AVirtualMachineForExploringSpaceTimeAndBeyond : The Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section- (L'eponge de Menger -iteration 5- avec une transformation conforme (4xO+1)/(1xO-1) dans l'ensemble des octonions -section tridimensionnelle-).

The Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section- [L'éponge de Menger -itération 5- avec une transformation conforme (4xO+1)/(1xO-1) dans l'ensemble des octonions -section tridimensionnelle-].

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...

Moreover an amazing cross-section can be made using the plane:

                    2X - 2Y + 2Z - 1 = 0

the origin of the coordinates being at the center of the main cube and the axis being parallel to its sides.

This process can be generalized in many different ways and for example:

                      3     3     3
                    2X  - 2Y  + 2Z  - 1 = 0

(the curved one) or again:

                          1  2         1  2         1  2    2
                    (X - ---)  + (Y - ---)  + (Z - ---)  = R
                          2            2            2

(the spherical one).

The "standard" Menger sponge:

See some related pictures (possibly including this one):

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Some non linear transformations of the "standard" Menger sponge:

See some related pictures (possibly including this one):

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The complement of the "standard" Menger sponge:

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Some extended Menger sponges: Two ways of extending the "standard" Menger sponge. On the one hand one can change the used volume (from a cube to a sphere for example). On the other hand one can change the rules of subdividing each cube as well as their numbers...

See some related pictures (possibly including this one):