Origami

Origami is the Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami, together with an animated crane constructed in Mathematica by L. Zamiatina.

To distinguish the two directions in which paper can be folded, the notations illustrated above are conventionally used in origami. A "mountain fold" is a fold in which a peak is formed, whereas a "valley fold" is a fold forming a trough.

The Season 2 episode "Judgment Call" (2006) of the television crime drama NUMB3RS features Charlie discussing the types of folds in origami.

Cube duplication and angle trisection can be solved using origami, although they cannot be solved using the traditional rules for geometric constructions. There are a number of recent very powerful results in origami mathematics. A very general result states that any planar straight-line drawing may be cut out of one sheet of paper by a single straight cut, after appropriate folding (Demaine et al. 1998, 1999; O'Rourke 1999). Another result is that any polyhedron may be wrapped with a sufficiently large square sheet of paper. This implies that any connected, planar, polygonal region may be covered by a flat origami folded from a single square of paper. Moreover, any 2-coloring of the faces may be realized with paper whose two sides are those colors (Demaine et al. 1999; O'Rourke 1999).

Huzita (1992) has formulated what is currently the most powerful known set of origami axioms (Hull).

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

2. Given two points and , we can fold onto .

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

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

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

6. Given two points and and two lines and , we can make a fold that places onto line and places onto line .

A seventh axiom overlooked by Huzita was subsequently discovered by Hatori in 2002 (Lang).

7. Given a point and two lines and , we can make a fold perpendicular to that places onto line .
Wolfram