Fractal Geometry of Nature

[Turtle]

Originally Posted by maddog

I felt bad I missed that show last week. We had just moved and had left our
TV. At least we have one now. I will have to try and watch it off of pbs.org
website.

...
Somebody earlier in this post mentioned Julia Sets and they simply where
the constant C in equation z <- z^2 + C is the coordinate in the iteration.

maddog

When you get a chance to watch the program, check me on a bit I recall hearing in it concerning Julia sets. To whit, that the Mandelbrot set contains all Julia sets. I may have to ask Rac for the book back if he still has it, as I don't recall that association from it.

[Qfwfq]

Originally Posted by Turtle

When you get a chance to watch the program, check me on a bit I recall hearing in it concerning Julia sets. To whit, that the Mandelbrot set contains all Julia sets.

Dubious statement.

It may be a sound-alike of "The set of all points for which J is connected is known as the Mandelbrot set." in this page on Julis sets, but the sentence seems to refer only to the case of meaning all values of c which give a connected Julia set for this form.

[Turtle]

Originally Posted by Qfwfq

Dubious statement.

It may be a sound-alike of "The set of all points for which J is connected is known as the Mandelbrot set." in this page on Julis sets, but the sentence seems to refer only to the case of meaning all values of c which give a connected Julia set for this form.

You didn't watch the show did you? At any rate, the show is the basis for my dubiosity and perhaps I missheard, or perhaps they misstated, or perhaps mathworld isn't the final word.

[Qfwfq]

Originally Posted by Turtle

You didn't watch the show did you?

No, I didn't, but the sentence I quoted seems to match up with what I can find about the topic although some sources can be confusing.

The fact is that the word contains in the show's sentence would need to be given a precise meaning. What Mathworld et al. say more exactly is presumeably what the show meant but said more vaguely.

But the king is not gallopping on the horse's back, it's the horse that's gallopping, with the king on its back!

[Turtle]

Originally Posted by Qfwfq

..But the king is not gallopping on the horse's back, it's the horse that's gallopping, with the king on its back!

Roger that. The King & his Horse go galloping everywhere together.

Here's the program link and I'll schedule some time this week to watch it again. NOVA | Hunting the Hidden Dimension | PBS Meantime Rac is going to try & root out Mandelbrot's Fractal Geometry of Nature for me and I hope to find an address for Benoit as I'd like to ask him about Bucky Fuller's work.

[maddog]

Originally Posted by Qfwfq

Dubious statement.

It may be a sound-alike of "The set of all points for which J is connected is known as the Mandelbrot set." in this page on Julis sets, but the sentence seems to refer only to the case of meaning all values of c which give a connected Julia set for this form.

As I understand it, I think you both have it right. The Julia Set is about a coordinate
point. So the Set of all Julia Sets would span the space and therefore would
be the Mandlebrot Set. Both are Iteration Functions Systems (IFS).

ps: I do have to start practicing that math/imath analogue to LaTeX. ;-)

maddog

[freeztar]

Originally Posted by Turtle

When you get a chance to watch the program, check me on a bit I recall hearing in it concerning Julia sets. To whit, that the Mandelbrot set contains all Julia sets. I may have to ask Rac for the book back if he still has it, as I don't recall that association from it.

"In 1980, he created an equation of his own*, one that combined all of the julia sets into a single image. When Mandelbrot iterated his equation, he got his own set of numbers. Graphed on a computer it was a kind of roadmap of all the julia sets and quickly became famous as the emblem of fractal geometry, the Mandelbrot set."

*

-From the second part of the episode on Nova's website

Off to watch part 3.

[Turtle]

Originally Posted by freeztar

"In 1980, he created an equation of his own*, one that combined all of the julia sets into a single image. When Mandelbrot iterated his equation, he got his own set of numbers. Graphed on a computer it was a kind of roadmap of all the julia sets and quickly became famous as the emblem of fractal geometry, the Mandelbrot set."

*

-From the second part of the episode on Nova's website

Off to watch part 3.

You rock! Again. I checked with Racoon & he thinks he still has Mandelbrot's book The Fractal Geometry of Nature somewhere and will look for it, and I wrote a communique de la electronique to Mr. Mandelbrot at Yale today inquiring if he had any communication with Fuller or familiarity/influence with/from Fuller's Synergetics:Explorations in the Geometry of Thinking. We shall see what we shall see.

[freeztar]

I still have issue with the quote though.

How could *all* Julia sets be represented in one image?

From the section just before they stated such, they identified a julia set as iterative functions (ie feedback loop of one equation). Is it not reasonable to assume that with infinite function equations one could also have infinite julia sets? How could Mandelbrot's 'set' incorporate 'infinite sets'?

[Turtle]

Originally Posted by freeztar

I still have issue with the quote though.

How could *all* Julia sets be represented in one image?

From the section just before they stated such, they identified a julia set as iterative functions (ie feedback loop of one equation). Is it not reasonable to assume that with infinite function equations one could also have infinite julia sets? How could Mandelbrot's 'set' incorporate 'infinite sets'?

I'll leave that one to Q or Maddog, other than to observe that only certain types of IFS's produce the Julia sets. Change the equation and get a different type of set. Earlier in the thread I give a link to some free software that originally came with Jame's Gleik's book Chaos, and it lets you mess about with a number of different types of fractals and their innards. I'll brb with it to post again. It's a peach I tell ya...a peach!

>> Chaos Downloads from Rudy Rucker's Wesbsite