Axiom |
Description |
Mathematical operation |
Axiom I |
Given two points, p1 and p2, there is a unique fold that passes through both of them |
This gives a line through p1 and p2 |
Axiom II |
Given two points, there is a fold that places p1 onto p2 |
This is finding the perpendicular bisector of the line through p1 and p2 |
Axiom III |
Given two lines, L1 and L2, there is a fold that places one onto the other |
This is like finding a bisector of the angle between L1 and L2 |
Axiom IV |
Given a line, L1, and a point, p1, there is a fold that is perpendicular to L1 and passes through p1 |
This is like finding a perpendicular line to L1 that passes through p1 |
Axiom V |
Given two points, p1 and p2, and a line, L1, there exists a fold that passes through p2 and places p1 onto L1 |
This is equivalent to finding the intersection of a line with a circle. The line is L1, and the circle is centered at p2 with a radius equal to the distance between p1 and p2 |
Axiom VI |
Given two points, p1 and p2, and two lines, L1 and L2, there is a fold that can place p1 and p2 onto L1 and L2 respectively |
This is like finding a line that is a tangent to two parabolas. The parabolas have foci at p1 and p2, and their directrices are defined by the lines L1 and L2 |