MATHEMATICS AND ORIGAMI

Mathematics and Origami

A conoid is a ruled surface with:

• Its own right directrix.


• A director plane not parallel to it.


• Another directrix, curved or straight.

Hence, the surface of Fig. 4 is not a conoid strictly speaking for, though it has the horizontal as
director plane, it has two curved directrices instead of having, at least, one straight.
To illustrate this, Figs. 5 and 6 show a couple of conoids: The first is a coiled stairwell and the
second is the Plücker ́s conoid. Both have the horizontal plane as director. The directrices of Fig. 5 are,
the vertical axis of a cylinder and a helix on its surface.

One of the directrices of Fig. 6 is a generatrix of the cylinder and the other is an ellipse on the
surface of said cylinder having one of its vertices on the straight directrix of that conoid. The genera-
trices of both conoids are outlined in Figs. 5 and 6.

A CONOID OF PAPER

Fig. 7 is a conoid obtainable by folding. It is a ruled surface whose generatrices rest at equal
intervals on the crossed diagonals of two squares. With a common side, these squares form a 90º dihe-
dral angle. The director plane, as shown in Fig. 8 is x = 0.
In Fig. 8 we can see the structure required designing a simple program of calculus that figures
out the length of mountain and valley folds. That will allow drawing the adjacent triangles of Fig. 9:
this figure is the conoid folding plan.
The inputs of that program are: the side of the square and the n parts to divide the diagonal. In

the drawing we took n = 8 and the value of the small sides of all triangles of Fig. 9 is
n

a 2

Let us calculate the length of both, a mountain and a valley fold, e.g., for points 7 and 3: m (7)
and v (3).

()

2 2

(^726) 





 + ×





= ×
n
a
n
a
m equivalent to: ()= ()()n−i+ 12 +i− 12
n
a
mi
()
2 2 2
(^325) 





 + ×





 +





= ×
n
a
n
a
n
a
v i.e.: () () ( )
2 2
i 1 1 n i
n
a
vi = − + + −