A random Menger Sponge -iteration 5- [Une éponge de Menger aléatoire -itération 5-].
A random Menger Sponge -iteration 5- [Une éponge de Menger aléatoire -itération 5-].
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 TTTwhere '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...
Iteration 2. | 
Iteration 3. | 
Iteration 4. | 
Iteration 5. |
RRR RRR RRR RRR RRR RRR RRR RRR RRRbut the probability of cube subdividing goes from 0 (biggest cubes) to 1 (smallest ones).