type in all data by hand, and rounding things off would save substantial
typing time. Lorenz figured, quite naturally, that rounding off should
make little difference to the computations—but as we saw in the table on
page 177, even a difference in the sixth decimal place can cause signifi-
cant changes in the values of later iterates, at least in the baker’s transfor-
mation. Lorenz was the first to document and describe a butterf ly effect
in a system in which variables changed gradually rather than discontinu-
ously. Lorenz is also responsible for the term butterf ly effect. At a 1972
meeting of the American Association for the Advancement of Science, he
presented a paper titled “Predictability: Does the Flap of a Butterf ly’s
Wings in Brazil Set Off a Tornado in Texas?” Later investigations were to
reveal that chaotic behavior frequently arose from nonlinear phenomena,
a common feature of many important systems. Linear phenomena are
those in which a simple multiple of an input results in a like multiple of
the output. (Hooke’s law is an example of a linear phenomenon. Apply 2
pounds of force to a spring and it stretches 1 inch; apply 8 pounds of force
to the spring and it stretches 4 inches.)
Extreme sensitivity to initial conditions was to be a much more perva-
sive phenomenon than originally suspected. Once chaotic behavior had
been described, it was not so surprising that complicated systems such as
the weather were subject to the butterf ly effect. In the mid-1980s, though,
it was shown that the orbit of the now-demoted planet Pluto was also cha-
otic.^7 The clockwork universe of Newton, in which the heavenly bodies
moved serenely in majestic and predictable orbits around the sun, had
given way to a much more helter-skelter scenario. Pluto turns out to be
eerily similar to the electron in Heisenberg’s uncertainty principle; we
may know where it is, but we don’t know where it’s going to be. Well, not
really: we don’t know where Pluto is going to be because we don’t know
where it and the other bodies in the solar system are (and how fast and in
what direction they are moving) with sufficient accuracy.
Strange Developments
Many systems exhibit periods of stability separated by episodes of transi-
tion between these periods. The geysers at Yellowstone Park are a good
example. Some, like Old Faithful, are very regular in the timing of their
eruptions; others are more erratic. A well-studied example in mathemati-
cal ecology is the interplay between the relative populations of predator
and prey, such as foxes and rabbits. The dynamics of how the fox and rab-
bit populations change is qualitatively straightforward. In the presence of
an adequate supply of food for the rabbits, the rabbit population will ex-
180 How Math Explains the World