24 Folds Create Smallest Origami Doughnut
Mathematician Richard Schwartz's groundbreaking proof requires a precise 24 folds to form 16 intersecting triangles
Mathematician Richard Schwartz of Brown University has discovered how to fold the smallest origami doughnut, a feat that requires at least 24 folds. The 24 folds create 16 triangles that intersect at eight vertices, resulting in a doughnut-shaped surface from a flat sheet of paper.
Schwartz used computer simulations and methods of analysis for his proof, which was published in the journal Proceedings of the National Academy of Sciences USA. His discovery is a significant milestone in the field of origami, which has been studied by mathematicians for decades.
The concept of creating a torus, or doughnut shape, from a flat sheet of paper has been explored by mathematicians since 1960, when Yuri Burago and Viktor Zalgaller determined the first examples of origami torus construction. More recently, researcher Vincent Tugayé constructed an origami torus with only nine vertices in 2025. However, Schwartz's proof shows that a torus with only seven vertices is impossible.
## Why it matters The discovery of the smallest origami doughnut has significant implications for our understanding of geometry and the principles of origami. By folding a flat sheet of paper into a doughnut shape, mathematicians can create certain curves that would be impossible to achieve otherwise. This has potential applications in various fields, including engineering and design.
The process of creating a torus from a flat sheet of paper is complex and requires careful folding to create the necessary curves. Schwartz's discovery provides a new understanding of the minimum number of folds required to achieve this feat, and his proof has been recognized as a significant contribution to the field of mathematics.
