Chapter 11.4, Problem 7E

Interpretation Introduction

Interpretation:

1. To sketch the sets ${\text{S}}_{\text{1}}{\text{, S}}_{\text{2}}{\text{, S}}_{\text{3}}{\text{, S}}_{\text{4}}$

2. To compute the box dimension of the limiting set ${\text{S}}_{\infty }$

3. To determine ${\text{S}}_{\infty }$ self-similar

Concept Introduction:

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

• Cantor set consist 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.