For the free pattern, click here, or open the supplementary file on the Bridges archive page.

Granny’s Double Torus is a twisty mathematical bauble constructed from eight seven-sided granny motifs.

The Four-Color Theorem tells us that every map in a plane can be colored with four colors in such a way that no two regions that share a border are the same color. (The history of the theorem and its controversial proof is fascinating in and of itself.)

On the other hand, maps on more complicated surfaces may require more colors. Among mathematicians and math hobbyists, it is well known that maps on a torus—the surface of a donut-shaped object—require up to seven colors. In fact, there are many examples of crocheted and knit seven-color torus maps.

For a double torus, the surface of a hypothetical two-holed donut, the number of colors required is eight. As I describe in two of my papers for the Bridges Conference, my 2014 paper and my 2020 paper, one way to construct a map that requires all eight colors is to stitch together eight heptagons in a very particular way. In Granny’s Double Torus, you literally do exactly this.

No one is more surprised than I that this turns out to be physically possible. The result is a delightfully twisted fabric model with the two holes of the double torus pointing in roughly perpendicular directions. Making the modules is fairly straightforward, especially if you have made traditional and solid granny motifs before. Assembling them is, well, challenging.

The pattern includes written instructions for seven-round motifs (easily adjusted to more or fewer rounds for the size of your choice), advice on blocking the motifs to shape, and step-by-step assembly instructions with photographs of each stage. Even with the instructions, you will probably find sewing the model together to be a bit of a puzzle. But I hope you will find it to be worth the trouble. I don’t think you can really understand Granny’s Double Torus until you make it.