MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

• Once Fig. 3 is folded we get a solid similar to that of Fig. 2; this is shown in Fig. 4 gather-
  ing to itself the cube of side l: This cube has obviously the same volume as the solid ob-
  tained from Fig. 3, and just half of the cube of side L.

SOLUTION 2

It is based on the orthogonal spiral of powers by H Huzita, as disclosed in Point

7.14.3. It is a matter of finding segment l such that its relation to another given segment

L will be =^32

l

L

. The process is as follows:

• To set a co-ordinates system XY (Fig.1) fixing in it points O (origin) and F (final) at dis-
  tances d,a to the co-ordinates origin, respectively. Condition: a = d / 2.


• To have available a pair of papers V,W (Fig. 2) with right angles.


• By try and error (a maximum of three attempts will be enough) get (Fig. 3):

To lie a side of V on O in such a way that the vertex of its right angle will rest on axle

X. The other side of V will intersect axle Y in a point where we shall position the vertex of

right angle W.

One of the sides of W will lie on the latter side of V, and the other must pass through F.

If that would not be the case, try out a new configuration: it is easy because the process is

fast convergent. Eventually, we get an orthogonal broken line starting at O and ending at F.

It has its two right angles lying on both co-ordinate axles (Fig. 4).

F

O

a

1

d

V

W

X

Y 2

L

Y

F X

O

V

W

3 4

F a

Y

X

d

O

l

b

c