The Menger Sponge -iteration 5- with a tridimensional non linear transformation [*L'éponge de Menger -itération 5- avec une transformation non linéaire tridimensionnelle*].

The "direct" tridimensional transformation uses the three pseudo-projections {PX(x,y),PY(x,y),PZ(x,y)} of the surfaces obtained by the interpolation between the Bonan-Jeener's triple Klein bottle and a sphere. These pseudo-projections define the transformation of the {X,Y,Z} tridimensional space.

See the result of the "inverse" transformation:

See the Menger sponge -iteration 5-:

The interpolation between the Bonan-Jeener's triple Klein bottle and a sphere (the so-called

- PX[x,y] = (the 'X' pseudo-projection of the interpolation between the Bonan-Jeener's triple Klein bottle and a sphere)
- PY[x,y] = (the 'Y' pseudo-projection of the interpolation between the Bonan-Jeener's triple Klein bottle and a sphere)
- PZ[x,y] = (the 'Z' pseudo-projection of the interpolation between the Bonan-Jeener's triple Klein bottle and a sphere)

[Plus d'informations à propos de la pseudo-projection des surfaces -en français/in french-]

[More information about the surface pseudo-projection -in english/en anglais-]

[Plus d'informations à propos de la génération d'entrelacs -en français/in french-]

[More information about the generation of intertwinings -in english/en anglais-]

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 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 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 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...

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).

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