MATHEMATICS AND ORIGAMI

Mathematics and Origami

of Fig. 6 (lines BH, BD, FH y FD in Fig. 8): The result is that in vertices B and F

certain tensions are produced and the paper replies with a minimum of deformation.

Let ́s see another example of a real surface, in this case by A. Ratner (“PAJARITA”,
special issue 1996). We begin with Fig. 2 (Point 7.15.2) to draw Fig. 9 that in turn is the folding
plan for Fig. 11.

It is advisable to produce in Fig. 9 as many valley folds as possible (horizontal seg-
ments), to reach near O (decreasing geometric progression).
Fig. 10 is deduced from Fig. 9: Triangles OAB and OA ́B ́ will become a pair of quasi-
cylindrical surfaces whose generatrices are the respective parallel lines to AB and A ́B ́.
They are not full cylindrical surfaces because though lines OA and OA ́ are free to take
its curvature, OB and OB ́ are restrained as open polygonal lines.
Both surfaces could be named spiroids because their directrices are not plane spirals but
helicoidals (Fig. 11)

I must say that Fig. 11 is so beautiful in reality that neither a photography nor a per-
spective can convey to the viewer the harmony it contains: it must be constructed! (what is very
easy, indeed).

A

O O

A

BB ́

A ́

9 10