Fractal Percolation, Porosity, and Dimension

Fractal Geometry and Porosity

A fractal is an object or a structure that is self-similar in all length scales. Fractal geometry is an excellent mathematical tool used in the study of irregular geometric objects. The concept of the fractal dimension, D, as a measure of complexity is defined. The concept of fractal geometry is closely linked to scale invariance, and it provides a framework for the analysis of natural phenomen...

Hausdorff Dimension and mean porosity

Percolation fractal exponents without fractal geometry

Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated to be the exact value for the d...

Computability and fractal dimension

This thesis combines computability theory and various notions of fractal dimension, mainly Hausdorff dimension. An algorithmic approach to Hausdorff measures makes it possible to define the Hausdorff dimension of individual points instead of sets in a metric space. This idea was first realized by Lutz (2000b). Working in the Cantor space 2ω of all infinite binary sequences, we study the theory ...

The fractal dimension of the minimum path in two - and three - dimensional percolation

We calculate the fractal dimension d, , , of the shortest path I between two points on a percolation cluster, where 1 rd”n and r is the Pythagorean distance between the points. We find d , , ,= 1.130*0.002 for d = 2 and 1.34i0.01 for d =3 . What is the length 1 of the shortest path or ‘chemical distance’ between two points of a random material? In general, I is greater than r, the Pythagorean d...