Designers and Artists: Ellie Baker and Eve Torrence

Materials and Methods: Sewn Spoonflower performance piqué fabric

Year of Creation: 2022

A map is *properly colored* if regions that share a border line are different colors. The Four Color Theorem states that any map on a plane or a sphere can be properly colored with at most four colors. In 1890, Heawood showed that \( \lfloor{{7 + \sqrt{1 + 48 g}} \over 2}\rfloor \) colors are *sufficient* to properly color any map on an orientable surface of genus \( g > 0\). For example, all maps on a three-holed torus \((g = 3)\) can be properly colored with at most nine colors. Later in 1890, Heffter showed that nine colors are also \emph{necessary} by using a table of numbers to describe a map on a three-holed torus requiring nine colors. We believe we are the first to use Heffter's numerical description to produce an easy-to-check physical model of Heffter's map.

In 2021, Carlo Séquin designed two more 9-color genus 3 maps that were distinct from Heffter's. Heffter's 1890 map has one valence-6 vertex that joins 5 regions, while one of Séquin's maps has one valence-6 vertex that joins 6 regions. Séquin's other map has three valence-4 vertices. All other vertices in each map have valence 3. Our fabric models of Séquin's maps are also displayed. The three flat fabric patterns we used to make the fabric maps are shown in the framed image.