The axioms assume that the operations are completed on a plane (i.e. a perfect piece of paper), and that all folds are linear.
The first seven axioms were first discovered by French folder and mathematician Jacques Justin in 1986.
[1] Axioms 1 through 6 were rediscovered by Japanese-Italian mathematician Humiaki Huzita and reported at the First International Conference on Origami in Education and Therapy in 1991.
Koshiro Hatori and Robert J. Lang also found axiom 7.
Use of neusis together with a compass and straightedge does allow trisection of an arbitrary angle.
Also, let u and v be the unit direction vectors of l1 and l2, respectively; that is: If the two lines are not parallel, their point of intersection is: where The direction of one of the bisectors is then: And the parametric equation of the fold is: A second bisector also exists, perpendicular to the first and passing through pint.
Folding along this second bisector will also achieve the desired result of placing l1 onto l2.
It may not be possible to perform one or the other of these folds, depending on the location of the intersection point.
This axiom is equivalent to finding the intersection of a line with a circle, so it may have 0, 1, or 2 solutions.
The line is defined by l1, and the circle has its center at p2, and a radius equal to the distance from p2 to p1.
If the discriminant is equal to 0, then there is a single solution, where the line is tangent to the circle.
And if the discriminant is greater than 0, there are two solutions, representing the two points of intersection.
Similarly, a fold F2(s) perpendicular to m2 through its midpoint will place p1 on the line at location d2.
[2] Given one point p and two lines l1 and l2 that aren't parallel, there is a fold that places p onto l1 and is perpendicular to l2.
This axiom was originally discovered by Jacques Justin in 1989 but was overlooked and was rediscovered by Koshiro Hatori in 2002.
The first three can be used with three given points not on a line to do what Alperin calls Thalian constructions.
[5] The first four axioms with two given points define a system weaker than compass and straightedge constructions: every shape that can be folded with those axioms can be constructed with compass and straightedge, but some things can be constructed by compass and straightedge that cannot be folded with those axioms.
Adding the neusis axiom 6, all compass-straightedge constructions, and more, can be made.
The existence of an eighth axiom was claimed by Lucero in 2017, which may be stated as: there is a fold along a given line l1.
[7] The new axiom was found after enumerating all possible incidences between constructible points and lines on a plane.
[8] Although it does not create a new line, it is nevertheless needed in actual paper folding when it is required to fold a layer of paper along a line marked on the layer immediately below.