Beautiful Equations Blog - The Maths of Origami. A flock of origami cranes hangs from a skylight.

Since the invention of origami, people have been folding paper to create art. Originating in 6th century Japan when monks introduced paper to the population, it was first used in religious ceremonies. Origami also had its place in the hundreds of beautiful paper butterflies that decorated weddings, and as small gifts exchanged by Samurai soldiers.

In recent years, the detail produced in origami designs has exploded. The humble paper crane has become an elegant dragon with hundreds of scales, and it’s mathematics that we have to thank for this.

Unfolding any pattern of origami reveals a square covered in the lines where it has been folded. We call this the structure’s ‘crease pattern’. In order to construct some figure, no matter how complex, it is this crease pattern that must first be determined. And so the mathematicians that first faced this problem did exactly what mathematicians do, they went about defining it.

It was discovered that any crease pattern must follow exactly four rules – or axioms, as they would be called in any other branch of maths.

1. Each section between creases can be coloured in using just two different colours, and no two sections of the same colour will share an edge (a concept that is familiar from graph theory)

2. The folds at each vertex have a direction: they will either stick up as a mountain fold, or down as a valley fold. The number of each type of fold must differ by exactly 2 for the design to be true

3. If you were to number each angle around a fold, the sum of the even angles would be 180 degrees (and so naturally would the sum of the odd angles)

4. No matter how you stack and fold the square, a sheet can never penetrate a fold.

 

These clearly defined laws have made so much more possible with origami thanks to its similarities with other areas of mathematics. For example, the area of paper needed for a ‘flap’ in origami is optimised by a circle, and therefore the design of the crease pattern for this flap becomes a circle packing problem. This familiar question was first solved many years ago (despite no apparent application) and so as is often the case with mathematics, these solutions will transfer across to the study of origami.

Even if the theory isn’t appealing, there is no denying the sheer beauty of some of the art produced by origami – and so also that of mathematics!

 

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