MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

7.10 PARABOLA ASSOCIATED TO THE FOLDING OF A QUADRATIC EQUATION

Point 1.2.4 explained how in folding a point over a straight line, the crease became the tangent

to a parabola whose focus was the point to be moved and its directrix was the line which re-

ceives the point. That was already proved in Point 4 when demonstrating Haga ́s theorem.

Now then, in fig 1 of Point 7.8 we have reproduced the same operation, so we can complete it

now by drawing the parabola with focus F ≡ A, directrix d, vertex V (midpoint of AO) and tan-

gent CE on point E whose abscissa is just 40.

The equation of this parabola is:

2

2 60

1

2

60

y x

×

= + ;^2

120

1

y= 30 + x (1)

This equation has the same structure as (1) in Point 4: we may observe that its first term has a

length dimension (L) whereas the coefficient ofx^2 has L−^1 as dimension.

Fig 1 of present Point 7.10 makes obvious that distances from E to F and from E to d are equal

because folding line EC is the symmetry axis. Eventually the result is the pair of tangents to a

parabola from point C.

Fig 1 also shows:

∆AOE’ is similar to ∆HVC and therefore

VC

2

40

40

60

= ;

3

40

VC=

∆VHC = ∆EHG, then:

EG = VC =

3

40

3

130

3

40

yE=EE'=E'G+GE= 30 + =

i.e. it is 











3

130

E 40 ,

Let ́s make now a translation of axes from the origin to D. Equation (1) will become:

-40 0

C(0,16.6)

A(0,60)

+40 X

d

F

D E

E'

V H G

1