MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

• To apply into both bases two opposed fine cardboard cones in order to conform the
  figure.


• Those cones will work as the coupling used in lathing: they adapt themselves to the
  natural shape of the quadric while allowing rotation to get its extreme position.
  Meanwhile the hyperboloid surface remains visible, and its stability may be guaran-
  teed adding the adequate weight to the inside of the upper cone (e.g. a necklace,
  some loose beads, a little chain, etc.).


• We have to stick rigorously to demand * and pre-conform the hyperboloid according
  to the need of a good coupling of the overlapped folds.

As already pointed out, the main form in Fig. 1 is ∆ABC that appears replicated in Figs.

3 and 4, though unnamed in the latter.

Angle in B must be obtuse (108º in our case) to produce an insinuated hyperboloid in its

natural form (Fig. 3).

The base BC is the side of the upper and lower polygons that, in the limit, represent two

circumferences (recall the end of Point 20.2). Sides AB and AC are, respectively, the moun-

tain and valley fold creases.

∆ABC is represented in Fig. 2 in connection with those two circumferences. It is essen-

tial to decide what the inclination of plane ABC will be with respect to the horizontal, for it

determines the altitude of the hyperboloid. In our case we have taken 80º for that angle, as

well in Fig. 2 as in Fig. 3. By so doing we get the so-called natural form of Fig. 3.

The hyperboloid we have just fabricated has 21 triangles with base AB (plus an extra

one for lateral closing). Therefore, starting with Fig. 2 we get Fig. 3 by rotating successively

said ∆ABC around O, the value of the revolving angle being 17. 142857 º

21

360

=. This contriv-

ance permits to draw the figure and shows that the hyperboloid is a quadric of revolution.

Fig. 4 derives from Fig. 5; in this, ∆ABC forms with the horizontal an angle of 50º in-

stead of 80º. To achieve that, the hyperboloid has to be revolved with the help of the cones re-

ferred to above, till the limit of torsion possibility.

We may observe then, two quite different sorts of rotation: the immaterial turning of

∆ABC around the hyperboloid axis, and that of torsion, up to the possible limit. Fig. 3 is the