Invertible Infinity: A Toroidal Fashion Statement

Ellie Baker and Charles Wampler
Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture
Pages 49–56 Regular Papers


We present the design of a uniquely constructed reversible “infinity scarf”—a specially made 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 shape 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 perfect equilateral triangle folding cannot. The triangle folding is a special case of what we refer to as “ribbon folding” and we show that our scarf and one made from a square are the only toroidal forms made from flexible but inextensible fabric that are invariant under inversion and fold in a planar ribbon loop. We present several fabric layouts that can be used to produce the scarf, along with sewing instructions, and show that, among all such layouts, the seam length of the hexagon is the shortest possible.