Chapter 11.3, Problem 9E

Interpretation Introduction

Interpretation:

To find the similarity dimension of $\text{3-}$ dimensional Menger sponge and N-dimensional Menger sponge.

Concept Introduction:

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

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

• The dimension of the Cantor set is not an integer, it is a fraction.

• The similarity dimension d is defined as

  $\text{d =}\frac{\text{ln m}}{\text{ln r}}$ the values of m and r are determined by inspection.

• Menger sponge is afractal curve. It is generalization of the one dimensional Cantor set and two dimensional Sierpinski carpet.