MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

8.3.4 PYTHAGOREAN UNITS

Demonstrations made in point 8.3.3 for the Pythagorean theorem are adequate for cutting and

folding. Now we are going to see Toshie Takahama ́s demonstration by means of a rhomboidal

unit. In turn, that unit is combined with some others identical to it to form the partial surfaces

that eventually will integrate the three squares associated to the right triangle.

In the present case, folding only is used. It is a rigorously geometric construction and has an

added pedagogic value, for 36 rhomboidal units being required, it allow a very interesting

teamwork.

These observations must be made:

• Demonstration is valid only for right triangles with their legs in the ratio
  2 2

1

1 :

• The pieces formed with the rhomboidal unit have a flattened-consistent structure with
  flattvolumelike shape. That ́s why, at the end, the initial geometric rigor is lost, but that does
  not lessen its interest neither geometric nor pedagogic.

Figs.1 and 2 shows the process to get the rhomboidal unit. From them on, it follows the con-

struction of the five pieces A, B, C, D, E.

Piece A (two units)

1

2

A

C

B

D

(^12)
3
4
6
5