Fractal classes

[Tglad]

Hi Alexis, yes it was intentional... I ended up with gas, solid and liquid from the last classification and so thought that fire might be the clue to how to complete it... the main difference with fire being that it moves and changes shape so I thought, maybe the extension is to include animated fractals.
 
I think its about recursive topology if there is such a subject. In topology two loops (handles) is different than one, but in a sponge which is basically made of loops it doesn't matter how dense and numerous the loops are, its the same class. I suppose because they all have infinite number of them.

 Tom

[reesej2]

There's a certain degree of topological thinking in your classification--as was said earlier, the "void" category has zero volume and the "solid" category has a complement with zero volume. The others are harder to speak of topologically, but I would draw the connected/disconnected connection. Also, I'm seeing a distinction between the categories with connected and disconnected complements. In general, it seems like we're classifying the complement of a fractal as much as we are the fractal itself.

A tree or a solid is topologically equivalent to a sphere, yes? But a sponge, shell, foam, or cluster most definitely are not. If I understand correctly, the sponge, shell, and foam are connected, but the cluster isn't. I think that the cluster, shell, and sponge are not compact.

I feel like the classification you've got is really intuitive, so if there aren't topological properties describing it then there should be.

[Prokofiev]

Volume, connectedness and simply-connectedness (no loops) of the set or of its complementary suffices to define most of these shapes.

The problem concern "tree" and "shell", which, to my opinion, share the same properties :
- Volume > 0 (the set and its complementary)
- Connected (the set and its complementary)
- Simply connected (no loops) (the set and its complementary).

If I refer to your definitions, Tom, it is quite difficult, for me, to translate into topological terms what you propose:
- Tree = "...Also can be thought of as many hills on fewer larger hills recursively"
- Shell = "...You can also think of it as recursively many indents inside fewer larger indents"
Take the mathematica logo, for instance : http://www.wolfram.com/mathematica/ is it a tree or a shell ? well...spikes => tree but surfaces => shell. 

[Tglad]

Tree, shell and solid might be in the same topological group, but that doesn't mean they should be in the same fractal class. Trees are like recursive edges and shells like recursive planes, or you could say trees are bumps on bumps and shells are dips in dips.
So this explosion is a tree (only bumps on bumps)

This cave is a shell (only indents in indents)

But a rock or a hilly terrain will usually involve dents and hills, so as a classification it is a tree+shell, just as a sponge including some branches would be a sponge+tree. Just like water is to the elements of the periodic table.

Here's perhaps a sturdier definition:
A tree is a fused unlooping hierarchy of convex solids.
A shell is the complement of a tree.

Does that sound like a better definition?

[reesej2]

I think that's reasonable. I agree, topological properties aren't enough for classifying fractals, in the same way that they aren't sufficient for classifying normal geometrical shapes. Kinda like trying to classify 3D objects as "curved versus flat".

[Tglad]

An interesting question is why fractals are so common in nature. One answer might be that recursive processes are very simple, so will be more common than objects that don't grow in a recursive way. You can imagine the number of genes needed to grow a branching tree shape being fewer than the number needed to grow something into the shape of a house or an electricity pylon.

Another answer would be that certain fractal forms are actually optimal, or 'better' in some way than non-fractal shapes. For example, a river is a tree form, which maximises the area covered for its length. Bird bones are a sponge, which optimises strength to weight ratio.
They are optimal in a broader sense than any individual shape is optimal, more like how an O(n) algorithm is 'better' than an O(n²) algorithm, regardless of the details of the algorithm itself.

So I think that each of the 7 classes of fractal each are optimal for a certain purpose, but I'm not sure exactly what they optimise, here's a guess:

Void-      maximum space
Cluster-  maximum coverage per volume?
Tree-     maximum connected coverage per volume?
Sponge- maximum strength per volume?
Shell-     maximum conduction of heat?
Foam-    don't know
Solid-     maximum connectedness