Carolyn Yakel and sarah-marie belcastro

"Dual Seven-Colored Tori"

Knitted and Crocheted cotton with polyester fiberfill, One torus has a 3.5 inch radius and the other has a 4 inch radius. Each is about 2.5 inches high, March 2006

Carolyn Yackel, Mercer University and sarah-marie belcastro, Smith College
(Carolyn made the crocheted piece and sarah-marie made the knitted piece.)

"These tori exhibit the Heawood bound for the torus, which implies that a graph on a torus requires at most seven colors in order to color the vertices so that no vertices connected by an edge are the same color. The dual notion is that any map of contiguous regions on the torus requires at most seven colors in order to color the regions so that no two regions sharing a boundary will have the same color. These dual tori exhibit cases in which the maximum number of colors (seven) is needed. The knitted torus, designed, engineered, and knitted by sarah-marie, shows the complete graph on seven vertices. The crocheted torus, designed and engineered jointly by sarah-marie and Carolyn and crocheted by Carolyn, shows the dual map consisting of seven regions, each of which shares a boundary with every other region."