Proving and Constraint Solving in Computational Origami - Artificial Intelligence and Symbolic Computation - page 135

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[];