I am not, by nature, an origamist. While I’m all for it in principle, I get frustrated at my inability to fold straight lines, and when I do succeed at making something, I never quite know what to do with it afterwards.

But the maths behind it is fascinating.

What is Kawasaki’s Theorem?

Kawasaki’s theorem explains when it’s possible (in principle) to fold a pattern flat at a given vertex, assuming no other folds in the paper.

To do this, there must be a difference of two between the number of valley folds and mountain folds (that’s Maetana’s theorem), and the alternating sum of the angles between adjacent folds must be zero.

For example, if your pattern calls for angles of 30º, 45º, 60º, 75º, 90º and 60º, you would work out $30 - 45 + 60 - 75 + 90 - 60$ and conclude that the pattern can be folded flat at this vertex.

In this example, the angles sum to 360º, but that’s not a condition of the theorem! Kawasaki’s theorem works on a curved surface where the angles at a point way sum to more or less than a full circle.

Why is it interesting?

Origami has mathematical applications I wouldn’t have expected before reading up about them – for example, trisecting an angle with straight-edge and compass is impossible, but with a sheet of paper and a few folds? It’s child’s play. More practically, origami informs how to package up (say) a solar sail so it can be transported into space without taking up much room, before unfolding elegantly and efficiently later in the voyage.

Understanding how origami works mathematically opens up new opportunities – both for art and for science.

Who is Toshikazu Kawasaki?

Toshikazu Kawasaki was born in Kurume, Fukuoka, Japan in 1955. He’s better known for his origami than for his maths, but teaches maths at Sasebo Technical Junior College in Nagasaki, Japan.