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### Origami Axioms with Pictures

Lord of the

Rings

Mantis

History of Origami

- Origami comes from the Japanese words “oru”, which means to fold and “kami” which means paper.
- Originated in China in the 1st or 2nd century. Moved to Japan in the 5th century.
- Originally only used by the wealthy since paper was so rare.
- The crane (one of the most popular shapes) was the first to have written instructions.
- There is a legend that if you fold 1000 cranes, you are granted one wish.

There is a theorem called Kawasaki's Theorem, which says that if the angles surrounding a single vertex in a flat origami crease pattern are a1, a2, a3, ..., a2n, then: a1 + a3 + a5 + ... + a2n-1 = 180anda2 + a4 + a6 + ... + a2n = 180

Origami Axioms

1. Given two points p1 and p2, we can fold a line connecting them.

2. Given two points p1and p2, we can fold p1 onto p2.

3. Given two lines l1 and l2, we can fold line l1 onto l2 .

4. Given a point p1and a line l1, we can make a fold perpendicular to l1 passing through the point p1.

5. Given two points p1 and p2and a line l1, we can make a fold that places p1 onto l1 and passes through the point p2.

6. Given two points p1and p2 and two lines l1 and l2, we can make a fold that places p1 onto line l1 and places p2onto line l2.

7. Given a point p1 and two lines l1 and l2, we can make a fold perpendicular to l2 that places p1onto line l1.

Compass and Straight Edge (S.E. & C.) Axioms

- Given two points we can draw a line connecting them.
- Given two (nonparallel) lines we can locate their point of intersection.
- Given a point p and a length r we can draw a circle with radius r centered at the point p.
- Given a circle we can locate its points of intersection with another circle or line.

- Given two points p1 and p2 we can fold a line connecting them.
- Given two points p1 and p2 we can fold p1 onto p2.
- Given two lines l1 and l2 we can fold line l1 onto l2.

- Given a point p1 and a line l1 we can make a fold perpendicular to l1 passing through the point p1.
- Given two points p1 and p2 and a line l1 we can make a fold that places p1 onto l1 and passes through the point p2.
- Given two points p1 and p2 and two lines l1 and l2 we can make a fold that places p1 onto line l1 and places p2 onto line l2.

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