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Fractal dimension

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A, possibly non-integer valued, dimension concept. Let be a metric space and a bounded subset. For each let be the minimal number of balls of radius necessary to cover . Then

is the fractal dimension of . It has also been called the capacity, the Mandelbrot dimension or the Shnirel'man–Kolmogorov dimension of .

One has

If denotes the Hausdorff dimension of , then .

References

[a1] B.B. Mandelbrot, "Form, chance and dimension" , Freeman (1977)
How to Cite This Entry:
Fractal dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractal_dimension&oldid=11902