By Chelsea Robson

What is Origami?

The Japanese art of folding paper

“Ori” is the Japanese word for folding and “kami” is the Japanese word for paper

History

Folding paper to make art actually began in China in the first or second century and spread to Japan in the sixth. The intriguing art of Origami later spread to the rest of the world.In Japan the most popular models were animals, however in other parts of the world they were creating other figures. The Moors, who were from Africa, created geometric figures from origami since their religion prohibited them from making animals.At first it was only the rich who took part in this cultural activity, since paper was expensive.The Japanese took the art of paper folding further and actually found useful purposes for their origami.The Samuri would exchange strips of fish or meat folded in paper, these gifts were know as noshi. At weddings the Shinto would wrap glasses of wine in paper forming butterflies that were meant to represent the bride and groom.

More History

For centuries there were no written instructions for the origami figures, the methods were just passed from generation to generation.The first book of how to make origami was published in 1797 it was called “How to Fold 1000 Cranes”. Another book was put into circulation in 1845 that contained a collection of approximately 150 origami models including many that are still popular today. The Crane is the most popular origami figure, it is a sacred bird in Japan and it represents peace. The Japanese believe that if you fold 1000 cranes you will be granted a wish.

Mathematics of Origami

There has been a lot of mathematical study in the area of origami which has led to several theorems and results. Thomas Hull has found links between origami and geometry, algebra, number theory and combinatorics.Some of these theorems and results are;

Flat-foldabilityHow to solve equations up to degree 4How to fold the side of a square into 3rds, 5ths

and 9ths (Haga Theorem)Other theorems have allowed paper folders to

form shapes such as equilateral triangles, pentagons, hexagons, and golden rectangles.

Flat-foldability

Most origami models are flat models. This means that they can be pressed flat without creating any new folds.

Another result from this is that you only need two colors to color all the regions making sure that no two touching regions are the same color.

The Waterbomb

The Crane

Huzita’s Axioms

(O1) Given two points p1 and p2, there is a unique fold that passes through both of them.(O2)Given two points p1 and p2, there is a unique fold that places p1 onto p2.(O3) Given two lines l1 and l2, there is a fold that places l1 onto l2.(O4) Given a point p1 and a line l1, there is a unique fold perpendicular to l1 that passes through point p1. (O5) Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2.(O6) 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.(O7) Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.

Modern Origami

Britney Gallivan

Modular Origami

Modular origami is also known as unit origami, since it involves combining several origami units to form a larger and more complex model.

Pentagon-Hexagon Zig-Zag unit (PhiZZ)

More Modular Origami

Kusudama – traditional Japanese model thought to be precurser to modern modular origami

Koch Snowflake

Lesser Stellated DodecahedronBuckyball

Cube Five intersecting tetrahedra

Modified Buckyball

TorusStellated Dodecahedron

Koch Star

Menger Cube

Penguin

Truncated Icosahedron

Sources

http://en.wikipedia.org/wiki/Huzita's_axiomshttp://kahuna.merrimack.edu/~thull/omfiles/geoconst.htmlhttp://www.seagrant.wisc.edu/Frogs/origami_instr3.htmlhttp://www.sciencenews.org/articles/20060617/mathtrek.asphttp://pomonahistorical.org/12times.htmhttp://www.merrimack.edu/~thull/phzig/phzig.htmlhttp://www.mathematische-basteleien.de/oricube.htmhttp://www.sadako.com/fold/folding.htmlhttp://hektor.umcs.lublin.pl/~mikosmul/origami/index.htmlhttp://library.thinkquest.org/5402/history.html