Paper Tori
Flat polyhedral tori in three-space with eight vertices, the fewest possible, and which shapes of flat torus they can take.
A paper torus is the mathematical idealization of folding a donut out of origami: a flat sheet, creased and glued into a torus without stretching. There has been a longstanding search for the simplest folding pattern, simplest meaning fewest vertices. The first constructions had thousands. Rich Schwartz then showed that seven is impossible and eight can be done.
But that realized just one torus, with a rather random shape. The finer question is, for each shape of torus, each parallelogram of paper, and in particular the square, how many vertices are needed? Doyle and Schwartz showed eight suffice for almost every shape, missing exactly two families that end at the square torus and the hexagonal torus, the two most symmetric shapes there are.
Fabian Lander and I started on this in June 2026, writing code to find eight-vertex paper tori numerically as a gift for Rich’s birthday conference. The search is tricky, since the tori we want are very constrained: flat, embedded, and of a prescribed shape all at once. We found tens of thousands, and the artwork above is one hundred of them. Fabian then proved rigorously that the square and hexagonal tori are realized with eight vertices, in a solo paper.
Our goal now is to fill in the two missing families. We have found the configurations that make the argument work along both, and are writing the paper. After that, we are teaming up with Rich Schwartz and Peter Doyle, whose work covers the interior shapes, to write a paper resolving the whole problem.
Status. The mathematics is done; we are writing it up.