sequence of numbers looks like
c
c2 + c
(c2 + c)2 + c
((c2 + c)2 + c)2 + c
(((c2 + c)2 + c)2 + c)2 + c
...
If that sequence converges, then c belongs to the Mandelbrot set and then you color it black.
 
OK, I was making it a bit more complicated and after watching the video and reading a few things, I've got a much better understanding. As I understand now, the black part is just the complex numbers that when run through successive iterations don't approach infinity, but instead follow some kind of pattern. The boundary is where things get interesting, because as you zoom in you get to see all the nifty shapes and patterns. The coloring is just based on the number of iterations. I think this is all correct, yes?

I don't understand the relevance of "orbit zero" though. Does that just refer to the first term you use for the squared term? With the later being the any complex number you decide to see if it's in the set?

z^2+z


And as an aside, this has all been quite captivating. I mean, I can see why my teacher skipped it, but that's too bad, it's really quite beautiful.
 
...I don't understand the relevance of "orbit zero" though. Does that just refer to the first term you use for the squared term? ...

The orbit is just the path connecting the different values. So choose a point c. The orbit of zero is the path connection the points
0
02 + c = c
c2 + c
(c2 + c)2 + c
((c2 + c)2 + c)2 + c
(((c2 + c)2 + c)2 + c)2 + c
...
see
http://www.geogebratube.org/student/m27369
for example
 
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