Categorising fractals

[Tglad]

I don't think that is a shell. A shell is 3d or higher and it encases a tree, it is the complement of a tree.

Here's a new one, it is a sponge, and maybe the simplest example. It also is the first fractal I know of that is its own complement, two of these could fit together to make a solid. As you can see from the table that is only possible with a (normal) sponge. 2nd pic is a cut through.
<sorry for poor render, too many UF parameters to find the fix to concentric circles>

[cKleinhuis]

this is a wonderful thread, i want to  realise a box-counting dimension fractal analysis on the current mandelbulb formulas, and i am wondering how
best to describe the outcome of the fractal dimension

a value close to an integer from the bottom direction is considered as a structure that nearly fills out the area of a higher dimension although it is just e.g. a line,
always having ( each structure with non int fractal dimension )  infinite surface, or length

a value close to an integer from the upper direction is considered as a structure that is nearly filling nothing of the area, but contains infnite lengthes or surfaces
as well

so, and the meaning of a fractal dimension close to the center ( with respect of the 2 possible directions, since fractal dimension is 1dimensional ) ist just
a nice bunch of organic surface, possesing an inifinte surface area

? i do not really know  how the middle (0.5) part can be best described

[Tglad]

Sorry, I'm not sure exactly what you mean, by upper direction and lower direction do you mean the rows and columns of the class table? That would seem to make sense. Lower rows correspond to more connectedness, and further right columns correspond to 'fatter' fractals that fill up more space.
The middle is perhaps the most interesting area because it is farthest from the more boring extremes of emptiness or solidness.

A good thing about classifying fractals is that you can then try and find fractals that don't fit into the categories.
The one below is interesting, like the sponge above it is its own complement, but I don't think it is a sponge. Its fractal border is the thin shell seen earlier in this thread.
A weird one.

[Tglad]

I've worked out the fractal above:
It has airtight pockets of air in it, so it is a foam
It has a mass so is more than a thin foam
It is only just airtight (the air pockets just touch) so is less than a 'normal' foam
So it is a composite foam.
Since it is its own complement, it is also an extended cluster.
So composite foam and extended cluster are the same thing... which slightly confuses the table.

With the additional special cases (of void, solid and partial solid) I was thinking today about a way to categorise the shapes more generally into a more complete table. I think I have the answer, so will post in a separate thread later.

[cKleinhuis]

can we say that a fractal dimension above x.5 is connected ?

[Tglad]

We can't say that, the fractal dimension is a separate property to the class (which is how it connects to itself).
You can have a cantor dust with fractal dimension 2.999 which is disconnected. And you can have foam of fractal dimension < 2.5 which is highly connected.

[visual.bermarte]

Hi! if we apply boolean operators to fractal A and to fractal B (let's say And(A,B)) and A belongs to category 5
(5 is just an example) and B belongs to category 9, is it correct to say that the category would be something like And(5,9)?
Last question: is it possible that a particular fractal/formula could change/morph from one category to another?
Thanks in advance

[Tglad]

I don't think so... for example tree AND sponge could look like a tree if the parts of the sponge happened to line up with the tree, or could be void (empty) if they didn't, or could be cluster if they just crossed at certain locations.
Perhaps there is an AND that can be done per iteration... in which case there might be some logic to the fractals that come out, but it would be logic on the fractal type (as in the v first table in this post) rather than on the class, which is a broader property.

Yes, no reason why a tree couldn't morph into a sponge for instance. Also the mandelbox seems to be an example of where it changes class from one location to another, examples at the bottom of here: https://sites.google.com/site/mandelbox/what-class-of-fractal

[cKleinhuis]

Hi! if we apply boolean operators to fractal A and to fractal B (let's say And(A,B)) and A belongs to category 5
(5 is just an example) and B belongs to category 9, is it correct to say that the category would be something like And(5,9)?
Last question: is it possible that a particular fractal/formula could change/morph from one category to another?
Thanks in advance

to your question about chaging/morphing ... in my eyes this is possible with all fractal types, concerning its starting parameters, for example
the julia set, which can have all the types, dust,connected,solid with just one plain parameter change ( the seed )

beside of that the use of hybrid fractals can define a fractal that uses formula A for the first 1000 iterations, and then formula B for
the following iterations, it is possible to have concrete varying fractal dimensions throughout the iteration ...