Geoweaving: Fold-Up Baskets from Dessins d'Enfants

I just gave a talk on this topic at Bridges Aalto 2022. Dessins d'enfants (children's drawings) are famous in number theory (be sure to catch Gareth Jones' short talk on that topic.) For our purposes, a dessin is a drawing on a closed surface such as the sphere (the only surface we will consider) drawn according to certain rules.

On the sphere, the prime rule is that the drawing must be connected, in other words, an ant scared to walk on the unmarked surface would still be able to walk from any part of the drawing to any other. That sort of drawing can be done by not lifting the pencil while drawing, or by drawing in any old way and going back later to add "bridges" for Mr. Ant.

Once you have a connected drawing:

1. Add black dots where lines end or cross.

2. Add white dots in the middle of the edges.

Now you have a proper dessin. There are a lot of dessins, and a more general way to make them. The next to last column in this table counts all dessins on the sphere having the same number of black-to-white edges.

The usefulness of dessins, for our purposes, is that they describe all the arrangements by which the Adams World Tile can seamlessly cover a closed surface (for us, the sphere.) What is the Adams World Tile? Oscar Sherman Adams invented it in 1929, calling his map projection "World in a Square II."

The beauty of the Adams World Tile is that multiple tiles can be tiled together to model a seamless periodic Earth.

Put a black dot at the South corner of the Adams World Tile, and a white dot at the North corner, and the dessin makes the arrangement of tiles explicit.

Maker details can be found in my slides for the Bridges talk and my paper in the Bridges Archive.