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