Chapter 11.4, Problem 5E

Interpretation Introduction

Interpretation:

To find the box dimension of the Menger hyper-sponge.

Concept Introduction:

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

• The similarity dimension d is defined as:

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

• The Menger sponge is a fractal curve. It is a generalization of one dimensional Cantor set and two dimensional Sierpinski carpets.

• For a N-dimensional hyper-sponge, the similarity dimension is given as:-

  $\text{d =}\frac{\text{ln m}}{\text{ln r}}=\frac{\ln \left(N{2}^{N-1}+2^N\right)}{\ln 3}$