Origami:
Using an Axiomatic System of Paper
Folding to Trisect the Angle
Agenda
• Euclidean Geometry
• Euclidean Constructions
• Origami and the Axiomatic System of Humiaki Huzita
• Trisecting the Angle
• Proof of Trisection
Euclid’s Postulates
1.
Between any two distinct points, a segment can be
constructed.
2.
Segments can be extended indefinitely.
Euclid’s Postulates (cont.)
3.
Given two points and a distance, a circle can be constructed with
the point as the center and the distance as the radius.
4.
All right angles are congruent.
Euclid’s Postulates (cont.)
5.
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)
Euclidean Constructions
Origami: Humiaki Huzita’s 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.
Origami: Humiaki Huzita’s Axiomatic
System (cont.)
3.
Given two constructed lines l1 and
l2, we can fold l1 onto l2.
4.
Given a constructed point P and a
constructed line l, we can construct
a perpendicular to l passing through
P.
Origami: Humiaki Huzita’s Axiomatic
System (cont.)
5.
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.
6.
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.
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').
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
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).
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
Other Origami Constructions
•
Doubling a Cube (construct cube roots)
•
The Margulis Napkin Problem
•
Quintinsection of an Angle