When I make things, I often stumble across fun puzzles and wind up not only with an interesting object, but also with a deeper understanding of some bit of mathematics. This infinity scarf is a favorite recent example. My experiences as a maker/crafter/artist/puzzler are also reflected in the book I coauthored with Susan Goldstine, “Crafting Conundrums: Puzzles and Patterns for the Bead Crochet Artist” (AK Peters/CRC Press 2014), in which we describe a new methodology for designing patterns for the craft of bead crochet.
The mathematical ideas incorporated into the design of this scarf were developed in collaboration with Charles Wampler. Thank you to Carol Maglitta for modeling.
This reversible infinity scarf is a specially constructed cloth torus such that its shape is invariant under inversion AND it folds flat into a six-layer equilateral triangle. Since the meridians and longitudes of a torus swap places under inversion, one might think the invariance property dictates construction from a square piece of fabric (with opposite edges sewn together). However, although inversion invariance can be achieved with a square construction, the equilateral triangle folding cannot. Can you figure out a possible shape for the flat fabric layout used? The fabric designs, both P6M wallpaper group patterns that I created with Richter-Gebert’s app iOrnament, are a clue, and permitted sewing the pattern to match at the seams.