Origami mathematics is the subset of mathematics that describes the underlying laws of origami. As part of mathematics, it is part of a deep, consistent (albeit incomplete) logical structure, but its applicability to real-world origami has its limits. Origami mathematics is always at most an approximation of real-world folding, and what one can construct, fold, or compute, using the operations of origami, depends, critically, on what one assumes as the underlying axioms, rules, or operations (depending on your choice of terminology).
One of the simplest sets of operations one can choose is the set of "Huzita-Justin Axioms"—a set of six (or is it seven?) basic operations that serve as the origami equivalents of compass and straightedge constructions from elementary geometry. Their analysis leads into some interesting mathematics of number fields and touches on Galois theory. While this exploration may seem a purely academic exercise, the HJAs provide a very real and practical tool for origami design: they are the basis of my ReferenceFinder tool for finding origami reference points.
The Huzita-Justin Axioms describe an extremely restrictive style of folding: only one fold at a time may be performed, and each fold must be unfolded before the next one is formed. Nearly all real-world origami lies outside of this mathematical domain. If we expand its boundaries just a bit, by allowing two simultaneous folds, still more becomes possible in origami geometric constructions. In this article, I describe how one form of "two-fold axiom"—making two simultaneous folds whose alignments rely on one another—leads to the solution for an exact angle quintisection.
These few examples barely scratch the surface of the broad world of origami mathematics. For a wider sampling, check out this page of links with a specifically mathematical/scientific bent.