Douglas G. Burkholder
Long-long ago in a place far-far away, my love of art was placed on the back burner as science and mathematics consumed my life. My love for geometry and my desire for visualization for the sake of mathematical understanding have always been central to my teaching and to comprehension of mathematical concepts. However, only decades later, have I begun to explore the artist within and attempted to merge art with my mathematics. Inspired by Kerry Mitchell’s award winning JMM 2015 artwork “Penrose Pursuit 2,” I began searching for beauty in non-periodic tilings, first with Radin-Conway’s Pinwheel tiling and now with Penrose’s Kite and Dart tiling.
Penrose Lace is a Penrose tiling containing 375,125 kites and 231,840 darts colored black, white, or half and half. I started with a tiling of this square with 2,973 larger Penrose tiles all colored white. The macro detail is formed by subdividing half darts and half kites into smaller components labeled 1-5. Then, similar to creating the Sierpiński triangle, every tile labeled k located between the kth pair of clockwise spirals is painted black. I repeated this subdivision a second time before switching the orientation of the spirals and create the micro detail by subdividing four additional times. Observe that clockwise spirals contain lace with similar macro detail and counterclockwise spirals contain lace with similar micro detail.
Penrose Skates was inspired by David Reimann’s "Binomial Pursuit" on the cover of the MAA Mathematics Magazine, June 2016. Starting from a square tiled with five Penrose kites, half darts and half kites are repeatedly subdivided into five smaller components. By labeling these five subcomponents A-E, then, similarly to creating the Sierpiński triangle, I alternately subdivide and remove all the components labeled B (half kites). After repeating this for five iterations I change to removing all the components labeled C (half darts) for seven more iterations. I ignore the tiny tiles remaining and focus on the negative space formed by the tiles removed. Pursuit curves are constructed in the triangular regions where each half tile was removed.