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3
Principles of Origami Construction
An origami is to be folded along a specified line on the origami called fold
line . The line segment of a fold line on the origami is called a crease ,since
the consecutive operation of a fold and an unfold along the same fold line makes
a crease on the origami.
A fold line can be determined by the points it passes through or by the points
(and/or lines) it brings together. As in Euclidean geometry, by specifying points,
lines, and their configuration, we have the following six basic fold operations
called origami axioms of Huzita [9, 8]. It is known that Huzita's origami Axiom
set is more powerful than the ruler-and-compass method in Euclidean geometry
[6]. Origami can construct objects that are impossible by the ruler-and-compass
method [4]. One of them is trisecting an angle, which we will show in this paper,
as our example of origami proving and solving.
3.1
Origami Axioms
Huzita's origami axioms are described in terms of the following fold operations:
(O1) Given two points P and Q , we can make a fold along the crease passing
through them.
(O2) Given two points P and Q , we can make a fold to bring one of the points
onto the other.
(O3) Given two lines m and n , we can make a fold to superpose the two lines.
(O4) Given a point P and a line m , we can make a fold along the crease that
is perpendicular to m and passes through P .
(O5) Given two points P and Q and a line m , either we can make a fold along
the crease that passes through Q , such that the fold superposes P onto m ,
or we can determine that the fold is impossible.
(O6) Given two points P and Q and two lines m and n ,eitherwecanmakea
fold along the crease, such that the fold superposes P and m ,and Q and n ,
simultaneously, or we can determine that the fold is impossible.
The operational meaning of these axioms is the following: finding crease(s)
and folding the origami along the crease.
Let us first formalize the process of finding the creases. Let O be an origami, P
and Q be points, and f , m and n be lines. The above fold operations can be stated
by the following logical formulas. They are the basis for the implementation of
the computational origami system.
f )
Axiom 1:
( P
f
Q
P,Q∈P O
f∈L O
Q )
Axiom 2:
(sym( P, f )
P,Q∈P O
f∈L O
( m
(dist( f, m )=dist( f, n ))
n )
( f
m )
m,n∈L O
f∈L O
bisect( m, n ))
Axiom 3:
¬
( m
n )
( f
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