Origami, the Japanese traditional art of paper folding, has gained the attention of researchers and engineers as a geometric tool to program the object’s mechanical properties. For example, by appropriately adding creases to the objects, we can flatten the things, or create a structure that behaves like a switch jumping two different states. In the present study, we focused on an interesting phenomenon, “undulation of shape,” observed in a kind of origami with creases aligned regularly, and clarified its hidden mathematical structure.

## Why is it important?

The mathematics of origami is based on the fact that a paper is inextensible and unshearable. Under this assumption, clarifying the possible folded state or possible folding motion of an object with creases is a significant question. However, solving the problem becomes more challenging as a crease pattern becomes complex. This time, we took a new approach by considering origami as a collection of simple units and describing their relationships as a "dynamical system." Through this approach, we unveiled the mathematical structures hidden behind the undulations of shapes. Viewing origami as a system rather than just gazing at the whole has the potential to lead to the discovery of new phenomena, including the undulations of shapes.