An extended Menger Sponge -iteration 5- displaying the 991818 first digits -base 2- of the Champernowne number (=0.1 10 11 100 101 110 111 1000...) -using all base 2 integer numbers- [Une éponge de Menger généralisée -itération 5- visualisant les 991818 premières 'décimales' -base 2- du nombre de Champernowne (=0.1 10 11 100 101 110 111 1000...) -utilisant tous les nombres entiers en base 2-].




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...


Obviously one can use a specific rule for each cube. The rule set "RRR RRR RRR RRR RRR RRR RRR RRR RRR" (each "R" uses the next digit -base 2- of the Champernowne number 0.1 10 11 100 101 110 111 1000...) is used for the following pictures:


   
   
   
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 the Champernowne number (0 1 10 11 100 101 110 111 1000 1001 1) For example the first triplet "011" appears on the front lower right as an empty/missing one and two full cubes.
   
   
   
432 digits.
   
   
   
7020 digits.
   
   
   
91719 digits.
   
   
   
991818 digits.


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


See some explanations regarding the digit encoding using the 'pi' digits:


   
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.
   
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.



(CMAP28 WWW site: this page was created on 06/19/2024 and last updated on 07/07/2024 12:19:31 -CEST-)



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