Fractal Geometry of Nature

[Qfwfq]

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'?

Indeed the program should have at least said "all quadratic Julia sets".

[freeztar]

Originally Posted by Qfwfq

Indeed the program should have at least said "all quadratic Julia sets".

Which begs the question...how many quadratic julia sets exist?

[Turtle]

I received a reply from Mr. Mandelbrot's office; he never met Mr. Fuller.

[Qfwfq]

Originally Posted by freeztar

Which begs the question...how many quadratic julia sets exist?

Plenty!

For every complex value there is a Julia set. I don't know if the map is injective but cardinality-wise my guess would be that it's anyway.

The actual point though was that not all Julia sets are the quadratic ones, whereas the Mandelbrot set concerns only these.

[maddog]

Originally Posted by Qfwfq

Plenty!

For every complex value there is a Julia set. I don't know if the map is injective but cardinality-wise my guess would be that it's anyway.

It is at least as you say injective, though I don't believe it is bijective.
Any one Julia Set so formed is likely to be in more than one algebraic map.
So 1-1 correspondence I think is out.

Originally Posted by Qfwfq

The actual point though was that not all Julia sets are the quadratic ones, whereas the Mandelbrot set concerns only these.

I am not sure of the exact definition of the Mandlebrot set being of only
quadratic equations. I had plotted many such maps base on nothing more
than the assumption that any Analytic Function over a piecewise defined
domain can form a Mandlebrot set.

For example or , etc.

maddog

[Qfwfq]

OK when saying "the" Mandelbrot set we usually mean "that" one i. e. the one given by the form so the relation is with the quadratic Julia sets. As for the map, it's definitely surjective over the quadatic Julia sets, by definition, so if the map is injective then it's also bijective.

[maddog]

Originally Posted by Qfwfq

OK when saying "the" Mandelbrot set we usually mean "that" one i. e. the one given by the form so the relation is with the quadratic Julia sets. As for the map, it's definitely surjective over the quadatic Julia sets, by definition, so if the map is injective then it's also bijective.

Like you, I don't know if Mandelbrot ever considered other analytic functions as being claim to being the set named after him or just referring to only quadratic function. As for the rest, I defer to your more concise definition. My Analysis course (injective + surjective = bijective) is a bit rusty.

I will admit I am always thinking in the largest of terms by using the largest container.

I conjectured a long time ago that an Iterative Function System f(z) can be formed from any analytic function being defined over a piecewise continuous domain.

maddog

[Thunderbird]

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