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 the Wolfram Language
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 .
and 
, we can fold a line connecting them.
and 
, we can fold 
onto 
.
and 
, we can fold line 
onto 
.
and a line 
,
we can make a fold perpendicular to 
passing through the point 
.
and 
and a line 
, we can make a fold that places 
onto 
and passes through the point 
.
and 
and two lines 
and 
,
we can make a fold that places 
onto line 
and places 
onto line 
.
and two lines 
and 
, we can make a fold perpendicular
to 
that places 
onto line 
.
Zamiatina, L. "Computer Simulations of Origami."