It's nice to see the axioms spelled out. I had only read about that Origami is more powerful than Euclid, but never had the details.
At this year's SF Documentary Film Festival I watched a great documentary about modern origami called Between the Folds. If you have the opportunity to see it you should take advantage.
The book to read about folding is Origami Design Secrets: Mathematical Methods for an Ancient Art by Robert J. Lang.
Don't miss Joseph Wu's origami web page, particularly the "gallery":