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# Royal Institution Masterclass, Manchester 2022

## Origami and Mathematics

As well being a delightful and enjoyable activity, paperfolding has a world of mathematics hiding inside the folds. This masterclass will give you a taste of why origami has captivated mathematicians, scientists and engineers at all levels as well as artists, magicians and educators.

Starter activity
A square is divided into a three by three grid.
How many squares are there? How many rectangles?How could you generalise this problem?

Here is some background information about paper, paperfolding and origami and mathematics.

Paper is usually said to be invented by Tsai Lun in China, 105 CE. However, paper may have been invented before then. Paper was used as packaging, clothing, for hygiene as well as writing and printing. Question: How many other uses for paper can you think of?

It seems reasonable to assume that paper was folded soon after it was invented. However, the earliest surviving records for paperfolding are from Japan, where origami means fold paper.

Four types of origami exist:

1. Ceremonial: O-shide paper streamers in Shinto shrines, Go-hei butterflies for weddings, Noshi gift wrappers, etc.
2. Practical: Wrappers, envelopes, boxes, etc.
3. Recreational: Cranes, boats, etc.
4. Creative: Creating new origami models for artistic and educational purposes, e.g. Akira Yoshizawa and Isao Honda.

Europe is believed to have a tradition of paperfolding that is separate from Asia, e.g. napkin folding, letter folds and playground folds.

In the last few decades origami has become a subject of research in mathematics, engineering and education. As well as using origami to learn mathematics, we can study the mathematics of origami. Here are two examples:

Plane geometry constructions Paperfolding is geometry in action: halving an angle is straightforward. However, dividing an angle into three equal parts is harder: a method for trisecting a right angle has been known for at least 100 years.

The mathematics of origami When we fold a flat origami model, what are the properties of the lines and angles when we unfold the paper and return to the starting shape?