Chapter 11.4, Problem 9E

Interpretation Introduction

Interpretation:

To find the dimension of the resulting fractal.

Concept Introduction:

A fractal slice of swiss cheese is constructed as: The unit square is divided into ${\text{p}}^{\text{2}}$ squares and ${\text{m}}^{\text{2}}$ squares are chosen at random and discarded. Here, $\text{p > m+1}$ and p, m are positive integers. The process is repeated for each remaining square of side $\frac{\text{1}}{\text{p}}$. Assuming that this process is repeated indefinitely.

The box dimension is a kind of measurement that involves covering the set with boxes of size $\text{ε}$.

The formula for the box dimension d is $\text{d =}\underset{\text{n}\to \infty }{\lim }\left(\frac{\text{ln N}\left(\text{ε}\right)}{\text{ln }\frac{\text{1}}{\text{ε}}}\right)$ .