Chapter 11.3, Problem 8E

Interpretation Introduction

Interpretation:

To analyse the case of the Sierpinski carpet and to draw the sketch of ${\text{S}}_{\text{3}}$.

To find the similarity dimension of the limiting fractal which is known as the Sierpinski carpet.

To show that the Sierpinski carpet has zero area.

Concept Introduction:

• Consider the process in which a closed unit box is divided into nine equalboxes, and the open central box is deleted. Then, this process is repeated for each of the eight remainingsub-boxes, and so on. The set created by this method is called the Sierpinski carpet.

• The similarity dimension dis the exponent defined by: ${\text{m=r}}^{\text{d}}$; therefore, $\text{d = }\frac{\ln m}{\ln r}$.