Origami originated in China, although it is now associated to Japan, around 200AD and was then known as *Zhezhi*. It was brought to Japan in the 6th century by Chinese Buddhist monks, where it obtained the name* origami* (”ori’ meaning fold and ‘gami’ meaning paper). Paper folding has become of great interest to mathematicians and in today’s post I will discuss the mathematics behind it.

### Flat Folding

The construction of origami models is often shown by the patterns left by the creases. The main question that mathematicians are interested in is looking at these crease patterns and answering whether they can be folded to a flat model, and if so, how to fold them. There are three rules for making flat-foldable origami crease patterns:

- Maekawa’s Theorem: at any vertex the number of valley (crease goes inwards) and mountain (crease goes outwards) folds always differs by two;
- Kawasaki’s Theorem: if you add up the angle measurements of every other angle around a point, the sum will be 180 degrees;
- A sheet can never penetrate a fold.

### Huzita-Hatori Axioms

Just as Euclid devised axioms for planar geometry, Humiaki Huzita and Koshiro Hatori created a complete set of axioms to describe the geometry of origami.

- Given two points
*p1*and*p2*, there is a unique fold that passes through both of them;

- Given two points
*p1*and*p2*, there is a unique fold that places*p1*into*p2*;

- Given two lines
*l1*and*l2*, there is a fold that places*l1*onto*l2*;

- Given a point
*p1*and a line*l1*, there is a unique fold perpendicular to*l1*that passes through point*p1*;

- Given two points
*p1*and*p2*and a line*l1*, there is a fold that places*p1*onto*l1*and passes through*p2*;

- Given two points
*p1*and*p2*and two lines*l1*and*l2*, there is a fold that places*p1*onto*l1*and*p2*onto*l2*;

- Given one point
*p*and two lines*l1*and*l2*, there is a fold that places*p*onto*l1*and is perpendicular to*l2*.

[Taken from plus.maths.org]

### Constructions

Origami has also been applied to solve geometric problems, which cannot be solved using a compass and unmarked ruler. I will highlight a few examples.

**Doubling the Cube**

This problem concerns a cube with a side length *s1* and of volume *V*. The challenge is to find a cube with side length *s2*, such that its volume is now *2V*. This construction is credited to Peter Messer and goes as follows:

**Trisecting an angle**

This construction is due to Hisashi Abe.

Any suggestions for future posts? M x

See: https://books.google.co.in/books?id=zJR2Rr_wuFQC&lpg=PR12&ots=DqltAIkde9&dq=origamics&pg=PP1#v=onepage&q&f=false

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Thank you for all these great links you send me 🙂 I really appreciate it!

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