Melissa Houck

Statement

I enjoy reading "popular math" books that bring mathematical concepts to life in layman's terms. As an actuary who uses Excel in my work, I find myself using the spreadsheet program to visualize and help me understand the topics I read about. Inspired after reading Chaos by James Gleick, I used Excel to create a rough facsimile of the familiar Mandelbrot fractal by using the mapping $f(z) = z^2 + c$, and iterating it several times, where the $z$'s are complex numbers. But why $z^2 + c$? Why not use a complex power? What about other functions, like a trig function? What if the starting grid were changed? What is the impact of making a small change in assumptions?

Artworks

I used Excel to create a grid of complex numbers, then iterated this grid several times using a mapping $z_2 = f(z_1)$, then used conditional formatting of the absolute value of the final resulting grid. My first print is the result of each iteration raising the previous grid of results to $(11+.1i)$ or $(-12-.05i)$ alternately. In addition, each iteration is multiplied by $(4-3i)$. The 2nd print changed the starting grid using the formula $(1-z^4)^{-1.5}$. The 3rd print used the formula $(1-z^4)^{+1.5}$ to change the starting grid. The fourth print substitutes a complex tangent for the third iteration.