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P∈P O ,m∈L O
Φ 1 ( P, Q, m )
(sym( P, f )
P,Q∈P O ,m∈L O
Φ 2 ( P, Q, m, n )
P,Q∈P O ,m,n∈L O
((sym( P, f )
(sym( Q, f )
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.
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
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).
PutPoint['E', Point[30, 100]];
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 .