A parallelepipedic extended Menger Sponge -iteration 4- [Une éponge de Menger généralisée parallélépipédique -itération 4- ]

A parallelepipedic extended Menger Sponge -iteration 4- [Une éponge de Menger généralisée parallélépipédique -itération 4-].




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

A parallelepipedic extended Menger Sponge -iteration 1- A parallelepipedic extended Menger Sponge -iteration 2- A parallelepipedic extended Menger Sponge -iteration 3- A parallelepipedic extended Menger Sponge -iteration 4- A parallelepipedic extended Menger Sponge -iteration 5- 
(SubdivisionRules=TTTTT TFTFT TTFTT TFTFT TTFTT TFTFT TTTTT  TFTFT FFFFF FTFTF FFFFF FTFTF FFFFF TFTFT  TTTTT TFTFT TTFTT TFTFT TTFTT TFTFT TTTTT)






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):

The Menger Sponge -iteration 0- The Menger Sponge -iteration 1- The Menger Sponge -iteration 2- The Menger Sponge -iteration 3- The Menger Sponge -iteration 4- The Menger Sponge -iteration 5-
The Menger Sponge -iteration 1- The Menger Sponge -iteration 2- The Menger Sponge -iteration 3- The Menger Sponge -iteration 4- The Menger Sponge -iteration 5- ==> A tridimensional intertwining made of the Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of the Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of the Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of the Menger Sponge -iteration 5- inside a sphere Untitled 0318 Untitled 0319
An amazing cross-section inside the Menger Sponge -iteration 1- An amazing cross-section inside the Menger Sponge -iteration 2- An amazing cross-section inside the Menger Sponge -iteration 3- An amazing cross-section inside the Menger Sponge -iteration 4- An amazing cross-section inside the Menger Sponge -iteration 5- ==> An amazing cross-section inside the Menger Sponge -iteration 5- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- An amazing cross-section inside the Menger Sponge -iteration 5- with a O^2conformal transformation in the octonionic space -tridimensional cross-section- An amazing cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section- Untitled 0234
Empty Empty Empty Empty Empty >>> A tridimensional intertwining made of an amazing cross-section inside the Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of an amazing cross-section inside the Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of an amazing cross-section inside the Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of an amazing cross-section inside the Menger Sponge -iteration 5- inside a sphere Untitled 0314 Untitled 0315
Empty Empty Empty Empty An amazing generalized cross-section inside the Menger Sponge -iteration 5- ==> An amazing generalized cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
Empty Empty Empty Empty A spherical cross-section inside the Menger Sponge -iteration 5- ==> A spherical cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
Empty Empty Empty Empty A double spherical cross-section inside the Menger Sponge -iteration 5-
Empty Empty Empty Empty A thin double spherical cross-section inside the Menger Sponge -iteration 5-
Empty Empty Empty Empty A 'pyramidal' cross-section inside the Menger Sponge -iteration 5-, 'Ô temps tes pyramides' -a Tribute to Jorge Luis Borges- ==> A 'pyramidal' cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-, 'Ô temps tes pyramides' -a Tribute to Jorge Luis Borges-
Empty Empty Empty Empty A 'double conic' cross-section inside the Menger Sponge -iteration 5-, 'Ô temps tes pyramides' -a Tribute to Jorge Luis Borges- ==> A 'double conic' cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
Empty Empty Empty Empty A 'conic' cross-section inside the Menger Sponge -iteration 5-, 'Ô temps tes pyramides' -a Tribute to Jorge Luis Borges- ==> A 'conic' cross-section inside the Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
Empty Empty Empty Empty The intersection of the Menger Sponge -iteration 5- and of a quadridimensional Calabi-Yau manifold -tridimensional representation- ==> The intersection of the Menger Sponge -iteration 5- and of a quadridimensional Calabi-Yau manifold -tridimensional representation- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-




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

See some related pictures (possibly including this one):

The Menger Sponge -iteration 1- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The Menger Sponge -iteration 2- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The Menger Sponge -iteration 3- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The Menger Sponge -iteration 4- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The Menger Sponge -iteration 5- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-
Empty Empty Empty Empty The Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-


Empty Empty Empty Empty The Menger Sponge -iteration 5- with a tridimensional non linear transformation ==> Untitled 0230
Empty Empty Empty Empty The Menger Sponge -iteration 5- with a tridimensional non linear transformation

Empty Empty Empty Empty An amazing cross-section inside the Menger Sponge -iteration 5- with a tridimensional non linear transformation ==> Untitled 0231 -a Tribute to Philippe Druillet-
Empty Empty Empty Empty An amazing cross-section inside the Menger Sponge -iteration 5- with a tridimensional non linear transformation




The complement of the "standard" Menger sponge:

See some related pictures (possibly including this one):

The complement of the Menger Sponge -iteration 1- The complement of the Menger Sponge -iteration 2- The complement of the Menger Sponge -iteration 3- The complement of the Menger Sponge -iteration 4- The complement of the Menger Sponge -iteration 5-
An amazing cross-section inside the complement of the Menger Sponge -iteration 1- An amazing cross-section inside the complement of the Menger Sponge -iteration 2- An amazing cross-section inside the complement of the Menger Sponge -iteration 3- An amazing cross-section inside the complement of the Menger Sponge -iteration 4- An amazing cross-section inside the complement of the Menger Sponge -iteration 5-




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):

An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- Empty Empty
An amazing cross-section inside an extended Menger Sponge -iteration 1- An amazing cross-section inside an extended Menger Sponge -iteration 2- An amazing cross-section inside an extended Menger Sponge -iteration 3- Empty Empty


An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section- An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
Empty Empty Empty Empty Empty >>> A tridimensional intertwining made of an extended Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of an extended Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of an extended Menger Sponge -iteration 5- inside a sphere A tridimensional intertwining made of an extended Menger Sponge -iteration 5- inside a sphere Untitled 0320 Untitled 0328
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-


An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An extended Menger Sponge -iteration 1- An extended Menger Sponge -iteration 2- An extended Menger Sponge -iteration 3- An extended Menger Sponge -iteration 4- An extended Menger Sponge -iteration 5- ==> An extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
A parallelepipedic extended Menger Sponge -iteration 1- A parallelepipedic extended Menger Sponge -iteration 2- A parallelepipedic extended Menger Sponge -iteration 3- A parallelepipedic extended Menger Sponge -iteration 4- A parallelepipedic extended Menger Sponge -iteration 5- ==> A parallelepipedic extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-


A full-random extended Menger Sponge -iteration 1- A full-random extended Menger Sponge -iteration 2- A full-random extended Menger Sponge -iteration 3- A full-random extended Menger Sponge -iteration 4- A full-random extended Menger Sponge -iteration 5- ==> A full-random extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section- Untitled 0272
A half-random extended Menger Sponge -iteration 1- A half-random extended Menger Sponge -iteration 2- A half-random extended Menger Sponge -iteration 3- A half-random extended Menger Sponge -iteration 4- A half-random extended Menger Sponge -iteration 5- ==> A half-random extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
An amazing cross-section inside an extended Menger Sponge -iteration 1- An amazing cross-section inside an extended Menger Sponge -iteration 2- An amazing cross-section inside an extended Menger Sponge -iteration 3- An amazing cross-section inside an extended Menger Sponge -iteration 4- An amazing cross-section inside an extended Menger Sponge -iteration 5- ==> An amazing cross-section inside an extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-
A half-random extended Menger Sponge -iteration 1- A half-random extended Menger Sponge -iteration 2- A half-random extended Menger Sponge -iteration 3- A half-random extended Menger Sponge -iteration 4- A half-random extended Menger Sponge -iteration 5- ==> A half-random extended Menger Sponge -iteration 5- with a (4xO+1)/(1xO-1) conformal transformation in the octonionic space -tridimensional cross-section-




The fractal "standard" Menger sponge:

See some related pictures (possibly including this one):

The eroded complement of the Menger Sponge -iteration 2- The eroded Menger Sponge -iteration 3-

The erosion of the Menger Sponge -iteration 1- The erosion of the Menger Sponge -iteration 2- The erosion of the Menger Sponge -iteration 3- The erosion of the Menger Sponge -iteration 4- The erosion of the Menger Sponge -iteration 5-  
The erosion of an amazing cross-section inside the Menger Sponge -iteration 1- The erosion of an amazing cross-section inside the Menger Sponge -iteration 2- The erosion of an amazing cross-section inside the Menger Sponge -iteration 3- The erosion of an amazing cross-section inside the Menger Sponge -iteration 4- The erosion of an amazing cross-section inside the Menger Sponge -iteration 5-  
The erosion of an amazing cross-section inside the Menger Sponge -iteration 1- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The erosion of an amazing cross-section inside the Menger Sponge -iteration 2- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The erosion of an amazing cross-section inside the Menger Sponge -iteration 3- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The erosion of an amazing cross-section inside the Menger Sponge -iteration 4- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section- The erosion of an amazing cross-section inside the Menger Sponge -iteration 5- with a 1/O conformal transformation in the octonionic space -tridimensional cross-section-




The "pi" Menger sponge:


The rule set "RRR RRR RRR RRR RRR RRR RRR RRR RRR" (each "R" uses the next digit -base 2- of 'pi' -the used colors code the reverse rank of the digits-) is used for the following pictures:


 An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'  
27 digits.

Each full cube codes a '1', when each empty/missing one codes a '0'. This picture displays the 27 first digits -base 2- of 'pi' (110 010 010 000 111 111 011 010 101). For example the first triplet "110" appears on the front lower left as two full cubes and an empty/missing one. 		 An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'  
378 digits. 		 An extended Menger Sponge -iteration 3- displaying the 5.508 first digits -base 2- of 'pi'  
5.508 digits. 		 An extended Menger Sponge -iteration 4- displaying the 53.919 first digits -base 2- of 'pi'  
53.919 digits. 		 An extended Menger Sponge -iteration 5- displaying the 1.168.749 first digits -base 2- of 'pi'  
1.168.749 digits.
 An extended Menger Sponge -iteration 6- displaying the 15.750.801 first digits -base 2- of 'pi'  
15.750.801 digits. 		 An extended Menger Sponge -iteration 7- displaying the 211.210.335 first digits -base 2- of 'pi'  
211.210.335 digits.


the digits base 2 are used with the following convention: 
                    0 --> F
                    1 --> T


See some explanations regarding the digit encoding:


 An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi'  
27 digits.

Each full cube codes a '1', when each empty/missing one codes a '0'. This picture displays the 27 first digits -base 2- of 'pi' (110 010 010 000 111 111 011 010 101). For example the first triplet "110" appears on the front lower left as two full cubes and an empty/missing one. The first digit ("1") is enhanced using a higher luminance. 		 An extended Menger Sponge -iteration 2- displaying the 378 first digits -base 2- of 'pi'  
378 digits.

Each full cube codes a '1', when each empty/missing one codes a '0'. This picture displays the 378 first digits -base 2- of 'pi' (110 010 010 000 111 111 011 010 101 000 100 010 000 101 101 000 110 000 100 011 010 011 000 100 110 001 100 110 001 010 001 011 100 000 001 101 110 000 011 100 110 100 010 010 100 100 000 010 010 011 100 000 100 010 001 010 011 001 111 100 110 001 110 100 000 000 100 000 101 110 111 110 101 001 100 011 101 100 010 011 100 110 110 010 001 001 010 001 010 010 100 000 100 001 111 001 100 011 100 011 010 000 000 100 110 111 011 110 111 110 010 101 000 110 011 011 001 111 001 101 001 110 100 100 001 100). The 3x3x3=27 digits following the first digit ("1") are enhanced using a higher luminance.









Definition of the Sierpinski carpet (related to the Cantor triadic set)

A square is cut into 3x3=9 identical smaller squares. Then the central subsquare -grey- is removed. At last this process is iterated recursively with the 9-1=8 remaining subsquares. The fractal dimension of the Sierpinski carpet is equal to: 
                     log(8)
                    -------- = 1.892789260714372
                     log(3)


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

The Sierpinski Carpet -iteration 0- The Sierpinski Carpet -iteration 1- The Sierpinski Carpet -iteration 2- The Sierpinski Carpet -iteration 3- The Sierpinski Carpet -iteration 4- The Sierpinski Carpet -iteration 5-  
Empty Empty Empty Empty Empty The Sierpinski Carpet -iteration 1 to 5-  
Empty Empty Empty The Sierpinski Carpet -iteration 3- with colors -a Tribute to Karl Menger and Piet Mondrian- Empty Empty


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