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Hi everyone, I decided it would be a good idea to try and categorise 3d fractals. I know there are types, such as menger sponge, but what are the different possibilities and can they be grouped?
So I made a list of all the symmetric fractals of a 3x3x3 cube, there are 81 fractals, then I tried naming and grouping them, what do you think of the names and groups? You can change and correct them here https://docs.google.com/document/edit?id=1bz0d4RZ9sg_MKJ-3CsIrLxbYUtYOTjKrGLqhsvq3nxE&authkey=COPg2ho&hl=en# (https://docs.google.com/document/edit?id=1bz0d4RZ9sg_MKJ-3CsIrLxbYUtYOTjKrGLqhsvq3nxE&authkey=COPg2ho&hl=en#)

[The cefm codes stand for the eight corner cubes (c), twelve edge cubes (e), six face cubes (f) and single middle cube (m). And the trinary number (such as 1201) is whether each cube (cefm) is either outside the set (0), inside the set (2) or is recursed (moved to the middle and scaled by 3).]

Nice initiative.
This also make me ask how to give a name to a (an hybrid) fractal? I think that, for the realm of 3D fractals,  the shortest name would be the generating algorithm (or a representation of it). Reminds me a little bit the story of carbon-based compounds naming :).
Is the "ternary code" a representation of the generating algorithm?

Thanks knighty, yes the ternary code defines how it was generated, its described a little better in the link.
What I want to do is to group the fractals and so name each class based on the way it self connects. For example, a tree fractal may look like the mandelbrot set, or a tree or a fern, but it just classed as 'tree'. A bit like topology where a cup and a donut are both just toruses. So each hybrid would also fit into one of these classes.

An interesting fractal in the list is 0111, below. I call it a shell and it is like branching planes. Its shape seems naturally stiff, and with some thought it looks like it encases a tree shape, it would also hold water.
Any animal that is tree-like in its topology could protect itself in a 'shell' fractal.. but not necessarily this particular one :)

That's my favorite Fractal.  I did hand drawings of this (2d), also with apophysis, and Photoshop.  I was going to do this with Mandelbulb3D as well but you beat me to it.  It's just a julia set pushed down on the x-axis right, or was that the y-axis?

It was my first fractal which I put into FractalForums gallery  :D. If I remember I rendered this by mistake when I tried to render Sierpinski triangle.

(http://www.fractalforums.com/gallery/1/thumb_640_22_11_09_10_07_04.jpg)
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=1082

NICE! :o  Can I get the code to that?  Please! :D  I'll give you credit, I just want to mess with it and see what I can get it to look like my other ones like this.

Where are there high resolution versions of the other ones on the chart? And what about the asymmetrical cubes?

I'm sorry. I didn't save that code. It will be difficult to remind because it was a year ago.

No problem, it just means I need to try and find it.  Thanks anyways.

Nice render Buddhi, I hadn't ever seen that fractal before. It definitely isn't a tree, and it isn't a sponge, it is in its own class of fractals, which seem to be rarely seen in fractal renders. There is one example in http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension (http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) the 3d greek cross fractal which is also a 'shell' (let me know if there's a better name).

A good natural example is cracks in rocks. 2d cracks are trees, but 3d cracks are not, they're usually fractal planes. The tree is actually the rock which is cracked.

A walnut is a 'tree' since the main lump is composed of 2 lumps, which are composed of many lumps. So the nutshell is a 'shell' as it encases a tree, and holds water.
The containing volume of the seashell below is a spiral which is a unary tree with each child segment rotated and connected by the tubes. So the seashell itself encases a 'tree' fractal, so is a 'shell' fractal.

It's a 3D version of these styles(   Vicsek fractal (http://"http://en.wikipedia.org/wiki/Vicsek_fractal") maybe) that I have produced:
Though my way was by crossing lines.  The bottom one was by Apophysis which was done by shifting a julia set down the axis.
(http://www.fractalforums.com/gallery/3/2367_23_07_10_4_08_32.jpeg)
(http://www.fractalforums.com/gallery/3/2367_30_07_10_2_02_35.jpeg)
(http://www.fractalforums.com/gallery/3/2367_30_07_10_3_10_59.jpeg)

@matsoljare- on the link you can copy and paste the images outside the table and enlarge them, only low quality 320x320 though. I only looked at symmetrical to get an idea of the basic flavours of fractal you get from rules on edge,corner,faces and middle, keeps it simple.
@thunderwave- nice image will you post your 3d version on the gallery?

So back to classes of fractal, there is another class you can see from the table (#39, 53, 77) which is almost never seen in fractal renders, it is the 'foam' fractal which is like lots of bubbles, or a solid with fractal air pockets in it. So it doesn't absorb water like a sponge, it doesn't hold water like a shell, but is airtight.

Like the other classes, the thin version has no volume (like soap foam), the normal version has volume (like pumice stone maybe) and the fat version it solid but containing fractal dots (cantor dust) inside; the air pockets have radius 0.

So here it is, my categorisation of fractals
(http://www.fractalforums.com/gallery/3/853_12_10_10_2_42_17.jpeg)
Each image is a single example for each class.
How does the vocabulary sound?
I think it covers all 81 types in the 3x3x3 table and covers those in http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension (http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension), so you can help me proof it by giving me a fractal and I'll see if it fits :D

This is the same as those pictures and the hand drawn version; I just gave a more interesting perspective on it, made with M3D.
(http://www.fractalforums.com/gallery/3/2367_12_10_10_7_03_42.jpeg)

Hello Tom,
Interresting work.  ;D
Correct me if i'm wrong : We have here the Menger sponge (=1100) and the Cantor dust (=1000), right ?
With your classification, the Hausdorff dimension is easy to calculate:
1) If cefm includes a "2" then dimH = 3. (It cannot be considered as fractal, by the way).
2) otherwise : dimH = log (8c+12e+6f+m)/ log(3)

Hi Prokofiev, thanks.
 100% correct about the sponge and the dust. Also nice work on the fractal dimension! So thin foam (1110) would have fractal dimension 2.97 about.
 Yes if includes 2 the dimension is 3, and so its volume isn't fractal, but its surface still is. So much like we coloquially call the mandelbrot set a fractal, I'm ok with calling these fractals, even though only part (the boundary) is fractal.
@thunderwave- nice fractal, that comes under 'thin tree'

An even more unusual and bizarre class emerges from the initial table, the fractal 1111. 0 means empty space that things can pass through, 2 means solid volume with a rigid shape, 1 means recurse and represents border points on the final fractal, e.g. a thin tree has no solid, just border points.
So 1111 is like a sea of border points. Like a solid it contains no gaps and has a known volume (27m^3) in this case; but it contains no solid, so nothing that is defined as rigid, like space, solids could move around in it. So to me it seems like a good description of liquid!
(This view works on the assumption that we can 'bend' the fractal freely as long as the measures of the shape don't change; the length of lines, the area of surfaces, the shape of solids etc.)

Such fractals exist elsewhere, all the space filling curves are seas of border points. I call them 'partial solids'.

Thanks Tglad!
You inspired another fractal from me type I call: "snowflake, fern, tree".

(http://www.fractalforums.com/gallery/3/2367_13_10_10_12_27_03.jpeg)
(http://www.fractalforums.com/gallery/3/2367_13_10_10_3_41_47.jpeg)

Tom, I don't think 1111 is different from 2222, They both lead to a simple cubic volume.
And if "1" means "recurse", then "1 represents border points on the final fractal" works only if the final fractal has no volume. I think

Other results concerning, this time, volume:
- If cefm contains no zero => volume= 1
- If cefm contains no "2" => volume=0 (except for 1111)
- if cefm contains only "0" and "2" => volume = (8c+12e+6f+m)/27  for the positions of the "2"s
- Who will come up with the right formula for the volume of a fractal containing "0", "1" and "2"s ?

the formula for the volume of a 3x3x3 "cefm set" is :

volume = n_2 / (27 - n_1)

where n_k = 8c_k+12e_k+6f_k+m_k ,
and c_k = 1 if there is a "k" in position "c", otherwise c_k = 0.
In other words, n_k is the number of "k-cubes" (cubes of type k)

Why ? This derives from the fact that:
- the volume of a "1-cube" is 1/27 of the whole set
- the volume of a "2-cube" is 1/27
so v = n_2/27 + v*n_1/27

Some examples, maybe ?
Full solid cube 2222, made of 27 "2-cubes" => n_2 = 27 and n_1=0 => v = 1
Menger Sponge 1100, made of 20 "1-cubes" => n_2 = 0 and n_1 =20 => v= 0
or the composite sponge 1022 (see attached image)=> n_1 = 8 and n_2 = 7 => v=0.3684

note : the volume of the 1111 set cannot be calculated that way, because of the division by zero. (still, its volume is 1)

Hey that's awesome Alexis, I think I'll add those formulas to the google doc.
I agree that 1111 is a dense volume, but of what? It isn't 2 which we define as a dense volume of rigid solid, or 0 which is a volume of empty space. The inverse of 2222 is 0000 whereas the inverse of 1111 is itself, it is not the opposite of empty, a class of its own I think.
We could equally define 0 as yellow and 2 as red, 1111 is not the same as a volume of red.
The boundary of a 4d object has a volume of border points so it can happen... but I certainly find it all quite puzzling  :hmh:

@thunderwave, that is a thin sponge, or thin tree if you consider the overlaps to be circumstantial.

So here's a question, which classes of fractal are in this picture? (answer in the gallery  :) )

Actually, Tom, I think you should try to define your categories precisely in terms of topology. I think it is a good approach.
Here are some basic ideas:
- Connected / disconnected set (totally or not)
- Holes / infinity of holes
- Branches / no branches
- "1 or 2-dimensionnal topology" (Ugly, i am not a specialist of topology, my idea is "tree" or "shell"-like structure). I think, by the way, that a totally disconnected set can be seen as a "0-dimensional topology".
- volume =0 or > 0
and think those criteria also for  
- its boundary and
- its complementary

for example:
- Cluster = disconnected (totally or not?)
- Tree => 1-dimensionnal topology + no holes + branches
- Sponge => 2-D topology + holes + complementary connected
- Shell => 2-D topology + no holes (=> complementary necessairily connected)
- Foam => 2-D topology + holes + complementary disconnected (totally?) => The opposite of the "Cluster"
We could add :
- "Lace" => 1-dimensionnal topology + holes + branches (like the newton fractal)
- "Path" => 1-dimensional topology + no holes + no branches (like the Koch curve)

From "thin" to "fat":
- "Thin" => volume=0
- "Fat" => volume of complementary = 0

That's a nice Topic. I think we could find help somewhere. Surely this thing has been studied before. I'll have a look.

I've seen ALL of those types in Mandelboxes with different scales, but couldn't find the 1D ones...

...until I tried partial Julias (update: z_n+1 = z_n + k*z₀, where k=0 for Julia, k=1 for original Mandelbox) and found the missing 1D ferns. ;D
scale=-1.77, k=0.623:
(http://www.ms.mff.cuni.cz/~kadlj3am/big/files/mandelbox_fern.jpg)

Nice find! To be clear with my words, the initial table are _types_ of fractal, the 5x5 table are of _classes_ of fractal, with one example fractal type in each.
So sierpinski tetrahedron, menger sponge and 2d newton are different types of fractal, but they are all in the 'thin sponge' class.

I also noticed many classes of fractal in the mandelbox, you're right I don't recall seeing thin trees... yet! It would be interesting to see how many exist in the basic mandelbox formula (where scale and minR can be chosen).

Alexis, I don't know much about topology either, but I have tried looking up fractal categories, and compared to euclidean geometry categorisation is very sparse.. there are type names (menger sponge etc) and methods (escape time, IFS etc) but not much more than that... at least not that I found.

A nice thing about these classes is you don't care about the exact shape, so you can describe the real world
(http://www.fractalforums.com/gallery/3/853_14_10_10_8_03_19.jpeg)

OK, clouds don't look like 3d cantor dust, but if you use a random version and make it denser by removing slices that are smaller, then you get something which varies in density and made of negligeable sized points (water droplets), which describes the structure of clouds well.

The top right is moon and stars, this is a perfect example of a cluster, a few very large spheres (e.g. betelgeuse), many smaller stars, and even more planets, moons, asteroids, rocks etc.

I classified the hills as a tree... this is because the big lump (the world) has smaller lumps on it (big hills) which each have many smaller lumps on them etc. So it is a (normal) tree but with short, wide branches. For a similar reason a solid koch snowflake is a tree, and so a koch curve is a 2d tree border.

So am I getting this right:
Our lungs could be considered a sponge and a shell, like veins (shell or tree or sponge, or all) and the Skin (Sponge or Shell)?  I think so many overlap, like the fern could be a tree.

A fern fractal (the type) is a tree (the class).
A fern plant is also a tree (the class)
Lungs are complicated things, but I think they are a type of shell, since the bit that isn't lung (the air) is in the shape of a tree.
Veins are trees... ok the viens connect to arteries to form a loop but this only happens once at the largest scale, it isn't fractal loops, so its a tree.
Skin doesn't have a fractal description that I can think of.

Thanks that clarifies a lot for me!

It took a while but was fun to do, the fractal table with real examples-
(http://www.fractalforums.com/gallery/3/853_16_10_10_2_29_36.jpeg)
hi res https://sites.google.com/site/mandelbox/classes-of-fractal (https://sites.google.com/site/mandelbox/classes-of-fractal)

Can you see the fractal in each image?

Awesome job Tglad!!  I see the fractal in most.  I had to zoom to see the others.

Now that I have a table of classes we can see how popular different classes of fractal are.
If I google image search '3d fractal', this is how many I find on the first few pages:

	thin	composite	normal	extended	fat
cluster	3	0	2	0	1
tree	3	5	14	0	0
sponge	12	0	0	0	0
shell	0	0	0	0	0
foam	0	0	0	0	0

Partial solid: 0

So we rarely see anything but trees or thin sponges.

Admittedly clusters, foams and 'fat' fractals are perhaps less interesting to render but it looks like shell fractals could be a big uncharted area of interesting shapes.

Perhaps people already live in a shell....  teehee... :D  I know I do.

I like the idea of exploring shells though.  Exactly how does one create them?  like this:
(http://www.fractalforums.com/gallery/3/2367_13_09_10_2_08_47.png)?

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>

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

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.

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.

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

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.

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

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 (https://sites.google.com/site/mandelbox/what-class-of-fractal)

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 ... :)