An amazing cross-section inside the Menger sponge -iteration 4- [*Une coupe très étonnante dans l'éponge de Menger -itération 4-*].

See some conformal transformations in the octonionic space -tridimensional cross-section-:

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(SubdivisionRules=TTT TFT TTT TFT FFF TFT TTT TFT TTT)

log(20) --------- = 2.726833027860842... log(3)The "standard" Menger sponge can be defined by means of

"standard" Menger sponge _____________________ / \ TTT TFT TTT TFT FFF TFT TTT TFT TTT \_/ Sierpinski carpetor again:

TTT TFT TTT TFT FFF TFT TTT TFT TTTwhere 'T' ('True') and 'F' ('False') means respectively "subdivide" and "do not subdivide". 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 complementalternates 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" or "do not subdivide" 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...

Moreover an amazing

2X - 2Y + 2Z - 1 = 0the origin of the coordinates being at the center of the main cube and the axis being parallel to its sides.

This process can be generalized in many different ways and for example:

3 3 3 2X - 2Y + 2Z - 1 = 0(the curved one) or again:

1 2 1 2 1 2 2 (X - ---) + (Y - ---) + (Z - ---) = R 2 2 2(the spherical one).

See some related pictures (possibly including this one):

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See some related pictures (possibly including this one):

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See some related pictures (possibly including this one):

See some related pictures (possibly including this one):

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See some related pictures (possibly including this one):

log(8) -------- = 1.892789260714372 log(3)

See the first objects of this family (including this one):

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