Chapter 11.4, Problem 4E

Interpretation Introduction

Interpretation:

To find the box dimension of the Cartesian product of the middle-middle-thirds' Cantor set with itself.

Concept Introduction:

• Fractals are the complex geometric shapes with fine structure at arbitrarily small scales.

• A Cantor set consists of an infinite number of infinitesimal pieces separated by gaps of various sizes.

• The capacity or box dimension of S given as,

  $d=\underset{\epsilon \to 0}{\lim }\frac{\text{ln}\left(\text{N}\left(\text{ε}\right)\right)}{\ln \left(\frac{\text{1}}{\text{ε}}\right)}$, if limit exists.