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If this interests you, be sure to check out Erik Demaine's work at MIT, Issei Yoshino's Super Complex Origami, HOYJO Takashi, Biruta Kresling's Keikki Bamboo folds, Robert Lang's Design Secrets of Origami, Robert Hull's Origami^3 compilation. Not all computational origami looks mathematical but the methods for getting to and end are clearly designed from step one. Quite frankly I understand very little of the math, but I can appreciate the elegance of an efficient fold.

Tom is definatly one of the leaders in this field. Those who haven't read his paper The Combinatorics of Flat Folds: a Survey are missing out.

You might also check out Robert Lang's upcoming book Origami Design Secrets: Mathematical Methods for an Ancient Art

There's a page here that descsribes Origami folds as an alternative to straight edge and compass contructions. You can trisect the angle using folds, interesting stuff

I should also plug hexaflexagon.sourceforge.net a little app that puts six pictures onto a foldable template

Origami math is cool... (check out the galleries!)
hyperbolic paraboloids are actually pretty easy and fun to make (and they drive the ladies wild ;).
The Five Intersecting Tetrahedra are neat too but can get really hard when you're putting in the last couple.
And there's plently of theoretical stuff; for example, you can axiomitize origami, and trisect angles and double cubes and stuff.
Some people have even made origami/combinatorial geometry courses.

Origami math is cool... (check out the galleries!)
hyperbolic paraboloids are actually pretty easy and fun to make (and they drive the ladies wild ;).
The Five Intersecting Tetrahedra are neat too but can get really hard when you're putting in the last couple.
And there's plently of theoretical stuff; for example, you can axiomitize origami, and trisect angles and double cubes and stuff.
Some people have even made origami/combinatorial geometry courses.

Origami math is cool... (check out the galleries!)
hyperbolic paraboloids are actually pretty easy and fun to make (and they drive the ladies wild ;).
The Five Intersecting Tetrahedra are neat too but can get really hard when you're putting in the last couple.
And there's plently of theoretical stuff; for example, you can axiomitize origami, and trisect angles and double cubes and stuff.
Some people have even made origami/combinatorial geometry courses.

Origami math is cool... (check out the galleries!)
hyperbolic paraboloids are actually pretty easy and fun to make (and they drive the ladies wild ;).
The Five Intersecting Tetrahedra are neat too but can get really hard when you're putting in the last couple.
And there's plently of theoretical stuff; for example, you can axiomitize origami, and trisect angles and double cubes and stuff.
Some people have even made origami/combinatorial geometry courses.

Origami math is cool... (check out the galleries!)
hyperbolic paraboloids are actually pretty easy and fun to make (and they drive the ladies wild ;).
The Five Intersecting Tetrahedra are neat too but can get really hard when you're putting in the last couple.
And there's plently of theoretical stuff; for example, you can axiomitize origami, and trisect angles and double cubes and stuff.
Some people have even made origami/combinatorial geometry courses.

Ahh, the many hours of meetings that I've spent folding paper. I did, however, tend to get in trouble for folding paper during middle and high school.

For some serious challenges, try memorizing more complicated models like Kawasaki's rose. (diagram) (makes a nice tip, too. the rose in Origami for the Connoisseur is easier to learn...) Or learn to make modular origami stuff (origami that uses multiple units that are [generally] all the same). (instructions)

A great place to start is Joseph Wu's Origami Page.

The myth that a thousand paper cranes will bring good luck and health is much older than Sadako's story, although she did try to fold 1,000 while she was sick with leukemia. She finished 644 before she died, and her classmates completed the rest. There are two books about her story: Sadoko and the Thousand Paper Cranes and Child of the Paper Crane .

It's also not true that "classical" origami is extremely restrictive. Most of the rules mentioned were added by outsiders. There are many very old designs (such as connected cranes) that require cutting. It is an interesting challenge to follow those kinds of restrictive rules, but they are not really requirments with a long history.

The Origami FYI covers these and many other interesting points.