Ellie Baker
One of my first forays into mathematical art, with Sophie Sommer and Susan Goldstine, involved creating seven-color torus maps in bead crochet. Since then, I've hankered to design seven-color torus infinity scarves as well. A seven-color torus map, which consists of seven "countries" that all border one another, demonstrates that at least seven colors are needed to paint maps on a torus (in contrast to maps on a plane or sphere, which are known to require no more than four). My first scarf used parallelograms for the countries, but I found the country shapes too boring. These scarf designs are the result of my efforts to fix that. Thank you to Kevin Lee for his helpful insights and modifications to his TesselManiac! software.
The black-outlined rectangle cut from the tessellated lizard pattern (left) can be used to form a seven-color torus map, if printed on fabric. The toroidal map is constructed by first sewing together the rectangle's left and right edges and then giving the resulting cylinder a half twist before sewing together the top and bottom edges. This tessellation, based on Escher's famous lizards, requires groups of three tiles to form the 7-color torus map. The print (left) uses one group of three for each "country." To achieve a scale better suited to the Mobius scarf (right), the fabric was printed using seven groups of three (or 21) tiles per country. The tessellation was created using TesselManiac! software by Kevin Lee.
The black-outlined rectangle cut from the tessellated bird pattern (upper left) can be used to form a seven-color torus map, if printed on fabric. The toroidal map is constructed by first sewing together the rectangle's left and right edges and then giving the resulting cylinder a half twist before sewing together the top and bottom edges. Using the same sewing instructions, the cream-colored rectangle cut from the lower left pattern forms a toroidal graph, fully connected on seven nodes, with no link crossings. These two fabric designs were used to make the invertible scarf shown right, which inverts (through a small hole in one seam) between the map and graph. The tessellation was designed using TesselManiac! software by Kevin Lee.