MATHEMATICS AND ORIGAMI

Mathematics and Origami

Let ́s figure out the three real roots of equation

t^3 +t^2 − 2 t− 1 = 0
that was already solved with a different method in Point 7.13 (Fig. 2). The vector sequence
will be: 1;1;-2;-1, as shown in present Fig. 2.

First vector starting at I will hit side y in such a way that it asks for line y ́. In final folding op-
eration y ́ will receive point I (see also 10.3.1, Heptagon).

On the other hand, last vector ending at F comes rebounded off side x asking, therefore, for line
x ́ to receive point F during folding operation. So the simultaneous folding will be:
I → y ́ ; F → x ́
What happens though, is that this folding can be performed in three different ways as shown in

Fig. 3. In it, dashed lines are, as usual, folding lines, and angles a, b, g lead to the solu-
tions of the equation.
You may notice that if we introduce in Fig. 3 the transformation
x ́→ OX ; y ́→ OY

Y Y ́

X ́

X

I

2

F

I

X

X ́

3

Y ́

Y