MATHEMATICS AND ORIGAMI

Mathematics and Origami

0. 1252249 ≠ 0. 122228 ≠ 0. 124226 = 0. 124226

Of course, to fold horizontally the given square through points E, G, etc. is equivalent to fold

through F, I, etc. (Thales theorem).

We are not going to show the folding process that, in fact, is identical to the exact solu-

tion. The only difference between the two solutions is the position of point E.

Exact solution

It keeps the correct point G of former solution, and produces a new point E which will

be also correct if we have applied the same process that gave G. In this case, vertical HI super-

sedes diagonal BC.

9.16 DIVISION IN n PARTS AFTER COROLLARY P.

The process is absolutely general, but for a better understanding we shall apply it to a

particular case, e.g. the division of a segment in 37 equal parts. Apparently the operation may

seem complicate once we are dealing with a high prime number, but we ́ll see that it is not so.

The first thing to do is to construct a square whose side is the given segment. We ́ll suppose the

problem solved and recall the biunivocal ratio a / x in Corollary P (see Point 5). That is, for

n

x

1

= , it is

1

2

+

=

n

a (see figs. 1 and 2).

Square nº 1, side AB (fig.3), has been produced.

1. We take for granted the final solution in that square nº 1: it is the point distant x AB
   37

1

= ,

from B.

10

6

G

E

7 8

G

E

G

E

9

G

E

11

G

G

E E

12

E G

C

B D

A =

1

G

D

G

B I

E

= C A H

(^2345)
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I
E E
G G G
E
H