MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

7.14.3 THE ORTHOGONAL SPIRAL OF POWERS (H.H.)
Observing the process pursued along the two previous Points, we see that successive powers of

a can be obtained without limit, by means of folding, i.e. we can get an and, conversely, an

1

, n

being any natural number.

It should be noted that lines y ́, x ́, y ́ ́, x ́ ́, etc. the receivers of folding points, are parallel to

the coordinate axes at a distance equal to that in between initial points and coordinate axes.

It is evident that if a <1, we have a closing orthogonal spiral, whereas the spiral opens if a >1.

The values of successive powers of a are measured along the coordinate axes: even in abscissas

and odd in ordinate. Figs. 1 and 2 show all that.

7.14.4 RESOLUTION OF A QUADRATIC EQUATION (H.H.)

First, let ́s figure out the quadratic equation with roots x 1 =1 and x 2 = -3

(x-1) (x+3) = 0 ; x^2 +2x-3 = 0 (1)

Fig. 1 shows the folding process to get its two roots:

• To set axes OX; OY.


• To draw x ́ distant one unit from OX. This is because the coefficient of greater degree –the
  2 nd- is 1.

1 o

a

a a

a

a

a

2

3

4

5

6

1

a^41

a

a^5

o

a^3

a^2

a

a^7

6

2

1

y

A O x

I

x ́

x x

F

2 1