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[7]). We believe that the rigor of paper folding and the beauty of origami art-
works enhance greatly when the paper folding is supported by a computer. In
our earlier work, computational origami performs paper folding by solving both
symbolically and numerically certain geometrical constraints, followed by the
visualization of origami by computer graphics tools. In this paper we extend the
system for proving the correctness of the origami constructions, as proposed in
our previous paper [3].
Origami is easy to practice. Origami is made even easier with a computer: we
can construct an origami by calling a sequence of origami folding functions on
the Mathematica Notebook [12] or by the interaction with our system, running
in the computing server, using the standard web browser. The constructed final
result, as well as the results of the intermediate steps, can be visualized and
manipulated. Moreover, in cases where a proof is needed, the system will produce
the proof of the correctness of the construction. Namely, for a given sequence
of origami construction steps and a given property, the system proves that the
resulting shape will satisfy the property (or disprove the property).
The rest of the paper is organized as follows. In sections 2 and 3 we present
a formal treatment of origami construction. In section 4 we discuss a method for
trisecting an angle by origami, which cannot be made using the traditional ruler-
and-compass method. The construction is followed by the proof of the correctness
of the construction in section 5. In section 6 we summarize our contributions and
point out some directions for future research.
2
Preliminaries
In this section we summarize basic notions and notations that are used to ex-
plain the principles of origami construction. We assume the basic notions from
elementary Euclidean geometry.
In our formalism origami is defined as a structure
O
=
χ,
R
,where
χ
is the
set of faces of the origami, and
be the set of origamis.
Origami construction is a finite sequence of origamis
O
0
,O
1
, ..., O
n
with
O
0
an
initial origami (usually square paper) and
O
i
+1
=
g
i
(
O
i
) for some function
g
i
defined by a particular fold.
R
is a relation on
χ
.Let
O
R
is the combination of overlay and neighborhood
relations on faces, but in this paper we will not elaborate it further.
Apoint
P
is said to be
on origami
O
=
χ,
R
if there exists a face
X
∈
χ
such that
P
is on
X
. Likewise, a line
l
is said to
pass through origami
O
=
χ,
R
if there exists a face
X
∈
χ
such that
l
passes through the interior of
X
.We
denote by
L
the set of lines, and by
P
the set of points. The set of points on
O
is denoted by
P
O
, and the set of lines on
O
by
L
O
. We abuse the notation
∩
and
∈
,todenoteby
P
∩
O
=
φ
(resp.
l
∩
O
=
φ
) the property that
P
(resp.
l
)
is on
O
,andby
P
f
the property that point
P
is on line
f
.
Furthermore, 'sym' is the function which computes the symmetric point of a
given point with respect to a given line, 'dist' is the function which computes the
distance between two parallel lines, and 'bisect' is the function which computes
the bisector(s) of two lines.
∈
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