Fractal classes

[Tglad]

Following on from the Categorising Fractals thread, this is an improved classification I think, all the fractals fit into the table nicely.
Firstly I have added void and solid to the class names to give the full range (even though void and solid aren't fractals as such).
Secondly I am actually using the class names for the size, so instead of thin tree and composite tree you write Void tree and Cluster tree respectively. So a Cluster tree is like a cluster of objects touching each other to form a tree. This format allows a few more classes to be identified it also makes it clear which fractals are actually the same, a Cluster tree is the same as a Tree cluster, in fact these reversed names are removed from the table to leave just the bottom triangle of 28 fractal classes.

So sponge-foam can equally be called foam-sponge etc, and a tree-tree, sponge-sponge etc can be shortened to just tree, sponge etc.

The complement of each fractal is now found by reflecting around the diagonal line.

Notice that the unusual special case of the partial solid is represented as void-solid or a solid-void, this covers space filling curves and fractal tilings. Being on the diagonal line it is its own inverse.

Another object which is its own inverse is the cluster-foam, an example of this is that tricky fractal at the end of the previous thread.

[Prokofiev]

Hello Tom,

So I would say, in topological terms:
- Void => volume = 0 = ok for the whole column
- Solid (anti-void) => volume of complementary = 0 (bubbles of volume 0) => Ok for the whole line
- Cluster => disconnected => Ok for the "cluster" line, but not for the column
- Foam (anti-cluster)=> complementary is disconnected => Ok for the whole line
- Tree => no loops => Ok for the line, but not for the column
- Shell (anti-tree) => complementary has no loops. => works for your Shell-Shell only, does not work for the rest
- Sponge => set and complementary have loops ??
That's all what I have in mind so far.
I am a bit puzzled by the combinations.  Or I am missing something ?
A Sponge-Cluster would be a disconnected set with loops ?...doesn't work.
How would you qualify the differences between your sponge-cluster and your shell tree ?

[Tglad]

Hi Alexis, helpful questions. Reading the name as Column-row here are some examples...
 a Tree-sponge is a fused tree shape that contacts itself to form a sponge, so water could pass through the gaps
 a Tree-shell is a fused tree shape that contacts itself to form a shell, it would hold water
 a Sponge-foam is a fused sponge shape that contacts itself to form a foam, so the gaps are just airtight
So the first word is the fused shape, dimension 3 with a fractal surface, defined by ignoring border-point contacts.
The second word is the more connected fractal that is defined by including border point connections.

So you could take a Tree-tree and turn it into a Tree-sponge by making the branches touch each other. It would only become a Sponge-sponge if the branches fused together, so no border points separating each branch pair.

"How would you qualify the differences between your sponge-cluster and your shell tree ?"
Reading column-row its cluster-sponge and tree-shell. A cluster-sponge is a cluster of solids that contact each other, leaving gaps so is porous, like pebbles on a beach. A tree-shell is a tree with volume that self contacts such that is becomes water-tight but contains no loops.

Actually I think the picture is wrong for the tree-shell, need to find an example for this.

[Prokofiev]

Thanks, 
let's see your answers more closely:

Gaps air tight = Complementary disconnected => ok
water can pass through = complementary is connected.=> ok
I understand "border-points contacts" => ok
So the first word is  defined by ignoring border-point contacts.=> ok
The second word is  defined by including border point connections.=> I get it

what do you call a "fused shape" exactly ?
isn't your void-shell a void-sponge ? it lets water pass through.
I don't se the tree in your tree-sponge (but I see the sponge), if i remove the border points I don't see the tree.
Can the sponge-shell hold water ? how is it a shell?

[Tglad]

The tree-tree in the table is an example of a "fused shape". I mean that it is made of one solid with no border points. It you take 2 pebbles and press them together they are connected (at border points) but not fused. They are not one solid and they can fall apart. The mandelbrot is not a fused shape (it is a cluster tree), the solid koch snowflake is fused (it is a tree tree).

"isn't your void-shell a void-sponge ? it lets water pass through."
No, it is water tight, if you put it on an angle it could act as a bucket. There is a very similar looking fractal though (#12 in the 3x3x3 table) which is a void sponge.
"I don't see the tree in your tree-sponge (but I see the sponge)"
This is not a very clear example, but it is actually 4 trees that touch each other. See how each corner looks like a castle with little turrets coming off bigger turrets, so ignoring contacts it is a fused tree.
"Can the sponge-shell hold water ? how is it a shell?"
Yes it would hold water like a bucket if tipped on an angle. There are no holes between the 8 octants. It is 8 shells facing outwards, each one like a bucket.

I have finally managed to produce this nice example of a tree shell. (previous table also updated).

[Prokofiev]

Your new tree-shell is nice ! And I see now how it can be called tree-shell. 
I think I understand more of your ideas now, but your concepts of classification are not easy to grasp (am I the only one ?)
And the "foam" line examples look quite unconvincing to me (so far). 

I think, to illustrate this concepts more clearly, a classification chart of 2D fractals would help.

[cKleinhuis]

i like this classification very much, because it is explaining the meaning of fractal dimension visually

[Bent-Winged Angel]

i like this classification very much, because it is explaining the meaning of fractal dimension visually

DITTO!  A pciture is worth a thousand words. I had no idea that my sponge was a sponge til tglad posted these.

[Tglad]

Good idea, so I made a 2d chart. Foam and shells don't exist in 2d.
It looks clearer to me, and each is the complement of its diagonally opposite. (the tree solid is a recentred complement of the void tree).

"I think I understand more of your ideas now, but your concepts of classification are not easy to grasp (am I the only one ?)"
I probably just need some more and better examples (I'm doing up the natural version now), I'm trying to cover all basic fractal shapes, so if you have any fractals which you're unsure which class it is, then post it here.

[Prokofiev]

Thanks, Tom. I think this clarifies a lot the concepts you chose !
I was wondering what your void-solid could be. Until I realized it could be a set composed of all the points with rational coordinates in the interval [0,1]x[0,1].
It is countable, so its Hausdorff dimension is zero => void.
yet, it is dense in this interval, its closure is the interval itself => solid.

Now some guesses (see my list if necessary: http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension):
the Hilbert curve => tree-solid ?
von Koch Curve (not the filled-in snowflake) => void tree ?
Apollonian gasket, Newton fractal => void-sponge ?
Douady rabbit => void-sponge ?
but filled-in Douady rabbit  => Cluster-solid or cluster-tree?
Sierpinski triangle => void sponge ?
but its complementary (the inner triangles) => Cluster solid ?

Now what if we consider those 2D fractals embedded in 3D space ?
Wouldn't the Hilbert curve be a tree-shell ? What about the filled-in Douady Rabbit or the filled-in Sierpinski triangle ?
I think we still lack precise definitions for all terms, (our eyes can betray us on the matter of fractals)........especially for the difficult ones : "Shell" and "foam".

[Tglad]

Thanks again for the helpful feedback Alexis  
You might be right about the void solid, I think it describes well the space filling curves and fractal tilings. You could say that there is no dense solid, and no 'dense' volume of space either. Every two 'inside' points have an outside point between them and visa versa.

the Hilbert curve => void-solid
von Koch Curve (not the filled-in snowflake) => surface of 2d tree-tree  (or void tree with no branches, i.e. 'order 0' tree)
Apollonian gasket, Newton fractal => void-sponge
Douady rabbit => surface of a cluster-tree (or void-sponge)
but filled-in Douady rabbit  => cluster-tree
Sierpinski triangle => void sponge
but its complementary (the inner triangles) => Cluster solid

"Now what if we consider those 2D fractals embedded in 3D space ?"
Most stuff stays the same, but for 2d we replaced 'shell' with 'solid', so back to 3d need to do the opposite, so:
the Hilbert curve => void-shell (shell has no branches, ie is an 'order 0' shell)
complementary Sierpinski triangle (the inner triangles) => Cluster shell (shell has no branches)

Yes, more precise definitions would be good, I think I am categorising recursive homeomorphisms. Perhaps another way to define it is:
                                                         tree-foam
                                                           ^     ^
complement of fractal is complement of this      This is the fractal topology

Anyway, whatever the exact definition, the table is really interesting, if I rotate it 45 degrees then it classifies real world structures in a very natural way:

(hi res https://docs.google.com/leaf?id=0B8S7Si-yu3DoNTNkMjNkYmQtMjAyYi00ZDJlLTg5MTUtODNkODFkY2MxOTg1&hl=en)

[Tglad]

I've been experimenting with different ways to render cubic fractals (they tend to cover most structures), and have made a few using paint3D.
These are both tree-shells, which is a self-complement class, the first looks a bit like the koch snowflake which is also in a self-complement class in 2d.
The second is actually a tiling tree shell. You can view them in real time 3d by downloading the trial version of paint3d and loading the voxel files: https://docs.google.com/leaf?id=0B8S7Si-yu3DoY2Q3ODBiOWMtZmY4YS00Mzc5LTk4MmItZmMyYWQ3ZGYwYWQ0&hl=en and https://docs.google.com/leaf?id=0B8S7Si-yu3DoMTJiM2VkMGEtYTVjNC00ODlkLTljYmQtMmQ1NGQ2YTc2NDA5&hl=en

A good natural example of a tree shell is mountains. If mountains were just small mounds on top of larger mounds etc they would be a tree tree, if they were small basins in larger basins etc they would be a shell shell, but mountains are made of mounds and basins so are a tree shell.
Rocks are also tree shells, because they have broken away from other rocks so contain bulges and dips at all scales.

[Tglad]

So these are the 7 forms that you get in 3d, I used a few different pictures for variety. Jotero I borrowed one of your fractals, hope thats ok 

[Tglad]

I have finally completed my classification by adding a whole new element (that extends the table into a 3d tetrahedron) animating fractals!
The obvious gap in fractals approximating nature is that things in nature change over time, so a good classification should classify not just the shape of objects but how they evolve over their lifespan. This is different to just classifying 4d fractals.

So this extended table now classifies things like bubbling mud, boiling lava, and even the life cycle of mushrooms.
Too much to stick in a post so I wrote it all here: https://sites.google.com/site/simplextable/


Any suggestions, questions or discrepencies do let me know. Not all the pictures are great example cases, hard to find in some cases.
   what an unusual hobby I have

[Prokofiev]

Hello Tom,
Well...another layer to your classification, introducing time.
I read it through very quickly, interesting. You define each term, clearly, now.
I still regret there is still no link to topology, so far. I'll give it a thought and give my feedback.
Interesting also that we find at the four corners of your 84-classification chart: fire, water, air and earth...that reminds me of something quite ancient  A look towards the ancient greek basic elements ? Was that intentionnal ?