MATHEMATICS AND ORIGAMI

Mathematics and Origami

To fold flat around O, we have these possibilities:

Fig. 1 is an enlargement of central node O in the square at left, with its four folds ex-

tended up to the sides of the square c. In said Fig.1 point A was marked to see what will happen

to it with the different folds: v transforms A to A ́; m transforms A ́ to A ́ ́; v ́, A ́ ́ to A ́ ́ ́, and

finally, v ́ ́ gives back A ́ ́ ́ to A.

This means that the product of several symmetries with respect to some axles (v, m, v ́,

v ́ ́ ) concurrent in O and making a perigon, is equivalent to a rotation of 360º around O (in this

case, concerning point A).

Let ́s fix now our attention in the transformation of A to A ́ ́. The conclusion is this: The

product of two symmetries with respect to axes (v; m) is a rotation of an angle double of that

formed by the two axes. This is because Ang. AOA ́ ́ = AOA ́ + AÓA ́ ́ = 2Ang. vm.

As soon as we begin to fold c, O is not coplanar any more with the four points A, but

still is the center of the sphere containing them. That sphere is the same all the time: center O

and radius OA. The 4 points A, are always coplanar but situated in different attitudes according

to folding progression.

When c is folded flat, points A, A ́, A ́ ́ y A ́ ́ ́ are coincident. In any previous position

lines AA ́ and A ́ ́A ́ ́ ́ meet forming a plane whose traces in c are ABA ́ ́ ́ and A ́B ́A ́ ́.

8.2.8.6 INCENTER AND HYPERBOLA

From the incenter theorem, Toshiyuki Meguro got at the necessary development to cre-

ate folding bases to allow the construction of different figures. It is not our intention now to go

deep into that development (see comprehensive article by Aníbal Voyer in Nº 68 of “PA-

c

A

A ́

A ́ ́

A ́ ́ ́

O

v

v ́ ́

m

v ́

1

B ́

B

not flattenable

mvv ́ ́v ́flattenable

not flattenable

flattenable

flattenable