Thank you, Yann Lanoiselée. could you recommend me some papers or textbooks on computing the fractal dimension of the complementary space?

A physical fractal dimension is a fractional degree of freedom, my fracture work has demonstrated this. Particles and pores that have physical density are not independent entities but are all manifestations of volumes. A particle can exhibit roundedness and fine scale roughness externally but a pore has a similar representation in a volume's surface. I'd suggest curvature might be the best way to understand and to use fractals more appropriately. I realize that opinions seem to favor the box measurements as the best way to measure D. I don't agree because if a "box" ruler is used to measure a heterogeneous stricture -whose geometric base is different on many scales, ie not. self-similar then wou'll have determine the entire geometric basis spectrum before you can do anything. In your situation I'd suggest that you consider how fluids flow through your bed, and find out if any dimensionless constants such as Reynolds number, ( you can't miss by checking on this

because you'll learn what's important in terms of your entire process-which you've not mentioned so far) I'd also like caution you that none, if any, of the fractal theorists have measured anything, nor did Beniot measure the coastline of England. It's very likely that opinions were made based on the "Box"Method's" statistics because there are no standards available. To be more specific - measure a structure of a fiber composite whose fibers are randomly oriented, you'll be measuring moments of the entanglements, and that wouldn't comply with the fact that fractals do 't have a first moment.

Dann Passoja

Yann Lanoiselée runs into trouble for D=3/2, as in that case 3-D=D. In fact, the fractal dimension of the union of two sets with fractal dimensions D_1 and D_2 has to be max(D_1,D_2) , as the volume of this within a box of size L grows as L^{D_1}+L^{D_2}. If this union is a cube of size L, as is the case with complementary sets, and D_1