An extended Menger Sponge -iteration 4- displaying the 108 first digits -base 2- of 'pi' [Une éponge de Menger généralisée -itération 4- visualisant les 108 premières 'décimales' -base 2- de 'pi'].




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. A rule set defined with the first digits -base 2- of 'pi' is used for the following pictures:



31x3=27 cubes ==> 1x33 =27 rules and 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.

32x3=729 cubes ==> 2x33 =54 rules and digits.

Each full cube of the preceding/right picture is subdivided into 27 smaller cubes and displays the same 27 next digits -base 2- of 'pi' (000 100 010 000 101 101 000 110 000).

33x3=19683 cubes ==> 3x33 =81 rules and digits.

Each full cube of the preceding/right picture is subdivided into 27 smaller cubes and displays the same 27 next digits -base 2- of 'pi' (100 011 010 011 000 100 110 001 100).

34x3=531441 cubes ==> 4x33 =108 rules and digits.

Each full cube of the preceding/right picture is subdivided into 27 smaller cubes and displays the same 27 next digits -base 2- of 'pi' (110 001 010 001 011 100 000 001 101).


The digits base 2 of 'pi' are used with the following convention:
                    0 --> F
                    1 --> T
(Please note that the preceding cube numbers include the full cubes as well as the empty/missing ones)


Here are the 108 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



(CMAP28 WWW site: this page was created on 06/16/2024 and last updated on 07/07/2024 12:19:07 -CEST-)



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