MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

6.7 DYNAMIC RECTANGLES. SQUARE ROOT OF SUCCESSIVE NATURAL NUMBERS.

PROCEDURE 1 (Fig. 1)

Let ́s start with a paper strip of length a and width one unit.

1- A → a; O → O. We get OB= 2.

2- B → a; O → O. We get C.

3- a → a; C → C. We get D: OD 1 1 1 () 2 3

2 2 2

= +OC = +OB = + =

4- D → a; O → O. We get E.

5- a → a; E → E. We get F: OF= 1 +OE^2 = 1 +OD^2 = 1 + 3 = 4

6- F → a; O → O. We get G.

7- a → a; G → G. We get H: OH= 1 +OG^2 = 1 +OF^2 = 1 + 4 = 5

... and so on.

PROCEDURE 2 (Fig. 2)

In this case foldings will be:

1- A → a; O → O. Getting OB= 2

2- A → OB; O → O. ,, Z

3- AO → AO; Z → Z ,, W

4- OW. Getting D; OD= 3

5- A → OD; O → O. Getting Y

6- AO → AO; Y → Y. ,, V

ABDFH

O C E G

1

(^357)
1
a
2 4 6
2
O
A
1
B DF
a
H
2
3
Z
Y
X
W
V
U
4
5
6
7
8
9
10