Measuring Fractal Sets

Fractal sets are irregular sets, exhibiting interesting properties. Some well-known fractal sets are the Cantor set, the Sierpinski triangle, and the von Koch snowflake. All of these examples are constructed iteratively. Furthermore, they have the property of self-similarity, that is, they contain scaled copies of themselves. Closely tied to the study of fractal sets is the study of measure. Measure, however, encompasses more than the traditional notions of length, area, and volume. Therefore, in this paper, we explore the Lebesgue measure and Hausdorff measure. Furthermore, tied closely to this is the notion of fractal sets. Therefore, we examine multiple self-similar fractal sets and their associated dimensions. We briefly touch upon other irregular sets that are not self-similar, namely Besicovitch sets. These are subsets of R^d that contain line segments in every direction. Along the way, we apply our ideas of measure to these sets, and we find their associated Hausdorff dimensions in R^2. Finally, we arrive at the open Kakeya Conjecture concerning their Hausdorff dimensions for d\u3e2