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 [quote="Thok"][quote="The Potter"]I believe that it is connected mostly because of the way it fills the space. Trailing lines that approach a point are normally connected. When there is just 5 or so of these finger it is pretty obvious. This image just has too many to easily count.[/quote] Being dense (aka filling space) and being connected are distinct mathematical concepts. The subset of the real line containing the irrational numbers is dense, but not connected. (There are irrational numbers in any open interval, no matter how small, but to travel from one irrational to another one you have to cross a rational number.) These difficulties are hard to see because they take place at a very fine grain level. You also should make sure you actually understand the precise mathematical definition of connected.[/quote]
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The Potter
Posted: Mon Jan 14, 2013 12:28 am    Post subject: 1

Oh, I see why the software is having trouble. The point is really close to -0.75 + 0i. This point is "neck" of the "bug" on the main Mandelbrot.
The Potter
Posted: Mon Jan 14, 2013 12:03 am    Post subject: 0

While I have been exploring fractals primarily as art, I have a reasonable understanding of the math behind it. But I am not ever going to be part of the math community. And yes, I do believe my understanding of being connected is consistent with your posts.

I wasn't searching for anything. But when was looking at the image I was a little confused-- it comprised of patterns that I see with obviously connected sets but the appearance of being disconnected.

Also I should note that after about 25k iteration level, there is a very sharp transition in amount of coloration. This explained my confusion perfectly-- the set is connected but became disconnected by limitations of the software.
Thok
Posted: Sun Jan 13, 2013 6:22 pm    Post subject: -1

Zag wrote:
I think his point, Thok, is that these are an expression of art and emotion, and he doesn't really care about the precise mathematics. It seems a valid way to think about it.

I just figured The Potter was searching for some meaning hiding behind extro's original statement.

Translating an intuitive concept like "connected" into mathematics turns out to be somewhat difficult. In the case of "connected", there are several different approaches that lead to different answers. The concept the mathematical community have decided best fits the term "connected" might be different than what The Potter would use.
Zag
Posted: Sun Jan 13, 2013 5:02 pm    Post subject: -2

I think his point, Thok, is that these are an expression of art and emotion, and he doesn't really care about the precise mathematics. It seems a valid way to think about it.
Thok
Posted: Sun Jan 13, 2013 3:24 pm    Post subject: -3

The Potter wrote:
I believe that it is connected mostly because of the way it fills the space. Trailing lines that approach a point are normally connected. When there is just 5 or so of these finger it is pretty obvious. This image just has too many to easily count.

Being dense (aka filling space) and being connected are distinct mathematical concepts. The subset of the real line containing the irrational numbers is dense, but not connected. (There are irrational numbers in any open interval, no matter how small, but to travel from one irrational to another one you have to cross a rational number.)

These difficulties are hard to see because they take place at a very fine grain level. You also should make sure you actually understand the precise mathematical definition of connected.
The Potter
Posted: Sun Jan 13, 2013 9:23 am    Post subject: -4

I believe that it is connected mostly because of the way it fills the space. Trailing lines that approach a point are normally connected. When there is just 5 or so of these finger it is pretty obvious. This image just has too many to easily count.
extropalopakettle
Posted: Sat Jan 12, 2013 4:41 pm    Post subject: -5

Lowering the number of iterations will show areas to be black that shouldn't be - areas very close to points in the set, but not in the set itself.

I tried checking C= -0.7472241 -0.06128591i, but I suspect there's some rounding there too, as what I got was a bit different.

Quote:
So there is a theorem that says if the point 0+0i is trapped, the set will be connected.

Which is also equivalent to saying if C is in the Mandelbrot set, the Julia set for C is connected.

If you take a C value from one of the tinier mini-Mandelbrots, and another C value from right nearby, but clearly outside the Mandelbrot set, the Julia sets for the two values will look identical until you magnify. Then you'll see one is black at the center, the other open.
The Potter
Posted: Sat Jan 12, 2013 8:35 am    Post subject: -6

Hmmm 2 people supported connected. One of them was me. There were 6 in favor of disconnected.

So there is a theorem that says if the point 0+0i is trapped, the set will be connected. So instead of some arbitrary location, lets center the image at 0+0i and see if it is black.

It is pretty but hard to tell exactly what is going on in the exact center. So I can zoom in as far as Fractal Domains can really handle, 1e-11.

Clearly the center is non-black. But there is something funky going on.
This fractal was using 30 000 iterations. So what happens if the iteration count is reduced to 20 000?

20 000 iterations

Suddenly the center of the image is black.

As far as I can see, the change is a result of rounding errors. I believe the Julia point C= -0.7472241 -0.06128591i is connected...unless we rely on rounded calcualtions. Which the first image did.

(When I posted this question, I didn't know the answer.)
The Ragin' South Asian
Posted: Fri Jan 11, 2013 7:14 pm    Post subject: -7

I don't really know what it means so I voted connected to feel less lonely.
The Potter
Posted: Fri Jan 11, 2013 5:11 am    Post subject: -8

extro...* wrote:
The disconnected Julia sets like the above[different fractal] are fascinating in how you can see it's made of a number of disconnected pieces, but magnifying any piece reveals that it too is created of disconnected pieces, and in fact there aren't even any two connected points in the whole set. Julia sets are either entirely connected, or entirely disconnected.

So lets look at this image:

So you think this one is entirely connected or entirely disconnected?