MATHEMATICS AND ORIGAMI

Mathematics and Origami

7.13 PARABOLAS ASSOCIATED TO THE FOLDING OF A COMPLETE EQUATION OF 3rd
DEGREE

Folding operation in fig 1 (Point 7.11) leads to get BF as a common tangent to these two pa-

rabolas:

PARABOLA FOCUS DIRECTRIX TANGENCY on BF

1DOX T 1

2COY T 2

In fig 1 of present Point 7.13 these two parabolas are shown overlapped with fig 2 (Point 7.11)

If equation

t^3 +pt^2 +qt+r= 0

has a negative discriminant, i.e.:

27

1

27

2 9 27

4

1

3 2

 +









 p − pq+ r

0

3

3

2 3

 <









 q−p

We have three different forms of folding simultaneously points C and D over the correspondent

axes.

Such is the case with the following equation that will be studied in another place:

t^3 +t^2 − 2 t− 1 = 0

Fig 2 of present Point 7.13 shows the two parabolas and the three common tangents. These tan-

gents are the symmetry axes in the simultaneous folds that carry focuses over directrices. Last

equation has, therefore, three real roots.

F

O

A

G

B

D(c,d)

X

Y

C(a,b)

par.1 par.1

par. 2

par. 2

T

T

1

2

1