MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

Fig. 4 is the pseudotetrahedron by T. Yenn. At left we have the folding plan, in the

center the solid form, and to the right the detail of ruled surfaces. Except for the small central

equilateral triangles, all the rest is made out of ruled surfaces of cylindrical generatrices that

give the impression of embossments and depressions.

Here it is an advice to paper folders relative to curve folding: pre-fold is eased with the

finger nail pressure.

It is relevant to recall that ruled surfaces with conic and cylindrical generatrices are

alike in this sense: all have a common point and rest on a directrix. In the second case, the

common point is the ideal point of one of the lines, for, being parallel, all these lines are paral-

lel to any plane containing one of them. A physical point is, obviously, the common point of

conic generatrices.

Fig. 7 is another example. I came across it when designing paper strips made with ar-

gentic rectangles to construct the perforated pentagonal-dodecahedron. The small rectangles in

Fig. 5 are argentic ones, and the oblong at right is the unit to draw 6. Folding 6 gives 7. By the

way, in all these figures (3,4,7) nothing is said on how to close the form: a practical resource is

to provide an extra unit to act as a glued closing lapjoin.

As can be seen, Fig. 8 is an enlarged view of the corresponding rhomb of Fig. 7. Rhomb

BDFH does not exist as such because the quadrilateral is not flat. Physically is made up by:

• two isosceles triangles AHG and CDE.


• the cylindrical surface ABCEFG bounded by the generatrices AG and CE, and the
  helix arcs AB, BC, EF, and FG. These helices are superimposed to mountain folds

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