MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

Play to two tablesides

In this case we shall fix our attention only in orthogonal billiards (Fig. 5)

Being placed balls B, R as indicated, we want to know the course that ball B will take to hit ball

R after touching two sides of the table.

Solution is in (Fig. 6):

• To draw x ́ , y ́ parallel to tablesides distant from them as much as B and R do to their re-
  spective tablesides.


• Produce simultaneous folds B → x ́ (A is got); R → y ́.


• Folding line determines the intermediate stage in the way of B to R.


• As that folding line is the axis of symmetry and the horizontal tableside is the media parallel
  in ∆ABC, right angle D sits on that horizontal tableside. The same applies to the lower tri-
  angle.

7.14.1 SQUARES AND SQUARE ROOTS (H. H.)

Let ́s get the square of a (Fig. 1):

• To start with points C (-1,0) and A (0,a).


• To fold: C → y ́ ; A → A.


• Folding line AB brings about B, whose abscissa is the square of a.

Justification:

∆ABC being a right-angled one, its altitude OA is the proportional media between OC and OB:

OA^2 = 1 ×a^2

You may observe that in this case, the square (a^2 ) is smaller than the number (a) since the

latter is smaller than 1.

5

6

B

R B

C

x ́

y ́

D

R

A

4

B

R