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Pentasia : An Aperiodic Surface in Modular Origami

Origami, the Japanese art of paper-folding, has numerous connections to mathematics, but some of the most direct appear in the genre of modular origami. In modular origami, one folds many sheets into identical units (or a few types of unit), and then fits the units together into larger constructions, most often, some polyhedral form. Modular origami is a diverse and dynamic field, with many practitioners (see, e.g., [12, 3]). While most modular origami is created primarily for its artistic or decorative value, it can be used effectively in mathematics education to provide physical models of geometric forms ranging from the Platonic solids to 900-unit pentagon-hexagon-heptagon torii [5]. As mathematicians have expanded their catalog of interesting solids and surfaces, origami designers have followed not far behind, rendering mathematical forms via folding, a notable recent example being a level-3 Menger Sponge folded from 66,048 business cards by Jeannine Mosely and co-workers [10]. In some cases, the origami explorations themselves can lead to new mathematical structures and/or insights. Mosely’s developments of business-card modulars led to the discovery of a new fractal polyhedron with a novel connection to the famous Snowflake curve [11]. One of the most popular geometric mathematical objects has been the sets of aperiodic tilings developed by Roger Penrose [14, 15], which acquired new significance with the discovery of quasi-crystals, their three-dimensional analogs in the physical world, in 1982 by Daniel Schechtman, who was awarded the 2011 Nobel Prize in Chemistry for his discovery. Penrose tilings, both kite-dart and rhomb tilings, have found application in a wide variety of decorative arts, as both purely 2D (flat) tilings, as well as quasi-three-dimensional structures, which are based on a 2D tiling but add some form of surface relief in the third dimension, for example, Anne Preston’s 2000 sculpture “You Were in Heaven,” installed at San Francisco International Airport [16]. Aperiodic tilings in general and Penrose tilings in particular seem like a natural fit to