Hidden Beauty in Penrose Tiling: Weavings & Lace

Douglas Burkholder
Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture
Pages 213–220 Regular Papers

Abstract

We explore Penroseā€™s tiling using kites and darts in search for hidden beauty. We focus on the iterative subdivision process that can be used to create Penrose tilings by subdividing half kites and half darts into smaller half kites and half darts. By selectively coloring the half darts and half kites, based only on their relative position in the subdivision process, we create 15 unexpected and distinctive patterns hidden within Penrose tiling. These patterns tend to have the appearance of a weaving. Alternately, by selectively discarding tiles as we recursively subdivide, we can obtain fractal patterns that appear lacelike. We finish by filling the negative spaces in the fractal patterns with pursuit curves.

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