From the text: The simple process of iteration can produce complex and beautiful figures. In this article, I present a set of tasks requiring students to use the geometric interpretation of complex number multiplication to construct linear iteration rules. When the outputs are plotted in the complex plane, the graphs trace pleasing designs reminiscent of hypotrochoids, pursuit curves, and star polygons. Students are challenged to duplicate designs, and in doing so they explore and generalize patterns, apply basic trigonometry and number theory ideas, and integrate their findings with the tools of iteration ‒ iteration rules, seeds, orbits, and fates. The first part of this article covers the basics of complex linear iteration and the geometric interpretation of complex number multiplication. Included are a number of relatively simple starter tasks for students to master before creating the beautiful designs presented in the second part of the article. The second part presents four sets of designs for students to create using their knowledge from part 1.