Circle Origami Axioms

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Truncated Icosahedron, by Bradford Hansen-Smith, wholemovement.com

Axioms are based on Huzita-Hatori Origami Axioms. All circle folds are based on one circle - the boundary of the paper.

Axiom 1

Given two points p1 and p2, there is a unique fold that passes through both of them.
 Folding a line through two points

Axiom 2

Given two points p1 and p2, there is a unique fold that places p1 onto p2.
 Folding a line putting one point on another

Axiom 3

Given two lines l1 and l2, there is a fold that places l1 onto l2.
 Folding a line putting one line on another

This is equivalent to finding a bisector of the angle between l1 and l2.

Axiom 2 - 3 - C

Given two points p1 and p2 on the circle, there is a unique fold that places p1 onto p2 and places the half-circles l1 and l2 defined by the fold onto one another.

This is equivalent to finding a bisector of the angle between radii to p1 and p2.

Axiom 4

Given a point p1 and a line l1, there is a unique fold perpendicular to l1 that passes through point p1.
 Folding through a point perpendicular to a line

This is equivalent to finding a perpendicular to l1 that passes through p1.

Axiom 4-C

Given a point p1, there is a unique fold that passes through point p1 and places the half-circles l1 and l2 defined by the fold onto one another.

This is equivalent to finding a diameter through p1.

Axiom 5

Given two points p1 and p2 and a line l1, there is a fold that places p1 onto l1 and passes through p2. If the distance between p1 and p2 is greater than the distance between p2 and l1, there are two such folds, if the distances are equal, one such fold. If the distance between p1 and p2 is smaller than the distance between p2 and l1, the fold is impossible.

This is equivalent to finding the intersection(s) of the circle centered in p2 with the radius equal to the distance between p1 and p2 with l1.

Axiom 5-C

Given two points p1 and p2, there is a fold that places p1 onto the circle and passes through p2. If the distance between p1 and p2 is greater than the distance between p2 and the circle, there are two such folds, if the distances are equal, one such fold. If the distance between p1 and p2 is smaller than the distance between p2 and the circle, the fold is impossible.

The two-solution situation is shown above.

Axiom 6

Given two points p1 and p2 and two lines l1 and l2, there is a fold that places p1 onto l1 and p2 onto l2.
 Huzita axiom 6.png

Axiom 1-6-C

Given two points p1 and p2, of which at least one is not on the circle, there are two and only two folds that place both points on the circle.

Need a drawing (Euclid's construction) here.

Axiom 7

Given one point p and two lines l1 and l2, there is a fold that places p onto l1 and is perpendicular to l2.
 Huzita-Hatori axiom 7.png

Axiom 7-C

Given one point p and a line l1, there is a fold that places p onto the circle and is perpendicular to l1. There is one such fold if p lies on the diameter perpendicular to l1, and two such folds otherwise.

This is equivalent to finding midpoints of segments between p and the intersections of the circle with the line parallel to l1 and passing through p.