Mathematician Solves 65-Year Paper Folding Puzzle

After 65 years of mathematicians trying to fold the tiniest origami doughnut, Richard Schwartz at Brown University just cracked the code.

The challenge sounds simple: how few folds does it take to transform a flat sheet of paper into a doughnut shape? But creating curves from something completely flat is surprisingly tricky, requiring precise mathematical arrangements that have stumped experts for decades.

Schwartz proved that the minimum is 24 folds creating 16 triangles meeting at 8 vertices. His paper, published in the Proceedings of the National Academy of Sciences, settles a question mathematicians have explored since 1960 when researchers first showed such folding was even possible.

The breakthrough almost didn't happen. Schwartz asked ChatGPT how many triangle arrangements were possible with eight vertices, and the AI confidently told him there were thousands. "I found that very discouraging because I didn't want to work my way through thousands of patterns," he admitted, nearly abandoning the project.

But the problem kept nagging at him. He grabbed a pencil and paper to work through the possibilities himself, suspecting ChatGPT might be wrong.

Turns out his instinct was right. When he finally checked the website ChatGPT had linked, he discovered there were only seven possible arrangements, not thousands. The AI had hallucinated the answer entirely.

From those seven options, Schwartz identified the most promising candidate. Using computer simulations, he successfully folded it into a torus, the mathematical term for a doughnut shape. The result looks more like a doghouse than a breakfast pastry, but mathematically it's perfect: a ring with a hole in the middle.

Why This Inspires

This story shows how human intuition still beats artificial intelligence when things get tough. Schwartz could have trusted the chatbot and given up, but his mathematical instinct told him something was off.

His persistence also demonstrates that sometimes the old-school approach works best. Armed with just pencil and paper, he solved what a sophisticated AI program got completely wrong.

The finding has practical applications beyond paper art. Understanding minimal folding patterns helps engineers design everything from deployable spacecraft components to medical devices that need to unfold inside the human body.

Schwartz hasn't actually folded his theoretical doughnut from real paper yet, freely admitting he's "terribly bad" at origami. But that's okay because the math is what matters, and the math is now beautifully solved.