MATHEMATICS AND ORIGAMI

Mathematics and Origami

• Start with a square with base AB = a and center O (Fig. 10).


• Rotate it clockwise on its own plane around O, an angle ω.


• Then move it vertically by the amount h.


• Repeat successively the operation n times by rotating ω and moving h.


• Thus we get Fig. 11 which appears with all its folds: AD (mountain), DB (valley)
  and DC (mountain). Fig. 12 shows them enlarged.

Besides a solid perspective, Fig. 11, is also the clue to the fold plan of Fig. 13.

We can note that all we need to draw Fig. 13 is to fix ABD (enlarged in Fig.14). Inci-
dentally, Fig. 11 is fully flattable if properly twisted.

We know the three sides of that triangle: AB is given; CAD in Fig. 11 can read BD and
DA. Then from triangle ABD we can complete Fig. 13 by extending and copying lines.
We should note two things:

• The three central vertical lines of Fig. 13 are of no use to construct Fig. 11. Only its
  bases AB and the other three are needed.


• The projection h ́ of AD over the vertical, times n, gives the altitude of the starting
  paper that is greater than nh. It means that the quadrangled prism originated by Fig.
  13 is contracted in its height while twisted to form Fig. 11.

A B

D

A B

14

13

11

B

A