Definition of fractal

The fractal is a geometric figure that is repeated in the same way at different scales. This definition does not exclude the possibility of finding distinct figures in the fractal (during magnification). The Mandelbrot set is one of these.


Benoit Mandelbrot's definition

Is defined as fractal if the Hausdorff-Besicovich dimension is greater than the topological dimension.

The formula to calculate the Hausdorff-Besicovitch dimension is: \; \frac{\log N}{\log l} \;

Where N is the total size of the fractal and l is a measure of the facet.


Example

Let's take the Von Koch's curve

Curva di Von Koch
We set: It=0 We set: It=1
Curva di Von Koch con It=0 Curva di Von Koch con It=1

NB: It mean "iteration"

Have you noticed? The size of the second figure has increased \; \frac{1}{3} \; and measures \; \frac{4}{3} \; units.

As the iteration increases, the size of the figure will increase up to \; \frac{4^{I_t}}{3^{I_t}} \;.

Applying the fractal dimension the result is: \; \frac{\log{4^{I_t}}}{\log{3^{I_t}}} \;.

Calculating the properties of logarithms, il risultato è: \; \frac{I_t \, \log{4}}{I_t \, \log{3}} \;.

By simplifying the fraction:

\; \frac{\log{4}}{\log{3}} \;

This is the Hausdoff's dimension of Von Koch's Curve.


Conclusions

This definition restricts very much the concept of fractal. Orbit fractals, for example, are excluded from this definition.

Today, fractal scientists are still finding an appropriate definition of fractal.

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