Fractal

Lesson

Dimensions

For several natural curves, the length increases indefinitely when tends to 0, following a linear relation in plot, which means that:

$L(\varepsilon) \sim \varepsilon^{1-\Delta}$

Where is the slope. So we have the following relations :

$L(\varepsilon) \sim \varepsilon^{1-\Delta} \quad and \quad L(\varepsilon) =p(\varepsilon)\varepsilon \\ \quad \\ p(\varepsilon) \sim \varepsilon^{\-Delta} \quad where \quad p(\varepsilon) \varepsilon^{\Delta} = C^{ste}$

The can be expressed with reference to a length such as:

$p(\varepsilon) \varepsilon^{\Delta} = \varepsilon_{0}^{\Delta} \\ \quad \\ \rightarrow p(\varepsilon) =\Big( \frac{\varepsilon_{0}}{\varepsilon_{0}} \Big)^{\Delta}$

In the case of , is the number of necessary segments to cover a line, and if is the number of necessary square of edge to overlap a surface and is the number of necessary cube of edge to overlap a volume.

For non-integer we are dealing with strange spatial quantities, which is measured with non-integer powers of the unit length . The fact that when is integer it merges with the Euclidean dimension of the measured object leads to conjecture that the objects which may be associated with a non-integer have a characteristic that we still call dimension since it generalizes this concept dimension non-integer called Fractal.

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