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• Print publication year: 2011
• Online publication date: October 2011

# 4 - Exploring Fractal Dimension, Area, and Volume

## Summary

Introduction

Fractals were conceived and introduced in the context of advanced mathematical research. However, the basic simplicity of their construction and numerous elementary representations lends itself to an introduction at the secondary school or beginning college curriculum. These fascinating geometric figures are, therefore, capable of enriching the contemporary mathematics curriculum in an important and significant manner, and, simultaneously, providing a bridge to the world of mathematics for the general public. In addition, there are numerous practical applications modeled by fractals.

In this article a fractal is considered to be a geometric shape that has the following properties:

The shape is self-similar,

The shape has fractal dimension, and

The shape is formed by iteration through infinitely many stages.

The word “fractal” is derived from the Latin word fractus meaning “broken” or “fractured”. The boundary or surface of a fractal is bent, twisted, broken, or fractured.

There are many surprises in generating fractals and investigating their properties. This paper will address the notion of self-similarity, as well as fractal dimension and area and volume for fractals.

Self-Similarity

Some fractals are strictly self-similar while others are approximately or quasi self-similar. At any level of magnification of a strictly self-similar fractal there is a smaller piece of the object that is a reduced copy of the whole fractal. For example, the Sierpiński Triangle or Sierpiński Gasket (Figure 4.1) is a strictly self-similar fractal. Notice that each of the three encircled portions of Figure 4.1 is identical to the whole figure when magnified by a factor of 2. The self- similarity is recursive.