MATHEMATICS AND ORIGAMI

Mathematics and Origami

• Now then, there are publications which affirm that
  7

3

GD= without any clarification

whether that is exact or approximate.

• We have just seen that, though with a great approximation, it does not exist accuracy: GD

measures

50

21

which differs from

49

21

7

3

=

9.13 THALES ́ THEOREM: DIVISION OF A RECTANGLE IN n EQUAL PARTS

In Solution 3 (Point 9.4) we saw in advance the present application. Sidney French, in

B.O.S monograph GEOMETRICAL DIVISION also develops this question which now we are

going to generalise.

It is well known that to divide in n equal parts by folding, is immediate for n equal to

the successive powers of 2: 4, 8, 16, 32 ...

For the other even values of n, e.g. 12, the operation is not immediate because each

forth has to be divided in 3 parts; for any odd value of n we need specific solutions. Somebody

could argue that for a division in 12 parts, we have already learned to divide by 3: that ́s true

but complicate. This is the reason why we shall develop now a general procedure.

Let ́s divide vertically in n = 12 parts the rectangle ABCD. We look for the power of 2

nearest to and greater than n. We find 16. Then we divide the rectangle horizontally in 16 parts

(fg. 1) producing only the indispensable folds. Counting 12 parts from C we get E.

Thales theorem transfers the equality of segments in CE, to DE.

Finally we fold the rectangle (fig. 2) by the perpendiculars to DC through points marked

in DE. It ́s easy to see that the process is good for any value of n.

P C

x

B

x

PC

= =

C

x

A A

D G E D G D

F

H H

A B

D C D C

12

E