MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

3

1 4 ( 1 )

1

x

x

− −

=

−

;

3

1 4 ( 2 )

2

x

x

− −

=

−

In general: x^2 + 4x + 3 = 0

7.14.5 RESOLUTION OF THE COMPLETE EQUATION OF THIRD DEGREE (H. H)

First of all we ́ll recall Fig. 6 (Point 7.14) to show how the 3rd degree equation is behind it.

That figure is now completed with Fig. 1 of present Point 7.14.5

Let ́s get a t expression just dependent of: balls coordinates (0,0) and (l,m) ; a angle (whose

tangent is t); the configuration of billiards table (a,b):

t

b

bt+ty=a− ; m at lt

t

at b bt

= − +

− −

2

2

()( )l−at^3 + m+bt^2 −at+b= 0

This means that the orientation given to the ball in O in order to hit the other one placed at (l,m)

after rebounding orthogonally on both tablesides, is the only real root of the equation just ob-

tained. And that is so because the equation has a positive discriminant, according to drawing

scale.

It is important to insist that lines which receive points (0,0) and (l,m) along the folding opera-

tion, are parallel to their respective tablesides, and distant from them as much as the balls are

distant from said tablesides.

2

Y

I

X

X ́

x O

A

2 x 1

F

t=tgα

z

b

t=

b y

a z

t

+

−

=

a l

m y

t

−

−

=

(l,m)

O(0,0)

Y

(a,y) X

(a,b)^1

(z,b)