The Menger Sponge iteration 3 [L'éponge de Menger itération 3].
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 277=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...
See the first objects of this family (including this one):
Iteration 1.

Iteration 2.

Iteration 3.

The used colors display the cube generation order: the first one front bottom left is dark blue when the last one rear top right is white.
(CMAP28 WWW site: this page was created on 09/17/2024 and last updated on 09/19/2024 18:29:09 CEST)
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