The Folding Mathematics
Archana S. Morye
TL;DR
The paper investigates origami as a bridge between art and mathematics, presenting origami as both a geometry education tool and a subject of formal mathematical analysis. It demonstrates folding-based proofs of classic results, such as the triangle angle sum of $180^\circ$ and the Pythagorean theorem, and introduces the Huzita-Hatori axioms that extend geometric construction beyond ruler-and-compass. A core contribution is the TreeMaker framework, which links uni-axial tree diagrams to crease-pattern generation and base realization, while revealing computational issues such as cycle packing. The discussion of open problems in creaseability, flat-foldability, and pattern verification positions origami as a rigorous, practically relevant area of mathematical research with potential engineering impact.
Abstract
Origami is the art of paper folding, and it borrows its name from two Japanese words \emph{ori} and \emph{kami}. In Japanese, {ori} means folding, and the paper is called {kami}. While origami is just a hobby to most, there is a lot more to it. If you fold a square sheet of paper into any of the traditional origami model (for example the flapping bird) and unfold it, you can see crease patterns. These crease patterns tell us that there is a lot of geometry hidden behind the folds. In this article, we investigate the symbiotic relationship between mathematics and origami. The first part of this article explores the utility of origami in education. We will see how origami could become an effective way of teaching methods of geometry, mainly because of its experiential nature. Complex origami patterns cannot be created out of thin air. They usually involve understanding deep mathematical theories and the ability to apply them to paper folding. In the second part of the article, we attempt to provide a glimpse of this beautiful connection between origami and mathematics.