“Unraveling” the mathematics of origami

The CAS Basic Research Prize (CAS Grunnforskningspris) was awarded for the first time this year in April as part of the Konkuransen unge forskere (KUF), designed to foster interest in basic research and underscore the intrinsic value of knowledge. The prize recognises projects driven by curiosity, aimed at challenging conventional theories and posing new and bold research questions.

The first recipient was high school student Toma Kamata Sydnes, whose project "How to Mathematically Evaluate Whether a Rigid Flat-Foldable Origami Design Will Unravel or Not, and Whether It Will Unravel Completely or Incompletely When Pulled from Two Points?" also garnered first place in the natural sciences and technology category. 

His project presents sophisticated mathematical theory and analysis, systematically investigating the mechanics of "pulling" on origami from two points while proposing new theorems and an "unravelability evaluation algorithm." Toma’s work was commended for its rigour, originality, thorough integration of relevant literature, and potential for future exploration.

From Youtube to Okinawa Institute of Science and Technology

Toma's journey into origami mathematics began young. While his Japanese mother introduced him to paper folding in kindergarten, his serious interest was sparked by a Veritasium YouTube video exploring the mathematics of origami and its engineering applications. This led him to investigate what he calls "unravelability"—how folded origami unfolds when pulled from different points and to what degree it unravels.

The research question emerged from practical experience. "When developing an origami designing algorithm, I observed how small changes in the crease pattern resulted in major changes in the degree of unraveling," Toma explains. He found this particularly noticeable with the traditional crane design, where "one fold is enough to make a fully unravelable design not unravel at all."

Developing mathematical theorems to predict origami behavior presented significant challenges. "The most time-consuming part was the initial stage where I had to postulate potential theorems and explanations from observations, as it required a lot of trial and error," he notes. Toma tested his theories using simple origami structures and the traditional crane, which he describes as having "a complex unraveling pattern, which changes easily, despite its simple crease pattern." The breakthrough came when he discovered how to exploit overlapping folds in his proofs he explains, "It was a big ‘aha moment’ as it was a proof I had been struggling with for a long time.” 

After winning second place at KUF last year for his development of a new origami designing algorithm, Toma was offered an internship at Okinawa Institute of Science and Technology where he worked on publishing his work. “The publication is currently in progress, with further developed mathematics that makes it easier for people to use”, Toma explains. “I plan to further develop the work on Origami Unraveling to introduce more complex mechanics as I continue my studies, and hope that the work on Origami Unraveling can also be published once it is complete”.

Beyond paper folding

The applications extend well beyond paper crafts. "My study is on how a folded flat sheet unravels, hence I believe it can be applied to study how biological structures unfold, such as leaves," Toma says. He also sees potential in aerospace engineering for analyzing unfolding structures and in developing medical tools and nanostructures.

Beyond origami mathematics, Toma is exploring other research directions with growing enthusiasm for probability theory. "One area of science and mathematics where my interest has been growing is how probability can be used to describe natural processes," he explains. "For example, in chemistry, reactions start and continue by chance, and in fluid dynamics, one can explain macroscopic changes using the probability of motion of individual microscopic units."

His fascination extends to quantum mechanics, where he notes "how the fundamental interactions of the universe, the quantum interactions, are probabilistic and how something by chance can have a profound effect." This interest was sparked by Katie Mack's book The End of Everything, which introduced him to the Vacuum Decay theory - the idea that "the universe could cease to exist at any moment just by chance”.

Representing Norway at the European finals in Riga in September is "a great honor to be given this opportunity," Toma reflects, adding that he is "excited to see what representatives from the other countries have done."

For other high school students considering original research, Toma offers practical advice: "Trying to do something new is scary, as it is unpredictable and unknown whether it will work out. However, I challenge everyone to use the opportunity when you can get help from teachers and have a motivating factor to give it a try."

The CAS Basic Research prize is a collaboration with the Young Academy of Norway (AYF) and Konkurransen unge forskere (KUF). 

You can read more about KUF and the research prize here (in Norwegian) >