method is depicted in Fig. 20.1. Similar figures can be found in books such as the one
by Stewart (2002).
The text of Poincaré used in the teaching sequence refers to a cone that one makes
stand on its apex—after which it falls (Poincaré 1914). Poincaré’s writing states that
such a cone, with a vertical axis and only gravity acting upon it, would—if it had
perfect symmetry—never fall. However, any degree of asymmetry would cause it to
lean and, eventually, to fall. Similarly, even in a case of perfect symmetry, any
outside force could be enough to disturb the balance.
Thus, this case shows that a small force can have a demonstrable effect—and the
force may be so small that it passes unnoticed, thus the effect appears to be caused by
nothing more than chance. Without full knowledge of the laws of nature and the state
of the “universe” (the environment, the natural world that the object belongs to) at
the moment of the trial, it would be impossible to predict exactly what the next
second of the universe will contain. According to Poincaré, even if we possessed full
knowledge of all the secrets of the “universe”, it is likely that we would still only be
able to gain an approximate understanding of its state at any given point (Poincaré
1948, 2001). This might allow us to create a similarly approximate prediction—but,
as tiny differences can lead to noticeable differences at a later point, even a small
mistake in the former could lead to a much greater one in the latter. This, Poincaré
concludes, makes prediction impossible, and leaves us with what we call “chance”
(Poincaré 1914, pp. 67–68).
After reading the excerpt from Poincaré, the students discuss the notions of
predictability or non-predictability as presented in the text above, as well as the
concept of sensitive dependence on initial conditions.
As a second example, students read in groups a famous extract from Lorenz
(2005a), in which he describes how even small rounded-up approximations in very
small decimal digits created Chaos in his computer’s generation of meteorological
values.
Graphically, what happened in the case of Lorentz is depicted in Fig. 20.2, which
is a drawn representation of the image created by him. It shows a section of a time
period of 15 months from the original point of time, divided into three segments of
5 months. The physical quantity that varies with time is a meteorological parameter.
The exact variable that Lorenz was measuring was the latitude of the strongest
winds from the west; a high value corresponded to low latitude. The diagram shows
a series of “episodes”, in which the value suddenly rose, remained at a high level for
approximately a month, and then suddenly dropped. However, these episodes were
not identical, and they did not last for equal lengths of time; their behaviour was
non-periodic.
Lorenz stated that, at some point, he repeated several computations to study this
phenomenon in more detail. He notes that, while inputting numbers, he took a break
of about an hour and returned to find that the computer had generated weather
simulations that bore no resemblance to the usual ones. Upon studying the results, he
realised that the values started to differ slightly at first, then more greatly, until they
were doubling in size around every 4 days, leaving them drastically different from
the original output after only 2 months. He soon realised what had happened; by
254
A. Gkiolmas et al.
by Stewart (2002).
The text of Poincaré used in the teaching sequence refers to a cone that one makes
stand on its apex—after which it falls (Poincaré 1914). Poincaré’s writing states that
such a cone, with a vertical axis and only gravity acting upon it, would—if it had
perfect symmetry—never fall. However, any degree of asymmetry would cause it to
lean and, eventually, to fall. Similarly, even in a case of perfect symmetry, any
outside force could be enough to disturb the balance.
Thus, this case shows that a small force can have a demonstrable effect—and the
force may be so small that it passes unnoticed, thus the effect appears to be caused by
nothing more than chance. Without full knowledge of the laws of nature and the state
of the “universe” (the environment, the natural world that the object belongs to) at
the moment of the trial, it would be impossible to predict exactly what the next
second of the universe will contain. According to Poincaré, even if we possessed full
knowledge of all the secrets of the “universe”, it is likely that we would still only be
able to gain an approximate understanding of its state at any given point (Poincaré
1948, 2001). This might allow us to create a similarly approximate prediction—but,
as tiny differences can lead to noticeable differences at a later point, even a small
mistake in the former could lead to a much greater one in the latter. This, Poincaré
concludes, makes prediction impossible, and leaves us with what we call “chance”
(Poincaré 1914, pp. 67–68).
After reading the excerpt from Poincaré, the students discuss the notions of
predictability or non-predictability as presented in the text above, as well as the
concept of sensitive dependence on initial conditions.
As a second example, students read in groups a famous extract from Lorenz
(2005a), in which he describes how even small rounded-up approximations in very
small decimal digits created Chaos in his computer’s generation of meteorological
values.
Graphically, what happened in the case of Lorentz is depicted in Fig. 20.2, which
is a drawn representation of the image created by him. It shows a section of a time
period of 15 months from the original point of time, divided into three segments of
5 months. The physical quantity that varies with time is a meteorological parameter.
The exact variable that Lorenz was measuring was the latitude of the strongest
winds from the west; a high value corresponded to low latitude. The diagram shows
a series of “episodes”, in which the value suddenly rose, remained at a high level for
approximately a month, and then suddenly dropped. However, these episodes were
not identical, and they did not last for equal lengths of time; their behaviour was
non-periodic.
Lorenz stated that, at some point, he repeated several computations to study this
phenomenon in more detail. He notes that, while inputting numbers, he took a break
of about an hour and returned to find that the computer had generated weather
simulations that bore no resemblance to the usual ones. Upon studying the results, he
realised that the values started to differ slightly at first, then more greatly, until they
were doubling in size around every 4 days, leaving them drastically different from
the original output after only 2 months. He soon realised what had happened; by
254
A. Gkiolmas et al.