On the elementary single-fold operations of origami: reflections and   incidence constraints on the plane
  This article reviews the so-called "axioms" of origami (paper folding), which are elementary single-fold operations to achieve incidences between points and lines in a sheet of paper. The geometry of reflections is applied, and exhaustive analysis of all possible incidences reveals a set of eight elementary operations. The set includes the previously known seven "axioms", plus the operation of folding along a given line. This operation has been ignored in past studies because it does not create a new line. However, completeness of the set and its regular application in practical origami dictate its inclusion. Formal definitions and conditions of existence of solutions are given for all the operations. 
Jorge C. Lucero
math.HO