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f
))

Axiom 4:

((
f

⊥

m
)

∧

(
P

∈

∀

P∈P
O
,m∈L
O

∃

f∈L
O

Φ
1
(
P, Q, m
)

m
))

Axiom 5:

⇒
∃

f∈L
O

((
Q

∈

f
)

∧

(sym(
P, f
)

∈

∀

P,Q∈P
O
,m∈L
O

Φ
2
(
P, Q, m, n
)

⇒

∀

P,Q∈P
O
,m,n∈L
O

n
))

Axiom 6:

((sym(
P, f
)

∈

m
)

∧

(sym(
Q, f
)

∈

∃

f∈L
O

In Axioms 5 and 6 we have to define the constraints that ensure the existence

of the fold line on the origami. Conditions
Φ
1
and
Φ
2
could be explicitly given

as boolean combinations of formulas of the form
A
=0or
A>
0, where the A's

are polynomials in the coordinates of
P, Q, m
(and
n
).

All the axioms have existential sub-formulas, hence the essence of an origami

construction is finding concrete terms for the existentially quantified variable
f
.

Our computational origami system returns the solution both symbolically and

numerically, depending on the input.

4

Trisecting an Angle

We give an example of trisecting an angle in our system. This example shows a

nontrivial use of Axiom 6. The method of construction is due to H. Abe as de-

scribed in [6, 5]. In the following, all the operations are performed by
Mathemat-

ica
function calls. Optional parameters can be specified by “
keyword

→

value
”.

Steps 1 and 2:
First, we define a square origami paper, whose corners are des-

ignated by the points
A
,
B
,
C
and
D
. The size may be arbitrary, but for our

example, let us fix it to 100 by 100. The new origami figure is created with

two differently colored surfaces: a light-gray front and a dark-gray back. We

then introduce an arbitrary point, say E at the coordinate (30
,
100), assum-

ing
A
is at (0
,
0).

→{

}

NewOrigami[Square[100, MarkPoints

'A','B','C','D'

],

→

'Step '];

PutPoint['E', Point[30, 100]];

FigureCaption

Our problem is to trisect the angle
EAB
. The method consists of the fol-

lowing seven steps (steps 3-9) of folds and unfolds.

Steps 3 and 4:
Wemakeafoldtobringpoint
A
to point
D
, to obtain the

perpendicular bisector of segment
AD
. This is the application of (O2). The

points
F
and
G
are automatically generated by the system. We unfold the

origami and obtain the crease
FG
.

FoldBring[A, D];

Unfold[];

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