Chapter 11.3, Problem 3E

Interpretation Introduction

Interpretation:

The “even-sevenths Cantor set” is constructed as: divide $\left[\text{0,1}\right]$ into seven pieces; delete pieces $\text{2, 4, and 6}$; and repeat on sub-intervals.

1. The similarity dimension of the set is to be found.

2. To find the similarity dimension of generalized Cantor set. Generalize the construction to any odd number of pieces, with even ones deleted.

Concept Introduction:

• Power law state that the dimension of the set equals the exponent $\text{d}$ is,

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

• The even-sevenths Cantor set constructed as by dividing $\left[\text{0,1}\right]$ in seven equal places and delete pieces $\text{2, 4, and 6}$; and repeat on sub intervals.