Adriel Alander: TBA
Time: Mon 2022-02-07 14.00 - 15.00
Respondent: Adriel Alander
Abstract: Fractals are undeniably popular, but what exactly is it that they are, and how do we construct them? Without giving a formal definition of a fractal, we will see that fractals are metric spaces that are characterized by having a fractional Hausdorff dimension, and an infinitely intricate structure that repeats on all scales. To generate examples of fractals, we use a technique known as “Iterative Function Systems”. It is based on the idea that the space of compact subsets of a complete metric space is itself a complete metric space. We will construct fractals as limits of Cauchy sequences in this “meta” space. By the means provided in M. F. Barnsley’s Fractals Everywhere we use the random iteration algorithm to render images for the Sierpinski triangle, the Barnsley fern, a fractal tree, Koch snowflake, and the dragon curve. We find their fractal dimensions to be 1.585, 1.45, 1.407, 1.262, and 1.524 respectively, where for the Koch snowflake and the dragon curve we specifically looked at the dimension of their boundaries.