Definition of the Menger sponge: 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 to the 27-7=20 remaining subcubes. The fractal dimension of the Menger sponge is equal to :

log(20)/log(3) = 2.726833027860842

Definition of the Menger sponge: 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 to the 27-7=20 remaining subcubes. The fractal dimension of the Menger sponge is equal to :

log(20)/log(3) = 2.726833027860842…

An amazing cross-section can be made using the plane:

Definition of the Menger sponge: 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 to the 27-7=20 remaining subcubes. The fractal dimension of the Menger sponge is equal to :

log(20)/log(3) = 2.726833027860842…

An amazing cross-section can be made using the plane:

Definition of the Menger sponge: 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 to the 27-7=20 remaining subcubes. The fractal dimension of the Menger sponge is equal to :

log(20)/log(3) = 2.726833027860842…

An amazing cross-section can be made using the plane:

Definition of the Menger sponge: 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 to the 27-7=20 remaining subcubes.

Definition of the Menger sponge: 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 to the 27-7=20 remaining subcubes.

An amazing cross-section can be made using the plane:

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

See http://www. lactamme. polytechnique.fr/Mosaic/descripteurs/NDimensionalDeterministicFractalSets.01.Ang.html for more information about this subject.

The Verhulst dynamics is defined using the following iteration:

                    X[0] = 0.5

                    X[n] = R.X[n-1].(1 - X[n-1])

Here, in this computation, the growing rate ‘R’ is no longer constant but changes its value periodically using the following arbitrary cycle:

R3 ==> R3 ==> R2 ==> R1 ==> R2 ==> R3 ==> R1 ==> R2 ==> R3 ==> R1 ==> R1 ==> R1

where {R1,R2,R3} are respectively the three coordinates of the current point inside the following domain [2.936,3.413]x[3.500,3.850]x[3.000,4.000].
Only the points corresponding to a dynamical system with a negative Lyapunov exponent are displayed.

The Verhulst dynamics is defined using the following iteration:

                    X[0] = 0.5

                    X[n] = R.X[n-1].(1 - X[n-1])

Here, in this computation, the growing rate ‘R’ is no longer constant but changes its value periodically using the following arbitrary cycle:

where {R1,R2,R3} are respectively the three coordinates of the current point inside the following domain [2.936,3.413]x[3.500,3.850]x[3.000,4.000].
Only the points corresponding to a dynamical system with a negative Lyapunov exponent are displayed.

The Verhulst dynamics is defined using the following iteration:

                    X[0] = 0.5

                    X[n] = R.X[n-1].(1 - X[n-1])

Here, in this computation, the growing rate ‘R’ is no longer constant but changes its value periodically using the following arbitrary cycle:

Tridimensional visualization of the Verhulst dynamics

The Verhulst dynamics is defined using the following iteration:

                    X[0] = 0.5

                    X[n] = R.X[n-1].(1 - X[n-1])

Here, in this computation, the growing rate ‘R’ is no longer constant but changes its value periodically using the following arbitrary cycle:

where {R1,R2,R3,R4,R5} are respectively the five coordinates of the current point inside the following domain [2.936,3.413]x[3.500,3.850]x[3.000,4.000]x[2.7]x[2.9]. Only the points corresponding to a dynamical system with a negative Lyapunov exponent are displayed.

The Verhulst dynamics is defined using the following iteration:

                    X[0] = 0.5

                    X[n] = R.X[n-1].(1 - X[n-1])

Here, in this computation, the growing rate ‘R’ is no longer constant but changes its value periodically using the following arbitrary cycle:

where {R1,R2,R3} are respectively the three coordinates of the current point inside the following domain [2.936,3.413]x[3.500,3.850]x[3.000,4.000]. Only the points corresponding to a dynamical system with a negative Lyapunov exponent are displayed.

The Verhulst dynamics is defined using the following iteration:

                    X[0] = 0.5

                    X[n] = R.X[n-1].(1 - X[n-1])

Here, in this computation, the growing rate ‘R’ is no longer constant but changes its value periodically using the following arbitrary cycle:

where {R1,R2,R3} are respectively the three coordinates of the current point inside the following domain [2.936,3.413]x[3.500,3.850]x[3.000,4.000]. Only the points corresponding to a dynamical system with a negative Lyapunov exponent are displayed.

The Verhulst dynamics is defined using the following iteration:

                    X[0] = 0.5

                    X[n] = R.X[n-1].(1 - X[n-1])

Here, in this computation, the growing rate ‘R’ is no longer constant but changes its value periodically using the following arbitrary cycle:

where {R1,R2,R3} are respectively the three coordinates of the current point inside the following domain [3.212,3.370]x[3.500,3.616]x[3.000,4.000]. Only the points corresponding to a dynamical system with a negative Lyapunov exponent are displayed.

(제목 번역: 3차원 힐베르트 곡선: 1-3단계)

(제목 번역: 3차원 힐베르트 곡선: 1-3단계)

(제목 번역: 2차원 힐베르트 곡선: 1-5단계)

(제목 번역: 2차원 힐베르트 곡선: 1-5단계)

(제목번역: 코흐 곡선의 첫 4단계 반복의 3차원 조망)

(제목번역: 코흐 곡선의 첫 4단계 반복)