Origami:

1. Origami: Using an Axiomatic System of Paper Folding to Trisect the Angle

2. Agenda • Euclidean Geometry • Euclidean Constructions • Origami and the Axiomatic System of Humiaki Huzita • Trisecting the Angle • Proof of Trisection

3. Euclid’s Postulates • Between any two distinct points, a segment can be constructed. • Segments can be extended indefinitely.

4. Given two points and a distance, a circle can be constructed with the point as the center and the distance as the radius. All right angles are congruent. Euclid’s Postulates (cont.)

5. Euclid’s Postulates (cont.) • Given two lines in the plane, if a third line l crosses the given lines such that the two interior angles on one side of l are less than two right angles, then the two lines if continued will meet on that side of l where the angles are less than two right angles. (Parallel Postulate)

6. Euclidean Constructions

7. Origami: Humiaki Huzita’s Axiomatic System • Given two constructed points P and Q, we can construct (fold) a line through them. • Given two constructed points P and Q, we can fold P onto Q.

8. Origami: Humiaki Huzita’s Axiomatic System (cont.) • Given two constructed lines l1 and l2, we can fold l1 onto l2. • Given a constructed point P and a constructed line l, we can construct a perpendicular to l passing through P.

9. Origami: Humiaki Huzita’s Axiomatic System (cont.) • Given two constructed points P and Q and a constructed line l, then whenever possible, the line through Q, which reflects P onto l, can be constructed. • Given two constructed points P and Q and two constructed lines l1 and l2, then whenever possible, a line that reflects P onto l1 and also reflects Q onto l2 can be constructed.

10. Trisecting the Angle Step 1: Create (fold) a line m that passes through the bottom right corner of your sheet of paper. Let be the given angle. Step 2: Create the lines l1 and l2 parallel to the bottom edge lb such that l1 is equidistant to l2 and lb. Step 3: Let P be the lower left vertex and let Q be the intersection of l2 and the left edge. Create the fold that places Q onto m (at Q') and P onto l1 (at P').

11. Trisecting the Angle (cont.) Step 4: Leaving the paper folded, create the line l3 by folding the paper along the folded-over portion of l1. Step 5: Create the line that passes through P an P'. The angle trisection is now complete

12. Proof of Angle Trisection We need to show that the triangles ∆PQ'R, ∆PP'R and ∆PP'S are congruent. Recall that l1 is the perpendicular bisector of the edge between P and Q. Then, • Segment Q'P' is a reflection of segment QP and l3 is the extension of the reflected line l1. So l3 is the perpendicular bisector of Q'P'. • ∆PQ'R = ∆PP'R (SAS congruence).

13. Proof of Angle Trisection (cont.) Let R` be the intersection of l1 and the left edge. From our construction we see that RP`P is the reflection of R`PP` across the fold created in Step 3. • RP'P = R'PP' and ∆P'PR' = ∆PP'S (SSS congruence). • ∆PP'S = ∆PP'R (SAS congruence). • ∆PP'S = ∆PP'R = ∆PQ'R

14. Other Origami Constructions • Doubling a Cube (construct cube roots) • The Margulis Napkin Problem • Quintinsection of an Angle