To turn a sheet of paper into a donut shape, you need to fold it into at least 16 triangles.
It is possible to make a torus by folding a sheet of paper just 24 times. The torus has a hole running through it in the center (shown in a computer-generated video).
Richard Evan Schwartz
Folding a flat piece of paper into a torus – a shape with a hole in the middle – requires origami skills. This is something that mathematician Richard Evan Schwartz misses. Still, it answered a lingering mathematical question about the process.
His work – done primarily on a computer rather than with paper – reveals the smallest number of folds needed to make a paper torus. The paper must have at least 24 pliesforming 16 triangles that meet at eight points, or vertices, reports Schwartz in the May 26 Proceedings of the National Academy of Sciences.
To make a torus from a piece of paper, you can roll it into a tube and bend the tube to connect its two ends, forming a donut. This would probably involve some inelegant creasing and potentially a few paper cuts.
A more refined method involves folding the paper into a set number of triangles, allowing it to fold elegantly into the desired shape. Schwartz, of Brown University in Providence, R.I., has an affinity for minimal mathematical objects; he had already found the shortest possible Möbius strip. So he wanted to know how to create a torus with the fewest folds, or equivalently, the smallest number of vertices.
To create a torus from a flat sheet of paper, a mathematical requirement must be met at each of the torus’s vertices – the places where the triangles meet. For each vertex of the torus, the angles of the triangles meeting there must total 360 degrees. Think of a sliced pizza. Add the angles at the end of each slice and you will get 360 degrees.
A torus with nine vertices meeting this condition had already been discovered. Mathematicians had also described a torus with only seven vertices, but no one knew if all seven could meet the requirement for a slice of pizza. Schwartz proved that one of the vertices would always fail this test, thus ruling out a paper torus with seven vertices.
Next, Schwartz used machine learning to see if an eight-vertex torus was possible. His program identified a folding pattern that worked. When built, it looks like a puppy tent with an extra flap inside. Schwartz created a template so anyone can fold their own, as long as their origami skills are up to par.
Bend your own torus
Mathematician Richard Evan Schwartz discovered how to make an origami torus with as little folding as possible. The tent-shaped object has a thin hole in the middle and requires only 24 folds that form 16 triangles meeting at eight vertices. Create your own minimal paper torus by printing this pattern and folding it as shown.
