This example shows that two points that start out very close together
can, after a limited number of iterations, occupy positions that are
quite distant from one another. This phenomenon has serious impli-
cations with regard to using mathematics to predict how systems will
behave.
If we were to multiply all the numbers in the above chart by 100, so we
could think of them as representing temperature, we can interpret the
chart as follows: we have a process in which if we start with a temperature
of 9.0171 degrees, after seventeen iterations the temperature is 89.3312
degrees; whereas if we start with a temperature of 9.0172 degrees, after
seventeen iterations we end up with a temperature of 2.4384 degrees. Un-
less we are in a laboratory exercising exquisite control over an experi-
ment, there is no way we can measure the temperature accurately to
.0001 degree. Thus, our inability to measure to exquisite accuracy makes
it impossible to render accurate predictions; small initial differences may
result in substantial subsequent ones. This phenomenon, one of the
centerpieces of the science of chaos, is technically known as “extreme
sensitivity to initial conditions,” but the colloquial expression “the butter-
f ly effect” describes it far more picturesequely: whether or not a butterf ly
f laps its wings in Brazil could determine whether there is a tornado in
Texas two weeks later.
A careful examination of the baker’s transformation reveals that it is the
cutting process that introduces this difficulty. If two points are both on
the left half of the dough, the baker’s transformation simply doubles the
distance between them—similarly for two points on the right half of the
dough. However, if two points are very close but one is on the left half of
the dough and the other on the right, the point on the left half ends up
very near the right end, but the point on the right half ends up very near
the left end. The baker’s transformation is an example of what is called
a discontinuous function—a function in which small differences in the
variable can result in large differences of the corresponding function
values. Although discontinuous functions occur in the real world—when
you turn on a light, it instantaneously goes from zero brightness to maxi-
mum brightness—it may be argued, with some justification, that natu-
ral physical processes are more gradual. When the temperature cools, it
does not drop from 70 degrees to 50 degrees instantaneously, like the
lightbulb—it goes from 70 degrees to 69.9999 degrees to 69.9998 de-
grees... to 50.0001 degrees to 50 degrees.^5 This is a continuous process;
small increases in time result in small changes in temperature. Nothing
chaotic there, right?
178 How Math Explains the World