CDS - Fractals • Cantor Set • Koch Set • Mandelbrot Set • Topological Dimension • Self-Similar Dimension

Remember:

Cantor Set:
Also called the “powder of cantor”.

Koch Curve:

Mandelbrot Set:

If we continue to scale as you can see we find the “original” form at all scales. A fractal has this characteristic: “It repeats itself at all scales”

• The bifurcatin diagram of the logistic map we have seen before.
• This diagram is a fractal.
• We we enter the chaotic regime, we actually have passed an infinite number of flip bifurcation (doubling the period each time).

Fractal Dimensions:

Topological Dimension:

• The intersection you find (open or closed curve) are at leased 1 or 2:
  • The least amount of point you can intersect is $1$.
  • For a closed line the minimum amount of intersection will be $2$ ⇒ topological dimension $2$.
• So as we have seen previously a set of point has topologial dimension $0$ ⇒ An open/closed curve has topologial dimension $1$

~Ex.: (in 3D):

• We are in 3D so now the circle becomes a sphere.
• The intersection between the sphere and the plane is a closed curve.
• So as we have seen previously an open/closed curve has topologial dimension $1$ ⇒ a plane has topological dimension $2$

Self-Similar Dimension:

• $r$ : scale.
• $N$ : copies of the first image.
• similarity dimension: $d$.
• If the self-similarity/similarity dimension is equal to the topological dimension ⇒ Then the object is not a fractal, and this is the case.
• This object has also topological dimension $=2$.

Fractal dimension for the Cantor set and the Koch curve:

• The topological dimension of both should be $1$.not-sure-about-this
• For the cantor set we have $d<1$, while for the Koch curve $d>1$.
• Take for example the Koch curve, it is so rough that it is more than a mono-dimensional curve. (a monodimensional curve has less “surface”)not-sure-about-this
  • While the cantor set is less (a monodimensional curve has more “surface”)not-sure-about-this

IMPORTANT

• Fractals dimensions are not integers.
• The definition of dimension only works for self-similar fractals.
  However not all fractals are self-similar, for those other definitions of dimensions also exists.
• Some fractals have still today unknown dimension.

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• Also called the “powder of cantor”.

• If we continue to scale as you can see we find the “original” form at all scales.
• A fractal has this characteristic: “It repeats itself at all scales”

• The bifurcatin diagram of the logistic map we have seen before.
• This diagram is a fractal.
• We we enter the chaotic regime, we actually have passed an infinite number of flip bifurcation (doubling the period each time).

• The first inspiration that Mandelbrot had for fractals.

• See the examples below.

• You can take the circles such that there are $0$ intersections.

• The intersection you find (open or closed curve) are at leased 1 or 2:
  • The least amount of point you can intersect is $1$.
  • For a closed line the minimum amount of intersection will be $2$ ⇒ topological dimension $2$.
• So as we have seen previously a set of point has topologial dimension $0$ ⇒ An open/closed curve has topologial dimension $1$

• We are in 3D so now the circle becomes a sphere.
• The intersection between the sphere and the plane is a closed curve.
• So as we have seen previously an open/closed curve has topologial dimension $1$ ⇒ a plane has topological dimension $2$

• We cannot use topological dimension for these sets.

• See the examples below.

• $r$ : scale.
• $N$ : copies of the first image.
• similarity dimension: $d$.
• If the self-similarity/similarity dimension is equal to the topological dimension ⇒ Then the object is not a fractal, and this is the case.
• This object has also topological dimension $=2$.

• The topological dimension of both should be $1$.not-sure-about-this
• For the cantor set we have $d<1$, while for the Koch curve $d>1$.
  • Take for example the Koch curve, it is so rough that it is more than a mono-dimensional curve. (a monodimensional curve has less “surface”)not-sure-about-this
  • While the cantor set is less (a monodimensional curve has more “surface”)not-sure-about-this