Chapter 11.4, Problem 3E

Interpretation Introduction

Interpretation:

To find the box dimension of the Menger sponge.

Concept Introduction:

• If $\text{S}$ is a subset of the D-dimensional Euclidean space and $\text{N}\left(\text{ε}\right)$ is the minimum number of D-dimensional cubes of side $\text{ε}$, then by the power law:

  $\text{N}\left(\text{ε}\right)\propto \frac{\text{1}}{{\text{ε}}^{\text{d}}}$

• The box dimension $\text{d}$ is given as:-

  $\text{d = }\underset{\text{ε}\to 0}{\text{lim}}\frac{\text{ln N}\left(\text{ε}\right)}{\ln \left(\frac{\text{1}}{\text{ε}}\right)}$