MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

15 FROM THE 2nd TO THE 3rd DIMENSION

Everything seen up to now has a bidimensional treatment, though we would have to in-

corporate certain nuances to this assertion. For example: the paper itself is three-dimensional

(2.500 times thinner than the small side of a DIN A4); the added volume when folding flat; the

flattvolumelike of unit figures in Point 8.3.4; the volume shaped as a bas-relief of tessellations

(case 3, Point 12).

We shall begin with the latter type of figures to jump to the 3rd dimension. Thus, we

have in Fig. 1 a development which, when folded as indicated leads to the elastic 3D composi-

tion of Fig. 2.

Just because of that elasticity it allows delightful transformations. Let ́s see several of

them:

As a matter of fact it is very easy to pass from the 2nd to the 3rd dimension; it suffices to

indicate in 2D that a fold has to be performed not at 180º as usual, but at any other angle speci-

fied there.

This becomes clear in the transition from Fig. 7 to 8: in Fig. 7 a rectangle receives three

cuts and then its central upper part is revolved 75º about AB. Fig. 8 is, in fact, a 3D figure.

Let us profit of that to show, again in 2D, a paradoxical figure. Fig. 11 results at the end

of process 8,9,10: In Fig.7 we made three cuts, whereas in Fig. 11 there appear to be four.

1

3

4

5

6

A B

3- Plan view of a quasi-ellipse.

4- Same of a quasi-hyperbola.

5- Face AB adapts itself to the

arc of any opposed curve (cir-

cumference, parabola, etc.).

6- Quasi-toroidal section.