MATHEMATICS AND ORIGAMI

Mathematics and Origami

ruler. That ́s why we recommend to perform several measures and then to get the mean

value. The best of all is to roll the paper in as many layers as possible and then to measure

the outer diameter by means of a calliper fitted up with a vernier (and of course, to obtain

the mean value of several measures).

• To measure also the remains AB. This can be easily done by using a small piece of paper
  introduced underneath it, and then rectified. What matters, as was recommended earlier is
  that part AB, as well as the rest of the layers will be tight fixed without any play at all.


• To take into consideration that length A ́B – AB equals n circumferences of which, the
  outer one has d as diameter, being d – 2a, d – 4a, etc. the successive diameters of the others.

Therefore we can write:
A ́B−AB=πd+π()()()d− 2 a +πd− 4 a +πd− 6 a.....+π[]d− 2 ()n− 1 a

nd []a a a ()n a

AB AB

2 2 3 ..... 1

́

− + + + + −

−

π=

At the denominator ́s subtrahend we can find the sum of all the terms of an arithmetic
progression whose value is (Point7.15.1):
()
2

an− 1 n

Therefore:

nd a()n n

AB AB

1

́

− −

−

π=

In our case we take for given:

A ́B = 297 ; AB = 13,5 ; n = 4 ; d = 23 ; a = 0,11 with this result:

()

3 , 1264

4 23 0 , 114 14

297 13 , 5

=

× − −

−

π= (1)

Comparing that result with the π value displayed in a pocket calculator (* 3,1415927)
we could be tempted to feel a sort of frustration; nevertheless, there is not any reason for dis-
couragement.

• Most likely, it is when measuring d where we introduce the main error: note that if we had
  taken 22,643 instead of 23 mm, the resulting value for π would have been *.


• The purpose of the experiment is to offer the order of magnitude of π. To obtain it very
  close to exactitude is highly difficult because it is an irrational number.


• In that respect we have to admit that the big error is in expression (1) which takes the form
  of

9068

28350

π= (2)

• The latter expression represents a rational number (what π is not) which, in turn, might take
  three different configurations: an integer, in the event of exact division (it ́s not our case),
  and a periodic or mixed-periodic fraction. The expression (2) belongs to one of the latter
  two.


• The man is in search of π over 4.000 years. By Euclid’s time it was well known that the
  value of π should be confined between 3 and 4. The reason: 3 is the ratio between an in-
  scribed hexagon ́s perimeter and its diameter; on the other hand 4 is the ratio between a cir-
  cumscribed square ́s perimeter and its diameter.


• Modern computers have made it possible to get π with more than 100.000 significative dig-
  its. It is a matter of time and memory applied to develop series such as ASN (x) (in turn
  obtained as an inversion of the sine series), in spite of its slow convergence. Or the series of
  the ATN(x). In any case, series development is the procedure to add significative figures to