Eve Torrence

In 1968 Ringel and Youngs showed that maps requiring $n = \lfloor{{7 + \sqrt{1 + 48 k} \over 2}}\rfloor$ colors exist for all $k$-holed tori. But their proof does not provide examples of maps we can see. Few physical examples of maps on higher genus surfaces exist. In 2022 Carlo Sequin designed a highly symmetric map on a 3-holed torus that requires $\lfloor{{7 + \sqrt{1 + 48 * 3} \over 2}}\rfloor = 9$ colors. The title comes from the cloud like structure and the thrill of being able to build this 3-dimensional model of Sequin’s 3-holed torus map out of 129 Sonobe units.

In 1890 Percy Heawood proved that any map on a k-holed torus can be properly colored with at most $n = \lfloor{{7 + \sqrt{1 + 48 k} \over 2}}\rfloor$ colors. Heawood published a map on a torus that required $\lfloor{{7 + \sqrt{1 + 48 * 1} \over 2}}\rfloor = 7$ colors but did not show that a map requiring $n$ colors exists on a k-holed torus for $k > 1$. This modular origami model shows a map on a 2-holed torus requiring $\lfloor{{7 + \sqrt{1 + 48 * 2} \over 2}}\rfloor = 8$ colors. The creases give the sculpture a frenzied look and the slight twist lends an off-balance feel to the structure. Directions for building this model, as well as two distinct seven-color torus maps, can be found in my 2022 Bridges workshop paper, in the link above.