An extended Menger Sponge -iteration 1- displaying the 27 first digits -base 2- of 'pi' [*Une éponge de Menger généralisée -itération 1- visualisant les 27 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:

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 06/15/2024 and last updated on 07/07/2024 12:19:04 -CEST-)

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