An extended Menger Sponge -iteration 5- [Une éponge de Menger généralisée -itération 5-].




Definition of the "standard" Menger sponge (related to the Cantor triadic set): A cube is cut into 3x3x3=27 identical smaller cubes. Then the 7 central subcubes (6 for each face and 1 at the center of the cube) are removed. At last this process is iterated recursively with the 27-7=20 remaining subcubes. The fractal dimension of the Menger sponge is equal to: 
                     log(20)
                    --------- = 2.726833027860842...
                     log(3)
The "standard" Menger sponge can be defined by means of subdivision rules. Here is the way how each of the 27 cubes of the "standard" Menger sponge at a given level is subdivided: 
                    
                   "standard"  Menger sponge
                     _____________________
                    /                     \
 
                    TTT       TFT       TTT
                    TFT       FFF       TFT
                    TTT       TFT       TTT
 
                    \_/
 
             Sierpinski carpet
or again: 
                    TTT TFT TTT  TFT FFF TFT  TTT TFT TTT
where 'T' ('True') and 'F' ('False') means respectively "subdivide the current cube" and "do not subdivide and destroy the current cube". The rules are repeated at each level, but they can be changed periodically and for example: 
                    
                    TTT TFT TTT  TFT FFF TFT  TTT TFT TTT   FFF FTF FFF  FTF TTT FTF  FFF FTF FFF
 
                    \___________________________________/   \___________________________________/
 
                          "standard"  Menger sponge                      complement
alternates the "standard" Menger sponge and its complement. Obviously many other rules do exist as shown below...

Beside 'F' and 'T' some other possibilities exist: 'R' that means "subdivide the current cube" or "do not subdivide and destroy the current cube" Randomly with a given threshold between 0 and 1 (0.5 being the default value) and 'S' that means "Stop subdividing". Obviously 'F', 'T', 'R' and 'S' can be mixed at will...






See the first objects of this family (including this one):


 
Iteration 0, 0 rule. 		 
Iteration 1, 27 rules. 		 
Iteration 2, 567 rules. 		 
Iteration 3, 11367 rules. 		 
Iteration 4, 227367 rules.
 
Iteration 5, 4547367 rules.



Here, the subdivision rules are: 
                    [R1] : TTT TFT TTT  TFT FFF TFT  TTT TFT TTT
except for the smallest cubes where they are: 
                    [R2] : FFF FTF FFF  FTF TTT FTF  FFF FTF FFF
Please note that R2 is the complement of R1...


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