Mathematics and Origami17 PAPER SURFACES
First of all let ́s recall how plane curves may be represented in two different manners
(Point 13.3.2, ellipse): by means of points (Fig. 4) or as the envelope of their tangents (Fig.2).
If we make use of an analogy, we shall call real surfaces those in which the paper is the
set of all the infinity of straight lines contained in it as generatrices (ruled surfaces).
This group covers Figs. 23 and 26 (Point 15), as they are described there. Likewise, Fig.
32 in said Point 15, and the cones shown in Figs. 1,2 and 3 (Point13). We ́ll come back on
these when dealing with quadrics.
Continuing with the analogy, we shall call virtual surfaces those that have to be guessed
as the envelope of a discrete amount of generatrices, which, in turn, are but intersections (fold-
ing lines) of paper planes.17.1 REAL SURFACES
Let ́s add some others to those already mentioned. In first place, Figs. 1,2 and 3, similar
to those designed by T. Tarnai.Fig. 1 is the folding diagram, and Fig. 3 is the result after folding. The latter may induce
to think that all the obtained surfaces are flat (distrust of retouched pictures), but it is not so. It
is pertinent to set this clear to avoid frustration when constructing forms.
It is evident that the surface of a triangle with straight sides, does is flat. They may also
be flat other polygonal surfaces (specially quadrilaterals) when none of their sides is curved.
Nevertheless, a paper surface bounded by any curve cannot be a spatial plane surface:
the natural paper docility leads to a composition of plane triangulations and conic surfaces
made out of straight line generatrices.
The latter has been disclosed in Fig. 2; not really in all the cases but only in those most
evident in order not to entangle the figure.
The fact that those surfaces do not become flat does not lessen the forms ́ beauty: they
may lead to very attractive models for stone sculptures or, with a bit more of difficulty, for steel
sheet works.1