MATHEMATICS AND ORIGAMI

Mathematics and Origami

Under these conditions AB = AF; in other words, A is a point on the parabola p which has focus F,
directrix IJ, vertex V (VI = VF) and passes through H (because HJ = HF). According to point 1.2.4, DE
is the tangent to parabola p in A.

Fig. 4 shows the structure of parabola p; its equation is:

2

2

1

y= +Kx ; IV =

2

1

As the parabola passes through H, we ́ll have:

x = 1 ; y = 1 ; replacing:

1 = +K

2

1

;

2

1

K=

Resultant equation of the parabola is

2

2

1

2

1

y= + x ; () 1 2

2

1

y= +x (1)

For the demonstrations we are after, now we already have the first motive: the parabola ́s
configuration. The second motive we need is the similarity of the three triangles we are dealing
with.
Those triangles are similar because they are right angled (three vertices of the square) and
each two of them have either acute angles, equal.
Both motives will allow us to define in a direct or indirect way, the nine sides as functions of
the independent variable x, which at the same time is one of the sides.

x () 1 2

2

1

y= +x z= y^2 −x^2

()

2 2

1

y x

x x

a

−

−

=

()

2 2

1

y x

y x

b

−

−

= c= 1 −x

d= 1 −b

()

x

y b

e

−

=

1

f= 1 −a−e

H

J

A

c

F

I

V

x

+X - X

+Y

p

B

4