MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

ration of the α angles adjacent to the sides of the triangle and for the corresponding triset of

parallel lines.

Note that in Figs. 22 and 25 all the sides of the triangles have folds of the same nature,

whereas that is not the case for Figs. 19 and 28.

Another peculiarity is this: as we said, the four figures at left are equal geometrically

speaking. The four to the right are also externally congruent (we can overlap in coincidence

their perimeters), though their internal foldings are all different.

Summarising: with α = 20º it has been possible to get flatness with any combination of

mountain / valley folds for the triangle. That is so because 0 < α ≤ 30º. If 30 < α ≤ 60º, flat-

tening will only be possible when there was mountain / valley alternation in the three sides of

the triangle. If 60 < α ≤ 90º, flatness is impossible.

Former conditions apply to triangles. For the octagon mentioned before, as the sum of

all its angles is 1080º, the limits for α change from 30; 60 to 67,5; 135. In fact, for Fig.1 (case 2

A, Point 12), it is α = 67,5º and all the eight sides of the octagon are mountain fold.

In practice, all the cases we have studied till now require that the paper will be subjected

to many folds to give way to 3D from 2D. We have even considered ruled developable sur-

faces. Right-oh!, but P. Jackson and A. Yoshizawa exploit to incredible limits of beauty the

obtention of 3D forms with the minimum of folds and the paper as developable means. To the

first of the authors belongs Fig. 32 obtained by a unique fold.

31 A

In effect, rounded angle α and γ are supplementary for

both are interior to the same side of a secant, therefore α and γ

will also be supplementary.

In all the cases one can observe the fulfilment of all the

flattening conditions by all the nodes.

It should be noted that, though in all the cases flattening

is reached, continuous docility (accordion-like) operates only

in Figs. 19 and 28. On the contrary, in the rest there appears a

forced docility. To keep flat the triangle and the surfaces be-

tween parallels, the other surfaces are compelled to adopt the

form of a ruled developable surface.

32