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
Euclids Postulates

1. Between any two distinct points, a segment can be
   constructed.
2. Segments can be extended indefinitely.

4
Euclids Postulates (cont.)

1. Given two points and a distance, a circle can be
   constructed with the point as the center and the
   distance as the radius.
2. All right angles are congruent.

5
Euclids Postulates (cont.)

1. 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 Huzitas Axiomatic System

1. Given two constructed points P and Q, we can
   construct (fold) a line through them.
2. Given two constructed points P and Q, we can fold
   P onto Q.

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Origami Humiaki Huzitas Axiomatic System (cont.)

1. Given two constructed lines l1 and l2, we can
   fold l1 onto l2.
2. Given a constructed point P and a constructed
   line l, we can construct a perpendicular to l
   passing through P.

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Origami Humiaki Huzitas 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.

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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').

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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

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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).

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Proof of Angle Trisection (cont.)

• Let R be the intersection of l1 and the left
  edge. From our construction we see that RPP
  is the reflection of RPP 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

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Other Origami Constructions

• Doubling a Cube (construct cube roots)
• The Margulis Napkin Problem
• Quintinsection of an Angle