Circle+origami+axioms

=Circle Origami Axioms=

Discussion participants
 * Bradford Hansen-Smith
 * Alexander Bogomolny
 * Maria Droujkova, Katherine Droujkov
 * Colin McAllister ([|see also this unit related to the discussion])

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 //p//1 and //p//2, there is a unique fold that passes through both of them.

Axiom 2
Given two points //p//1 and //p//2, there is a unique fold that places //p//1 onto //p//2.

Axiom 3
Given two lines //l//1 and //l//2, there is a fold that places //l//1 onto //l//2. This is equivalent to finding a bisector of the angle between //l//1 and //l//2.

Axiom 2 - 3 - C
Given two points //p//1 and //p//2 on the circle, there is a unique fold that places //p//1 onto //p//2 and places the half-circles //l//1 and //l//2 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 //p//1 and a line //l//1, there is a unique fold perpendicular to //l//1 that passes through point //p//1. This is equivalent to finding a perpendicular to //l//1 that passes through //p//1.

Axiom 4-C
Given a point //p//1, there is a unique fold that passes through point //p//1 and places the half-circles //l//1 and //l//2 defined by the fold onto one another. This is equivalent to finding a diameter through //p//1.

Axiom 5
Given two points //p//1 and //p//2 and a line //l//1, there is a fold that places //p//1 onto //l//1 and passes through //p//2. 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 //p//1 and //p//2, there is a fold that places //p//1 onto the circle and passes through //p//2. 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 //p//1 and //p//2 and two lines //l//1 and //l//2, there is a fold that places //p//1 onto //l//1 and //p//2 onto //l//2.

Axiom 1-6-C
Given two points //p//1 and //p//2, 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 //l//1 and //l//2, there is a fold that places //p// onto //l//1 and is perpendicular to //l//2.

Axiom 7-C
Given one point //p// and a line //l//1, there is a fold that places //p// onto the circle and is perpendicular to //l//1. 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//.