Fractal

Lesson

Introduction
Fractals
Fractal dimension
- Hausdorff dimension
- Von Koch snowflake
- Cantor set
- Sierpinski carpet
- Box counting
- Co-dimension
Experimental fractal dimension estimation
Exercises
To go further
Conclusion

Sierpinski carpet

The construction of this object starts from the iteration of an equilateral triangle with side . The process is then repeated indefinitely on every remaining equilateral triangle.

The surface of the object obtained at the iteration is equal to:

$S_n = \frac{a\sqrt{3}}{3} \Big(\frac{3}{4} \Big)^{n+1}$

If we apply the technique of overlapping already mentioned above for calculating the fractal dimension of this object, one can found the fractal dimension:

$3 = 2^{\Delta} \quad and \quad \Delta = \frac{\log 3}{\log 2} = 1,5850$

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